Source code for geotecha.speccon.integrals

# geotecha - A software suite for geotechncial engineering
# Copyright (C) 2018  Rohan T. Walker (rtrwalker@gmail.com)
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see http://www.gnu.org/licenses/gpl.html.
"""Integrals and eigenvalues for consolidation using spectral methods.

Most of the functions have been generated using
geotecha.speccon.integrals_generate_code functions using sympy.

"""

from __future__ import division, print_function
import numpy as np
import math

from geotecha.piecewise.piecewise_linear_1d import segment_containing_also_segments_less_than_xi
from geotecha.piecewise.piecewise_linear_1d import segments_between_xi_and_xj
import geotecha.piecewise.piecewise_linear_1d as pwise
from geotecha.piecewise.piecewise_linear_1d import PolyLine

MUST_TRY_FORTRAN=False #use = True when debugging the fortran extensions.

def m_from_sin_mx(i, boundary=0):
    """Sine series eigenvalue of boundary value problem on [0, 1]


    `i` th eigenvalue, M,  of f(x) = sin(M*x) that satisfies:
        f(0) = 0, f(1) = 0; for `boundary` = 0 i.e. PTPB
        f(0) = 0; f'(1) = 0; for `boundary` = 1 i.e. PTIB

    Parameters
    ----------
    i : ``int``
        Eigenvalue number in series to return.
    boundary : {0, 1}, optional
        Boundary condition.
        For 'Pervious Top Pervious Bottom (PTPB)', boundary = 0.
        For 'Pervious Top Impervious Bottom (PTIB)', boundary = 1.
        Default boundary=0.

    Returns
    -------
    out : ``float``
        The `i` th eigenvalue

    """

    from math import pi

    if boundary not in {0, 1}:
        raise ValueError('boundary = {}; must be 0 or 1.'.format(boundary))

    return pi * (i + 1 - boundary / 2.0)

def pdim1sin_af_linear(m, a, **kwargs):
    """Integration of sin(mi * z) * a(z) * sin(mj * z)
    between ztop and zbot where a(z) is piecewise linear.

    Wrapper for dim1sin_af_linear with PolyLine input.

    Calculation of integrals is performed at each element of a square symmetric
    matrix (size depends on size of `m`)


    Parameters
    ----------
    m : ``list`` of ``float``
       Eeigenvlaues of BVP. Generate with geoteca.speccon.m_from_sin_mx.
    a : PolyLine object
        PolyLine defining piecewise linear relationship.
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A square symmetric matrix, size determined by size of `m`

    See Also
    --------
    m_from_sin_mx : Used to generate 'm' input parameter.
    dim1sin_af_linear : Wrapped function.
        inputs.

    Notes
    -----
    The `dim1sin_af_linear` matrix, :math:`A` is given by:

    .. math:: \\mathbf{A}_{i,j}=\\int_{0}^1{{a\\left(z\\right)}\\phi_i\\phi_j\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)` is a piecewise linear function
    with respect to :math:`z`, that within a layer are defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.


s

    """

    return dim1sin_af_linear(m, a.y1, a.y2, a.x1, a.x2, **kwargs)


def pdim1sin_abf_linear(m, a, b, **kwargs):
    """Integration of sin(mi * z) * a(z) * a(z) * sin(mj * z)
    between ztop and zbot where a(z) is piecewise linear.

    Wrapper for dim1sin_abf_linear to allow PolyLine inputs.


    Calulation of integrals is performed at each element of a square symmetric
    matrix (size depends on size of `m`).

    Parameters
    ----------
    m : ``list`` of ``float``
        eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    a, b : PolyLine object
        PolyLine defining piecewise linear relationship.
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A square symmetric matrix, size determined by size of `m`.

    See Also
    -------
    dim1sin_abf_linear : Wrapped function.


    Notes
    -----
    The `dim1sin_abf_linear` matrix, :math:`A` is given by:

    .. math:: \\mathbf{A}_{i,j}=\\int_{0}^1{{a\\left(z\\right)}{b\\left(z\\right)}\\phi_i\\phi_j\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)` and :math:`b\\left(z\\right)` are piecewise
    linear functions with respect to :math:`z`, that within a layer are defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.

    """
    a, b = pwise.polyline_make_x_common(a,b)
    return dim1sin_abf_linear(m, a.y1, a.y2, b.y1, b.y2, a.x1, a.x2, **kwargs)


def pdim1sin_D_aDf_linear(m, a, **kwargs):
    """Integration of sin(mi * z) * D[a(z) * D[sin(mj * z),z],z]
    between ztop and zbot where a(z) is piecewise linear functions of z.


    Wrapper for dim1sin_D_aDf_linear to allow PolyLine inputs.


    Calulation of integrals is performed at each element of a square symmetric
    matrix (size depends on size of `m`)

    Parameters
    ----------
    m : ``list`` of ``float``
        eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    a : PolyLine object
        PolyLine defining piecewise linear relationship.
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A square symmetric matrix, size determined by size of `m`.

    See Also
    --------
    dim1sin_D_aDf_linear : Wrapped function.


    Notes
    -----
    The `dim1sin_D_aDf_linear` matrix, :math:`A` is given by:

    .. math:: \\mathbf{A}_{i,j}=\\int_{0}^1{\\frac{d}{dz}\\left({a\\left(z\\right)}\\frac{d\\phi_j}{dz}\\right)\\phi_i\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)` is a piecewise
    linear functions with respect to :math:`z`, that within a layer is defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.

    To make the above integration simpler we integate by parts to get:

    .. math:: \\mathbf{A}_{i,j}= \\left.\\phi_i{a\\left(z\\right)}\\frac{d\\phi_j}{dz}\\right|_{z=0}^{z=1} -\\int_{0}^1{{a\\left(z\\right)}\\frac{d\\phi_j}{dz}\\frac{d\\phi_i}{dz}\\,dz}

    In this case the sine basis functions means the left term in the above
    equation is zero, leaving us with

    .. math:: \\mathbf{A}_{i,j}= -\\int_{0}^1{{a\\left(z\\right)}\\frac{d\\phi_j}{dz}\\frac{d\\phi_i}{dz}\\,dz}


    """

    return dim1sin_D_aDf_linear(m, a.y1, a.y2, a.x1, a.x2, **kwargs)


def pdim1sin_ab_linear(m, a, b, **kwargs):
    """Integration of sin(mi * z) * a(z) * b(z)
    between ztop and zbot where a(z) and b(z) are piecewise linear functions of z.

    Wrapper for dim1sin_ab_linear to allow PolyLine inputs.


    Calulation of integrals is performed at each element of a 1d array
    (size depends on size of `m`).

    Parameters
    ----------
    m : ``list`` of ``float``
        eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    a, b : PolyLine object
        PolyLine defining piecewise linear relationship.
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A 1d array size determined by size of `m`. Treat as column
        vector.

    See Also
    --------
    dim1sin_ab_linear : Wrapped function.


    Notes
    -----
    The `dim1sin_ab_linear` which should be treated as a column vector,
    :math:`A` is given by:

    .. math:: \\mathbf{A}_{i}=\\int_{0}^1{{a\\left(z\\right)}{b\\left(z\\right)}\\phi_i\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)` and :math:`b\\left(z\\right)` are piecewise
    linear functions with respect to :math:`z`, that within a layer are defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.

    """
    a, b = pwise.polyline_make_x_common(a,b)
    return dim1sin_ab_linear(m, a.y1, a.y2, b.y1, b.y2, a.x1, a.x2, **kwargs)


def pdim1sin_abc_linear(m, a, b, c, **kwargs):
    """Integrations of sin(mi * z) * a(z) * b(z) * c(z)
    between ztop and zbot where a(z), b(z), c(z) are piecewise linear functions of z.

    Wrapper for dim1sin_abc_linear to allow PolyLine inputs.


    Calulation of integrals is performed at each element of a 1d array
    (size depends on size of `m`).

    Parameters
    ----------
    m : ``list`` of ``float``
        eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    a, b, c : PolyLine object
        PolyLine defining piecewise linear relationship.
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A 1d array size determined by size of `m`. Treat as column
        vector.

    See Also
    --------
    dim1sin_abc_linear : Wrapped function.



    Notes
    -----
    The `dim1sin_abc_linear` which should be treated as a column vector,
    :math:`A` is given by:

    .. math:: \\mathbf{A}_{i}=\\int_{0}^1{{a\\left(z\\right)}{b\\left(z\\right)}{c\\left(z\\right)}\\phi_i\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)`, :math:`b\\left(z\\right)`, and
    :math:`c\\left(z\\right)` are piecewise linear functions
    with respect to :math:`z`, that within a layer are defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.

    """
    a, b, c = pwise.polyline_make_x_common(a,b,c)
    return dim1sin_abc_linear(m, a.y1, a.y2, b.y1, b.y2,c.y1, c.y2, a.x1, a.x2, **kwargs)


def pdim1sin_D_aDb_linear(m, a, b, **kwargs):
    """Integrations of `sin(mi * z) * D[a(z) * D[b(z), z], z]`
    between ztop and zbot where a(z) is a piecewise linear function of z,
    and b(z) is a linear function of z.

    Wrapper for dim1sin_D_aDb_linear to allow PolyLine inputs.

    Calulation of integrals is performed at each element of a 1d array
    (size depends on size of `m`).

    .. warning::
        The functions produced are set up to accept the b(z) input as
        piecewise linear, i.e. b.x1, b.x2, b.y1, b.y2 etc. It is up to the user to
        ensure that the b.x1 and b.x2 are such that they define a continuous
        linear function. eg. to define b(z)=z+1 then use
        b.x1=[0,0.4], b.x2=[0.4, 1], b.y1=[1,1.4], b.y2=[1.4,2].


    Parameters
    ----------
    m : ``list`` of ``float``
        eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    a, b : PolyLine object
        PolyLine defining piecewise linear relationship.
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A 1d array size determined by size of `m`. Treat as column
        vector.

    See Also
    --------
    dim1sin_D_aDb_linear : Wrapped function.


    Notes
    -----
    The `dim1sin_D_aDb_linear` which should be treated as a column vector,
    :math:`A` is given by:

    .. math:: \\mathbf{A}_{i}=\\int_{0}^1{\\frac{d}{dz}\\left({a\\left(z\\right)}\\frac{d}{dz}{b\\left(z\\right)}\\right)\\phi_i\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)` is a piecewise
    linear functions with respect to :math:`z`, that within a layer is defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.

    :math:`b\\left(z\\right)` is a linear function of :math:`z` defined by

    .. math:: b\\left(z\\right) = b_t+\\left({b_b-b_t}\\right)z

    with :math:`t` and :math:`b` subscripts now representing 'top' and
    'bottom' of the profile respectively.

    Using the product rule for differentiation the above integral can be split
    into:

    .. math:: \\mathbf{A}_{i}=\\int_{0}^1{\\frac{da\\left(z\\right)}{dz}\\frac{db\\left(z\\right)}{dz}\\phi_i\\,dz} +
                              \\int_{0}^1{a\\left(z\\right)\\frac{d^2b\\left(z\\right)}{dz^2}\\phi_i\\,dz}

    The right hand term is zero because :math:`b\\left(z\\right)` is a
    continuous linear function so it's second derivative is zero.  The
    first derivative of :math:`b\\left(z\\right)` is a constant so the
    left term can be integrated by parts to give:

    .. math:: \\mathbf{A}_{i}=\\frac{db\\left(z\\right)}{dz}\\left(
                \\left.\\phi_i{a\\left(z\\right)}\\right|_{z=0}^{z=1} -
                -\\int_{0}^1{{a\\left(z\\right)}\\frac{d\\phi_i}{dz}\\,dz}
                \\right)



    """
    a, b = pwise.polyline_make_x_common(a,b)
    return dim1sin_D_aDb_linear(m, a.y1, a.y2, b.y1, b.y2, a.x1, a.x2, **kwargs)


def pEDload_linear(a, eigs, tvals, dT=1.0, **kwargs):
    """Integration of D[load(tau), tau] * exp(dT * eig * (t-tau)) between
    [0, t], where load(tau) is piecewise linear.

    Wrapper for EDload_linear to allow PolyLine inputs.

    Performs integrations involving a piecewise linear load.  A 2d array of
    dimensions A[len(tvals), len(eigs)]
    is produced where the 'i'th row of A contains the diagonal elements of the
    spectral 'E' matrix calculated for the time value tvals[i]. i.e. rows of
    this matrix will be assembled into the diagonal matrix 'E' elsewhere.

    Parameters
    ----------
    a : PolyLine
        PolyLine object representing piecewise linear load vs time.
    eigs : 1d numpy.ndarray
        List of eigenvalues.
    tvals : 1d numpy.ndarray`
        List of time values to calculate integral at.
    dT : ``float``, optional
        Time factor multiple (Default dT=1.0).
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.


    Returns
    -------
    A : numpy.ndarray
        A 2d array of dimesnions A[len(tvals), len(eigs)].
        The 'i'th row of A is the diagonal elements of the spectral 'E' matrix
        calculated for the time tvals[i].


    See Also
    --------
    EDload_linear : Wrapped function.


    Notes
    -----
    Assuming the load are formulated as the product of separate time and depth
    dependant functions:

    .. math:: \\sigma\\left({Z,t}\\right)=\\sigma\\left({Z}\\right)\\sigma\\left({t}\\right)

    the solution to the consolidation equation using the spectral method has
    the form:

    .. math:: u\\left(Z,t\\right)=\\mathbf{\\Phi v E}\\left(\\mathbf{\\Gamma v}\\right)^{-1}\\mathbf{\\theta}

    The matrix :math:`E` is a time dependent diagonal matrix due to time
    dependant loadings.  The version of :math:`E` calculated here in
    `EDload_linear` is from loading terms in the governing equation that are NOT
    differentiated wrt :math:`t`.
    The diagonal elements of :math:`E` are given by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{0}^t{\\frac{d{\\sigma\\left(\\tau\\right)}}{d\\tau}{\\exp\\left({(dT\\left(t-\\tau\\right)\\lambda_i}\\right)}\\,d\\tau}

    where

     :math:`\\lambda_i` is the `ith` eigenvalue of the problem,
     :math:`dT` is a time factor for numerical convienience,
     :math:`\\sigma\left(\\tau\\right)` is the time dependant portion of the loading function.

    When the time dependant loading term :math:`\\sigma\\left(\\tau\\right)` is
    piecewise in time. The contribution of each load segment is found by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{t_s}^{t_f}{{\\sigma\\left(\\tau\\right)}\\exp\\left({dT\\left(t-\\tau\\right)*\\lambda_i}\\right)\\,d\\tau}

    where

    .. math:: t_s = \\min\\left(t,t_{increment\\:start}\\right)

    .. math:: t_f = \\min\\left(t,t_{increment\\:end}\\right)

    (note that this function,`EDload_linear`, rather than use :math:`t_s` and
    :math:`t_f`,
    explicitly finds increments that the current time falls in, falls after,
    and falls before and treates each case on it's own.)

    Each :math:`t` value of interest requires a separate diagonal matrix
    :math:`E`.  To use space more efficiently and to facilitate numpy
    broadcasting when using the results of the function, the diagonal elements
    of :math:`E` for each time value `t` value are stored in the rows of
    array :math:`A` returned by `EDload_linear`.  Thus:

    .. math:: \\mathbf{A}=\\left(\\begin{matrix}E_{0,0}(t_0)&E_{1,1}(t_0)& \cdots & E_{neig-1,neig-1}(t_0)\\\ E_{0,0}(t_1)&E_{1,1}(t_1)& \\cdots & E_{neig-1,neig-1}(t_1)\\\ \\vdots&\\vdots&\\ddots&\\vdots \\\ E_{0,0}(t_m)&E_{1,1}(t_m)& \cdots & E_{neig-1,neig-1}(t_m)\\end{matrix}\\right)


    """

    return EDload_linear(a.x, a.y, eigs, tvals, dT, **kwargs)


def pEload_linear(a, eigs, tvals, dT=1.0, **kwargs):
    """Integration of load(tau) * exp(dT * eig * (t-tau)) between [0, t], where
    load(tau) is piecewise linear.

    Wrapper for Eload_linear to allow PolyLine inputs.

    Performs integrations involving a piecewise linear load.  A 2d array of
    dimensions A[len(tvals), len(eigs)]
    is produced where the 'i'th row of A contains the diagonal elements of the
    spectral 'E' matrix calculated for the time value tvals[i]. i.e. rows of
    this matrix will be assembled into the diagonal matrix 'E' elsewhere.

    Parameters
    ----------
    a : PolyLine
        PolyLine object representing piecewise linear load vs time.
    eigs : 1d numpy.ndarray
        List of eigenvalues.
    tvals : 1d numpy.ndarray`
        List of time values to calculate integral at.
    dT : ``float``, optional
        Time factor multiple (Default dT=1.0).
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.


    Returns
    -------
    A : numpy.ndarray
        A 2d array of dimesnions A[len(tvals), len(eigs)].
        The 'i'th row of A is the diagonal elements of the spectral 'E' matrix
        calculated for the time tvals[i].


    See Also
    --------
    Eload_linear : Wrapped function, see for equations and other related
        functions.

    Notes
    -----
    Assuming the load are formulated as the product of separate time and depth
    dependant functions:

    .. math:: \\sigma\\left({Z,t}\\right)=\\sigma\\left({Z}\\right)\\sigma\\left({t}\\right)

    the solution to the consolidation equation using the spectral method has
    the form:

    .. math:: u\\left(Z,t\\right)=\\mathbf{\\Phi v E}\\left(\\mathbf{\\Gamma v}\\right)^{-1}\\mathbf{\\theta}

    The matrix :math:`E` is a time dependent diagonal matrix due to time
    dependant loadings.  The version of :math:`E` calculated here in
    `Eload_linear` is from loading terms in the governing equation that are NOT
    differentiated wrt :math:`t`.
    The diagonal elements of :math:`E` are given by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{0}^t{{\\sigma\\left(\\tau\\right)}{\exp\\left({(dT\\left(t-\\tau\\right)\\lambda_i}\\right)}\\,d\\tau}

    where

     :math:`\\lambda_i` is the `ith` eigenvalue of the problem,
     :math:`dT` is a time factor for numerical convienience,
     :math:`\\sigma\left(\\tau\\right)` is the time dependant portion of the loading function.

    When the time dependant loading term :math:`\\sigma\\left(\\tau\\right)` is
    piecewise in time. The contribution of each load segment is found by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{t_s}^{t_f}{{\\sigma\\left(\\tau\\right)}\\exp\\left({dT\\left(t-\\tau\\right)*\\lambda_i}\\right)\\,d\\tau}

    where

    .. math:: t_s = \\min\\left(t,t_{increment\\:start}\\right)

    .. math:: t_f = \\min\\left(t,t_{increment\\:end}\\right)

    (note that this function,`Eload_linear`, rather than use :math:`t_s` and
    :math:`t_f`,
    explicitly finds increments that the current time falls in, falls after,
    and falls before and treates each case on it's own.)

    Each :math:`t` value of interest requires a separate diagonal matrix
    :math:`E`.  To use space more efficiently and to facilitate numpy
    broadcasting when using the results of the function, the diagonal elements
    of :math:`E` for each time value `t` value are stored in the rows of
    array :math:`A` returned by `Eload_linear`.  Thus:

    .. math:: \\mathbf{A}=\\left(\\begin{matrix}E_{0,0}(t_0)&E_{1,1}(t_0)& \cdots & E_{neig-1,neig-1}(t_0)\\\ E_{0,0}(t_1)&E_{1,1}(t_1)& \\cdots & E_{neig-1,neig-1}(t_1)\\\ \\vdots&\\vdots&\\ddots&\\vdots \\\ E_{0,0}(t_m)&E_{1,1}(t_m)& \cdots & E_{neig-1,neig-1}(t_m)\\end{matrix}\\right)

    """

    return Eload_linear(a.x, a.y, eigs, tvals, dT, **kwargs)


def pEDload_coslinear(a, omega, phase, eigs, tvals, dT=1.0, **kwargs):
    """Integration of D[cos(omega*tau+phase)*load(tau), tau] * exp(dT * eig * (t-tau)) between [0, t], where
    load(tau) is piecewise linear.

    Wrapper for EDload_coslinear to allow PolyLine inputs.

    Performs integrations involving a piecewise linear load.  A 2d array of
    dimensions A[len(tvals), len(eigs)]
    is produced where the 'i'th row of A contains the diagonal elements of the
    spectral 'E' matrix calculated for the time value tvals[i]. i.e. rows of
    this matrix will be assembled into the diagonal matrix 'E' elsewhere.

    Parameters
    ----------
    a : PolyLine
        PolyLine object representing piecewise linear load vs time.
        List of load magnitudes.
    omega, phase : float
        Parameters that describe a cyclic load cos(omega * t + phase).
    eigs : 1d numpy.ndarray
        List of eigenvalues.
    tvals : 1d numpy.ndarray`
        List of time values to calculate integral at.
    dT : ``float``, optional
        Time factor multiple (Default dT=1.0).
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.


    Returns
    -------
    A : numpy.ndarray
        A 2d array of dimesnions A[len(tvals), len(eigs)].
        The 'i'th row of A is the diagonal elements of the spectral 'E' matrix
        calculated for the time tvals[i].


    Notes
    -----
    Assuming the load are formulated as the product of separate time and depth
    dependant functions:

    .. math:: \\sigma\\left({Z,t}\\right)=\\sigma\\left({Z}\\right)\\sigma\\left({t}\\right)

    the solution to the consolidation equation using the spectral method has
    the form:

    .. math:: u\\left(Z,t\\right)=\\mathbf{\\Phi v E}\\left(\\mathbf{\\Gamma v}\\right)^{-1}\\mathbf{\\theta}

    The matrix :math:`E` is a time dependent diagonal matrix due to time
    dependant loadings.  The version of :math:`E` calculated here in
    `EDload_coslinear` is from loading terms in the governing equation that are NOT
    differentiated wrt :math:`t`.
    The diagonal elements of :math:`E` are given by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{0}^t{\\frac{d{{\\cos\\left(\\omega\\tau+\\textrm{phase}\\right)}\\sigma\\left(\\tau\\right)}}{d\\tau}{\\exp\\left({(dT\\left(t-\\tau\\right)\\lambda_i}\\right)}\\,d\\tau}

    where

     :math:`\\lambda_i` is the `ith` eigenvalue of the problem,
     :math:`dT` is a time factor for numerical convienience,
     :math:`\\sigma\left(\\tau\\right)` is the time dependant portion of the loading function.

    When the time dependant loading term :math:`\\sigma\\left(\\tau\\right)` is
    piecewise in time. The contribution of each load segment is found by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{t_s}^{t_f}{\\frac{d{{\\cos\\left(\\omega\\tau+\\textrm{phase}\\right)}\\sigma\\left(\\tau\\right)}}{d\\tau}\\exp\\left({dT\\left(t-\\tau\\right)*\\lambda_i}\\right)\\,d\\tau}

    where

    .. math:: t_s = \\min\\left(t,t_{increment\\:start}\\right)

    .. math:: t_f = \\min\\left(t,t_{increment\\:end}\\right)

    (note that this function,`EDload_coslinear`, rather than use :math:`t_s` and
    :math:`t_f`,
    explicitly finds increments that the current time falls in, falls after,
    and falls before and treates each case on it's own.)

    Each :math:`t` value of interest requires a separate diagonal matrix
    :math:`E`.  To use space more efficiently and to facilitate numpy
    broadcasting when using the results of the function, the diagonal elements
    of :math:`E` for each time value `t` value are stored in the rows of
    array :math:`A` returned by `EDload_coslinear`.  Thus:

    .. math:: \\mathbf{A}=\\left(\\begin{matrix}E_{0,0}(t_0)&E_{1,1}(t_0)& \cdots & E_{neig-1,neig-1}(t_0)\\\ E_{0,0}(t_1)&E_{1,1}(t_1)& \\cdots & E_{neig-1,neig-1}(t_1)\\\ \\vdots&\\vdots&\\ddots&\\vdots \\\ E_{0,0}(t_m)&E_{1,1}(t_m)& \cdots & E_{neig-1,neig-1}(t_m)\\end{matrix}\\right)

    See Also
    --------
    pEDload_coslinear : Same function with PolyLine inputs.

    """

    return EDload_coslinear(a.x, a.y, omega, phase, eigs, tvals, dT, **kwargs)


def pEload_coslinear(a, omega, phase, eigs, tvals, dT=1.0, **kwargs):
    """Integration of cos(omega*tau+phase)*load(tau) * exp(dT * eig * (t-tau)) between [0, t], where
    load(tau) is piecewise linear.


    Wrapper for Eload_coslinear to allow PolyLine inputs.


    Performs integrations involving a piecewise linear load.  A 2d array of
    dimensions A[len(tvals), len(eigs)]
    is produced where the 'i'th row of A contains the diagonal elements of the
    spectral 'E' matrix calculated for the time value tvals[i]. i.e. rows of
    this matrix will be assembled into the diagonal matrix 'E' elsewhere.

    Parameters
    ----------
    a : PolyLine
        PolyLine object representing piecewise linear load vs time.
    omega, phase : float
        Parameters that describe a cyclic load cos(omega * t + phase).
    eigs : 1d numpy.ndarray
        List of eigenvalues.
    tvals : 1d numpy.ndarray`
        List of time values to calculate integral at.
    dT : ``float``, optional
        Time factor multiple (Default dT=1.0).
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.


    Returns
    -------
    A : numpy.ndarray
        A 2d array of dimesnions A[len(tvals), len(eigs)].
        The 'i'th row of A is the diagonal elements of the spectral 'E' matrix
        calculated for the time tvals[i].


    Notes
    -----
    Assuming the load are formulated as the product of separate time and depth
    dependant functions:

    .. math:: \\sigma\\left({Z,t}\\right)=\\sigma\\left({Z}\\right)\\sigma\\left({t}\\right)

    the solution to the consolidation equation using the spectral method has
    the form:

    .. math:: u\\left(Z,t\\right)=\\mathbf{\\Phi v E}\\left(\\mathbf{\\Gamma v}\\right)^{-1}\\mathbf{\\theta}

    The matrix :math:`E` is a time dependent diagonal matrix due to time
    dependant loadings.  The version of :math:`E` calculated here in
    `Eload_coslinear` is from loading terms in the governing equation that are NOT
    differentiated wrt :math:`t`.
    The diagonal elements of :math:`E` are given by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{0}^t{{\\cos\\left(\\omega\\tau+\\textrm{phase}\\right)}{\\sigma\\left(\\tau\\right)}{\exp\\left({(dT\\left(t-\\tau\\right)\\lambda_i}\\right)}\\,d\\tau}

    where

     :math:`\\lambda_i` is the `ith` eigenvalue of the problem,
     :math:`dT` is a time factor for numerical convienience,
     :math:`\\sigma\left(\\tau\\right)` is the time dependant portion of the loading function.

    When the time dependant loading term :math:`\\sigma\\left(\\tau\\right)` is
    piecewise in time. The contribution of each load segment is found by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{t_s}^{t_f}{{\\cos\\left(\\omega\\tau+\\textrm{phase}\\right)}{\\sigma\\left(\\tau\\right)}\\exp\\left({dT\\left(t-\\tau\\right)*\\lambda_i}\\right)\\,d\\tau}

    where

    .. math:: t_s = \\min\\left(t,t_{increment\\:start}\\right)

    .. math:: t_f = \\min\\left(t,t_{increment\\:end}\\right)

    (note that this function,`Eload_coslinear`, rather than use :math:`t_s` and
    :math:`t_f`,
    explicitly finds increments that the current time falls in, falls after,
    and falls before and treates each case on it's own.)

    Each :math:`t` value of interest requires a separate diagonal matrix
    :math:`E`.  To use space more efficiently and to facilitate numpy
    broadcasting when using the results of the function, the diagonal elements
    of :math:`E` for each time value `t` value are stored in the rows of
    array :math:`A` returned by `Eload_coslinear`.  Thus:

    .. math:: \\mathbf{A}=\\left(\\begin{matrix}E_{0,0}(t_0)&E_{1,1}(t_0)& \cdots & E_{neig-1,neig-1}(t_0)\\\ E_{0,0}(t_1)&E_{1,1}(t_1)& \\cdots & E_{neig-1,neig-1}(t_1)\\\ \\vdots&\\vdots&\\ddots&\\vdots \\\ E_{0,0}(t_m)&E_{1,1}(t_m)& \cdots & E_{neig-1,neig-1}(t_m)\\end{matrix}\\right)

    See Also
    --------
    Eload_coslinear : Wrapped function.

    """

    return Eload_coslinear(a.x, a.y, omega, phase, eigs, tvals, dT, **kwargs)


def pEload_sinlinear(a, omega, phase, eigs, tvals, dT=1.0, **kwargs):
    """Integration of sin(omega*tau+phase)*load(tau) * exp(dT * eig * (t-tau)) between [0, t], where
    load(tau) is piecewise linear.


    Wrapper for Eload_sinlinear to allow PolyLine inputs.


    Performs integrations involving a piecewise linear load.  A 2d array of
    dimensions A[len(tvals), len(eigs)]
    is produced where the 'i'th row of A contains the diagonal elements of the
    spectral 'E' matrix calculated for the time value tvals[i]. i.e. rows of
    this matrix will be assembled into the diagonal matrix 'E' elsewhere.

    Parameters
    ----------
    a : PolyLine
        PolyLine object representing piecewise linear load vs time.
    omega, phase : float
        Parameters that describe a cyclic load sin(omega * t + phase).
    eigs : 1d numpy.ndarray
        List of eigenvalues.
    tvals : 1d numpy.ndarray`
        List of time values to calculate integral at.
    dT : ``float``, optional
        Time factor multiple (Default dT=1.0).
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.


    Returns
    -------
    A : numpy.ndarray
        A 2d array of dimesnions A[len(tvals), len(eigs)].
        The 'i'th row of A is the diagonal elements of the spectral 'E' matrix
        calculated for the time tvals[i].


    Notes
    -----
    Assuming the load are formulated as the product of separate time and depth
    dependant functions:

    .. math:: \\sigma\\left({Z,t}\\right)=\\sigma\\left({Z}\\right)\\sigma\\left({t}\\right)

    the solution to the consolidation equation using the spectral method has
    the form:

    .. math:: u\\left(Z,t\\right)=\\mathbf{\\Phi v E}\\left(\\mathbf{\\Gamma v}\\right)^{-1}\\mathbf{\\theta}

    The matrix :math:`E` is a time dependent diagonal matrix due to time
    dependant loadings.  The version of :math:`E` calculated here in
    `Eload_sinlinear` is from loading terms in the governing equation that are NOT
    differentiated wrt :math:`t`.
    The diagonal elements of :math:`E` are given by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{0}^t{{\\sin\\left(\\omega\\tau+\\textrm{phase}\\right)}{\\sigma\\left(\\tau\\right)}{\exp\\left({(dT\\left(t-\\tau\\right)\\lambda_i}\\right)}\\,d\\tau}

    where

     :math:`\\lambda_i` is the `ith` eigenvalue of the problem,
     :math:`dT` is a time factor for numerical convienience,
     :math:`\\sigma\left(\\tau\\right)` is the time dependant portion of the loading function.

    When the time dependant loading term :math:`\\sigma\\left(\\tau\\right)` is
    piecewise in time. The contribution of each load segment is found by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{t_s}^{t_f}{{\\sin\\left(\\omega\\tau+\\textrm{phase}\\right)}{\\sigma\\left(\\tau\\right)}\\exp\\left({dT\\left(t-\\tau\\right)*\\lambda_i}\\right)\\,d\\tau}

    where

    .. math:: t_s = \\min\\left(t,t_{increment\\:start}\\right)

    .. math:: t_f = \\min\\left(t,t_{increment\\:end}\\right)

    (note that this function,`Eload_sinlinear`, rather than use :math:`t_s` and
    :math:`t_f`,
    explicitly finds increments that the current time falls in, falls after,
    and falls before and treates each case on it's own.)

    Each :math:`t` value of interest requires a separate diagonal matrix
    :math:`E`.  To use space more efficiently and to facilitate numpy
    broadcasting when using the results of the function, the diagonal elements
    of :math:`E` for each time value `t` value are stored in the rows of
    array :math:`A` returned by `Eload_sinlinear`.  Thus:

    .. math:: \\mathbf{A}=\\left(\\begin{matrix}E_{0,0}(t_0)&E_{1,1}(t_0)& \cdots & E_{neig-1,neig-1}(t_0)\\\ E_{0,0}(t_1)&E_{1,1}(t_1)& \\cdots & E_{neig-1,neig-1}(t_1)\\\ \\vdots&\\vdots&\\ddots&\\vdots \\\ E_{0,0}(t_m)&E_{1,1}(t_m)& \cdots & E_{neig-1,neig-1}(t_m)\\end{matrix}\\right)

    See Also
    --------
    Eload_sinlinear : Wrapped function.

    """

    return Eload_sinlinear(a.x, a.y, omega, phase, eigs, tvals, dT, **kwargs)


def dim1sin(m, z):
    """sin(m*z) for each combination of m and z

    Calculates array A[len(z), len(m)]

    Parameters
    ----------
    m : ``list`` of ``float``
        Eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    z : ``list`` of ``float``
        Normalised depth or z-coordinate should be between 0 and 1.

    Returns
    -------
    A : numpy.ndarray
        A 1d array size A[len(z), len(m)].

    See Also
    --------
    m_from_sin_mx : used to generate 'm' input parameter

    Notes
    -----
    .. math:: \\mathbf{A}_{i,j}=\\sin\\left({m_j, z_i}\\right)

    """

    m = np.asarray(m)
    z = np.asarray(z)

    return np.sin(z[:, np.newaxis]*m[np.newaxis, :])


def dim1sin_avg_between(m, z):
    """Average of sin(m * z) between z1 and z2

    Integrate(sin(m*z), (z, z1, z2))/(z2-z1) for each combination of m
    and [z1,z2].

    Calculates array A[len(z), len(m)].

    Parameters
    ----------
    m : ``list`` of ``float``
        Eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    z : ``list`` of ``list`` of float``
        Normalised depth or z-coordinate pair should be two values
        between 0 and 1..

    Returns
    -------
    A : numpy.ndarray
        A 1d array size A[len(z), len(m)]

    See Also
    --------
    m_from_sin_mx : used to generate 'm' input parameter

    Notes
    -----
    .. math:: \\mathbf{A}_{i,j}=\\frac{1}{z_{2i}-z_{1i}}\\int_{z_{1i}}^{z_{2i}}{\\sin\\left({m_j, z}\\right)\\,dz}

    """

    from numpy import sin, cos
    m = np.asarray(m)
    z = np.asarray(z)

    a = z[:,0][:,np.newaxis]
    b = z[:,1][:,np.newaxis]
    mm = m[np.newaxis,:]

    return (cos(a*mm)-cos(b*mm))/mm/(b-a)


def pdim1sin_a_linear_between(m, a, z, **kwargs):
    """Integrations of `sin(mi * z) * a(z)`
    between [z1, z2] where a(z) is a piecewise linear functions of z.

    Wrapper for dim1sin_a_linear_between to allow PolyLine inputs.

    Calculates array A[len(z), len(m)].


    Parameters
    ----------
    m : ``list`` of ``float``
        Eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    a : PolyLine object
        PolyLine defining piecewise linear relationship.
    z : ``list`` of ``list`` of float``
        Normalised depth or z-coordinate pair should be two values
        between 0 and 1.

    Returns
    -------
    A : numpy.ndarray
        A 2d array size of size [len(z), len(m)].

    See Also
    --------
    dim1sin_a_linear_between : Wrapped function.


    Notes
    -----
    The `dim1sin_a_linear_between`, :math:`A`, is given by:

    .. math:: \\mathbf{A}_{i,j}=
                \\int_{z_1}^{z_2}{{a\\left(z\\right)}\\phi_j\\,dz}

    where the basis function :math:`\\phi_j` is given by:

    .. math:: \\phi_j\\left(z\\right)=\\sin\\left({m_j}z\\right)

    and :math:`a\\left(z\\right)` is a piecewise
    linear functions with respect to :math:`z`, that within a layer are defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.

    """

    return dim1sin_a_linear_between(m, a.y1, a.y2, a.x1, a.x2, z, **kwargs)


def dim1sin_a_linear_between(m, at, ab, zt, zb, z):
    """Integrations of `sin(mi * z) * a(z)`
    between [z1, z2] where a(z) is a piecewise linear functions of z.

    Calculates array A[len(z), len(m)].


    Parameters
    ----------
    m : ``list`` of ``float``
        Eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    at : ``list`` of ``float``
        Property at top of each layer.
    ab : ``list`` of ``float``
        Property at bottom of each layer.
    zt : ``list`` of ``float``
        Normalised depth or z-coordinate at top of each layer. `zt[0]` = 0
    zb : ``list`` of ``float``
        Normalised depth or z-coordinate at bottom of each layer. `zt[-1]` = 1
    z : ``list`` of ``list`` of float``
        Normalised depth or z-coordinate pair should be two values
        between 0 and 1.

    Returns
    -------
    A : numpy.ndarray
        A 2d array size of size [len(z), len(m)].

    See Also
    --------
    m_from_sin_mx : Used to generate 'm' input parameter.
    geotecha.speccon.integrals_generate_code.dim1sin_a_linear_between : Generation
        of code for this function.
    pdim1sin_a_linear_between : Same function with PolyLine
        inputs.
    geotecha.piecewise.piecewise_linear_1d.segments_between_xi_and_xj : Used in the
        function.

    Notes
    -----
    The `dim1sin_a_linear_between`, :math:`A`, is given by:

    .. math:: \\mathbf{A}_{i,j}=
                \\int_{z_1}^{z_2}{{a\\left(z\\right)}\\phi_j\\,dz}

    where the basis function :math:`\\phi_j` is given by:

    .. math:: \\phi_j\\left(z\\right)=\\sin\\left({m_j}z\\right)

    and :math:`a\\left(z\\right)` is a piecewise
    linear functions with respect to :math:`z`, that within a layer are defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.


    """


    #import numpy as np #import this globally
    #import math #import this globally

    sin=math.sin
    cos=math.cos
    m = np.asarray(m)
    at = np.asarray(at)
    ab = np.asarray(ab)
    zt = np.asarray(zt)
    zb = np.asarray(zb)

    z = np.atleast_2d(z)

    z1 = z[:,0]
    z2 = z[:,1]

    z_for_interp = np.zeros(len(zt)+1)
    z_for_interp[:-1] = zt[:]
    z_for_interp[-1]=zb[-1]


    (segment_both,
     segment_z1_only,
     segment_z2_only,
     segments_between) = segments_between_xi_and_xj(z_for_interp, z1, z2)

    nz = len(z)
    neig = len(m)

    A = np.zeros((nz,neig), dtype=float)
    for i in range(nz):
        for layer in segment_both[i]:
            a_slope = (ab[layer] - at[layer]) / (zb[layer] - zt[layer])
            for j in range(neig):
                A[i,j] += (-a_slope*m[j]**(-2)*sin(m[j]*z1[i]) + a_slope*m[j]**(-2)*sin(m[j]*z2[i]) +
                    a_slope*m[j]**(-1)*z1[i]*cos(m[j]*z1[i]) -
                    a_slope*m[j]**(-1)*z2[i]*cos(m[j]*z2[i]) -
                    a_slope*zt[layer]*m[j]**(-1)*cos(m[j]*z1[i]) +
                    a_slope*zt[layer]*m[j]**(-1)*cos(m[j]*z2[i]) +
                    at[layer]*m[j]**(-1)*cos(m[j]*z1[i]) - at[layer]*m[j]**(-1)*cos(m[j]*z2[i]))
        for layer in segment_z1_only[i]:
            a_slope = (ab[layer] - at[layer]) / (zb[layer] - zt[layer])
            for j in range(neig):
                A[i,j] += (-a_slope*m[j]**(-2)*sin(m[j]*z1[i]) + a_slope*m[j]**(-2)*sin(zb[layer]*m[j]) +
                    a_slope*m[j]**(-1)*z1[i]*cos(m[j]*z1[i]) -
                    a_slope*zb[layer]*m[j]**(-1)*cos(zb[layer]*m[j]) -
                    a_slope*zt[layer]*m[j]**(-1)*cos(m[j]*z1[i]) +
                    a_slope*zt[layer]*m[j]**(-1)*cos(zb[layer]*m[j]) +
                    at[layer]*m[j]**(-1)*cos(m[j]*z1[i]) - at[layer]*m[j]**(-1)*cos(zb[layer]*m[j]))
        for layer in segments_between[i]:
            a_slope = (ab[layer] - at[layer]) / (zb[layer] - zt[layer])
            for j in range(neig):
                A[i,j] += (a_slope*m[j]**(-2)*sin(zb[layer]*m[j]) - a_slope*m[j]**(-2)*sin(zt[layer]*m[j]) -
                    a_slope*zb[layer]*m[j]**(-1)*cos(zb[layer]*m[j]) +
                    a_slope*zt[layer]*m[j]**(-1)*cos(zb[layer]*m[j]) -
                    at[layer]*m[j]**(-1)*cos(zb[layer]*m[j]) +
                    at[layer]*m[j]**(-1)*cos(zt[layer]*m[j]))
        for layer in segment_z2_only[i]:
            a_slope = (ab[layer] - at[layer]) / (zb[layer] - zt[layer])
            for j in range(neig):
                A[i,j] += (a_slope*m[j]**(-2)*sin(m[j]*z2[i]) - a_slope*m[j]**(-2)*sin(zt[layer]*m[j]) -
                    a_slope*m[j]**(-1)*z2[i]*cos(m[j]*z2[i]) +
                    a_slope*zt[layer]*m[j]**(-1)*cos(m[j]*z2[i]) -
                    at[layer]*m[j]**(-1)*cos(m[j]*z2[i]) + at[layer]*m[j]**(-1)*cos(zt[layer]*m[j]))
    return A


def dim1sin_af_linear(m, at, ab, zt, zb, implementation='vectorized'):
    """Integration of sin(mi * z) * a(z) * sin(mj * z)
    between ztop and zbot where a(z) is piecewise linear.


    Calculation of integrals is performed at each element of a square symmetric
    matrix (size depends on size of `m`)

    Parameters
    ----------
    m : ``list`` of ``float``
        eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    at : ``list`` of ``float``
        Property at top of each layer.
    ab : ``list`` of ``float``
        Property at bottom of each layer.
    zt : ``list`` of ``float``
        Normalised depth or z-coordinate at top of each layer. `zt[0]` = 0
    zb : ``list`` of ``float``
        Normalised depth or z-coordinate at bottom of each layer. `zt[-1]` = 1
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A square symmetric matrix, size determined by size of `m`.

    See Also
    --------
    m_from_sin_mx : Used to generate 'm' input parameter.
    geotecha.integrals_generate_code.dim1sin_af_linear : Generation
        of code for this function.
    pdim1sin_af_linear : Same function with PolyLine
        inputs.
    geotecha.speccon.ext_integrals.dim1sin_af_linear : Equivalent fortran
        function.

    Notes
    -----
    The `dim1sin_af_linear` matrix, :math:`A` is given by:

    .. math:: \\mathbf{A}_{i,j}=\\int_{0}^1{{a\\left(z\\right)}\\phi_i\\phi_j\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)` is a piecewise linear function
    with respect to :math:`z`, that within a layer are defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.




    """
    #import numpy as np #import this at module level
    #import math #import this at module level

    m = np.asarray(m)
    at = np.asarray(at)
    ab = np.asarray(ab)
    zt = np.asarray(zt)
    zb = np.asarray(zb)

    neig = len(m)

    if implementation == 'scalar':
        sin = math.sin
        cos = math.cos
        A = np.zeros([neig, neig], float)
        nlayers = len(zt)
        for layer in range(nlayers):
            a_slope = (ab[layer] - at[layer]) / (zb[layer] - zt[layer])
            for i in range(neig):
                A[i, i] += (a_slope*m[i]**(-2)*sin(m[i]*zb[layer])**2/4 - a_slope*m[i]**(-2)*sin(m[i]*zt[layer])**2/4 +
                    a_slope*m[i]**(-1)*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]/2 -
                    a_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])/2 -
                    a_slope*zb[layer]*sin(m[i]*zb[layer])**2*zt[layer]/2 -
                    a_slope*zb[layer]*cos(m[i]*zb[layer])**2*zt[layer]/2 +
                    a_slope*zb[layer]**2*sin(m[i]*zb[layer])**2/4 +
                    a_slope*zb[layer]**2*cos(m[i]*zb[layer])**2/4 +
                    a_slope*zt[layer]**2*sin(m[i]*zt[layer])**2/4 +
                    a_slope*zt[layer]**2*cos(m[i]*zt[layer])**2/4 -
                    m[i]**(-1)*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])*at[layer]/2 +
                    m[i]**(-1)*cos(m[i]*zt[layer])*sin(m[i]*zt[layer])*at[layer]/2 +
                    zb[layer]*sin(m[i]*zb[layer])**2*at[layer]/2 +
                    zb[layer]*cos(m[i]*zb[layer])**2*at[layer]/2 -
                    zt[layer]*sin(m[i]*zt[layer])**2*at[layer]/2 -
                    zt[layer]*cos(m[i]*zt[layer])**2*at[layer]/2)
            for i in range(neig-1):
                for j in range(i + 1, neig):
                    A[i, j] += (a_slope*m[i]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*sin(m[i]*zb[layer])*sin(m[j]*zb[layer]) -
                        a_slope*m[i]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*sin(m[i]*zt[layer])*sin(m[j]*zt[layer]) +
                        a_slope*m[i]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer] -
                        a_slope*m[i]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[j]*zb[layer]) +
                        2*a_slope*m[j]*m[i]*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zb[layer])*cos(m[j]*zb[layer]) -
                        2*a_slope*m[j]*m[i]*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zt[layer])*cos(m[j]*zt[layer]) -
                        a_slope*m[j]*m[i]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer] +
                        a_slope*m[j]*m[i]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*zb[layer]*cos(m[j]*zb[layer])*sin(m[i]*zb[layer]) +
                        a_slope*m[j]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*sin(m[i]*zb[layer])*sin(m[j]*zb[layer]) -
                        a_slope*m[j]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*sin(m[i]*zt[layer])*sin(m[j]*zt[layer]) -
                        a_slope*m[j]**2*m[i]*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer] +
                        a_slope*m[j]**2*m[i]*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[j]*zb[layer]) +
                        a_slope*m[j]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer] -
                        a_slope*m[j]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*zb[layer]*cos(m[j]*zb[layer])*sin(m[i]*zb[layer]) -
                        m[i]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*at[layer] +
                        m[i]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zt[layer])*sin(m[j]*zt[layer])*at[layer] +
                        m[j]*m[i]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*at[layer] -
                        m[j]*m[i]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[j]*zt[layer])*sin(m[i]*zt[layer])*at[layer] +
                        m[j]**2*m[i]*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*at[layer] -
                        m[j]**2*m[i]*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zt[layer])*sin(m[j]*zt[layer])*at[layer] -
                        m[j]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*at[layer] +
                        m[j]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[j]*zt[layer])*sin(m[i]*zt[layer])*at[layer])

        #A is symmetric
        for i in range(neig - 1):
            for j in range(i + 1, neig):
                A[j, i] = A[i, j]

    elif implementation == 'fortran':
        if MUST_TRY_FORTRAN:
            import geotecha.speccon.ext_integrals as ext_integ
            A = ext_integ.dim1sin_af_linear(m, at, ab, zt, zb)
        try:
            import geotecha.speccon.ext_integrals as ext_integ
            A = ext_integ.dim1sin_af_linear(m, at, ab, zt, zb)
        except ImportError:
            A = dim1sin_af_linear(m, at, ab, zt, zb, implementation='vectorized')

    else:#default is 'vectorized' using numpy
        sin = np.sin
        cos = np.cos
        A = np.zeros([neig, neig], float)

        diag =  np.diag_indices(neig)
        triu = np.triu_indices(neig, k = 1)
        tril = (triu[1], triu[0])

        a_slope = (ab - at) / (zb - zt)

        mi = m[:, np.newaxis]
        A[diag] = np.sum(-a_slope*zb*sin(mi*zb)**2*zt/2 - a_slope*zb*cos(mi*zb)**2*zt/2 + a_slope*zb**2*sin(mi*zb)**2/4 +
            a_slope*zb**2*cos(mi*zb)**2/4 + a_slope*zt**2*sin(mi*zt)**2/4 +
            a_slope*zt**2*cos(mi*zt)**2/4 + a_slope*cos(mi*zb)*sin(mi*zb)*zt/(2*mi) -
            a_slope*zb*cos(mi*zb)*sin(mi*zb)/(2*mi) + a_slope*sin(mi*zb)**2/(4*mi**2) -
            a_slope*sin(mi*zt)**2/(4*mi**2) + zb*sin(mi*zb)**2*at/2 + zb*cos(mi*zb)**2*at/2 -
            zt*sin(mi*zt)**2*at/2 - zt*cos(mi*zt)**2*at/2 - cos(mi*zb)*sin(mi*zb)*at/(2*mi) +
            cos(mi*zt)*sin(mi*zt)*at/(2*mi), axis=1)

        mi = m[triu[0]][:, np.newaxis]
        mj = m[triu[1]][:, np.newaxis]
        A[triu] = np.sum(a_slope*mi**3*cos(mi*zb)*sin(mj*zb)*zt/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            a_slope*mi**3*zb*cos(mi*zb)*sin(mj*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            a_slope*mi**2*mj*cos(mj*zb)*sin(mi*zb)*zt/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            a_slope*mi**2*mj*zb*cos(mj*zb)*sin(mi*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            a_slope*mi**2*sin(mi*zb)*sin(mj*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            a_slope*mi**2*sin(mi*zt)*sin(mj*zt)/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            a_slope*mi*mj**2*cos(mi*zb)*sin(mj*zb)*zt/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            a_slope*mi*mj**2*zb*cos(mi*zb)*sin(mj*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            2*a_slope*mi*mj*cos(mi*zb)*cos(mj*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            2*a_slope*mi*mj*cos(mi*zt)*cos(mj*zt)/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            a_slope*mj**3*cos(mj*zb)*sin(mi*zb)*zt/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            a_slope*mj**3*zb*cos(mj*zb)*sin(mi*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            a_slope*mj**2*sin(mi*zb)*sin(mj*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            a_slope*mj**2*sin(mi*zt)*sin(mj*zt)/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            mi**3*cos(mi*zb)*sin(mj*zb)*at/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            mi**3*cos(mi*zt)*sin(mj*zt)*at/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            mi**2*mj*cos(mj*zb)*sin(mi*zb)*at/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            mi**2*mj*cos(mj*zt)*sin(mi*zt)*at/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            mi*mj**2*cos(mi*zb)*sin(mj*zb)*at/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            mi*mj**2*cos(mi*zt)*sin(mj*zt)*at/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            mj**3*cos(mj*zb)*sin(mi*zb)*at/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            mj**3*cos(mj*zt)*sin(mi*zt)*at/(mi**4 - 2*mi**2*mj**2 + mj**4), axis=1)
        #A is symmetric
        A[tril] = A[triu]

    return A


def dim1sin_abf_linear(m, at, ab, bt, bb,  zt, zb, implementation='vectorized'):
    """Integration of sin(mi * z) * a(z) * a(z) * sin(mj * z)
    between ztop and zbot where a(z) is piecewise linear.

    Calulation of integrals is performed at each element of a square symmetric
    matrix (size depends on size of `m`).

    Parameters
    ----------
    m : ``list`` of ``float``
        eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    at : ``list`` of ``float``
        Property at top of each layer.
    ab : ``list`` of ``float``
        Property at bottom of each layer.
    bt : ``list`` of ``float``
        2nd property at top of each layer.
    bb : ``list`` of ``float``
        2nd property at bottom of each layer.
    zt : ``list`` of ``float``
        Normalised depth or z-coordinate at top of each layer. `zt[0]` = 0
    zb : ``list`` of ``float``
        Normalised depth or z-coordinate at bottom of each layer. `zt[-1]` = 1
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A square symmetric matrix, size determined by size of `m`.

    See Also
    --------
    m_from_sin_mx : Used to generate 'm' input parameter.
    geotecha.integrals_generate_code.dim1sin_abf_linear : Generation
        of code for this function.
    pdim1sin_abf_linear : Same function with PolyLine
        inputs.
    geotecha.speccon.ext_integrals.dim1sin_abf_linear : Equivalent fortran
        function.

    Notes
    -----
    The `dim1sin_abf_linear` matrix, :math:`A` is given by:

    .. math:: \\mathbf{A}_{i,j}=\\int_{0}^1{{a\\left(z\\right)}{b\\left(z\\right)}\\phi_i\\phi_j\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)` and :math:`b\\left(z\\right)` are piecewise
    linear functions with respect to :math:`z`, that within a layer are defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.

    """

    #import numpy as np #import this at module level
    #import math #import this at module level

    m = np.asarray(m)
    at = np.asarray(at)
    ab = np.asarray(ab)
    bt = np.asarray(bt)
    bb = np.asarray(bb)
    zt = np.asarray(zt)
    zb = np.asarray(zb)

    neig = len(m)

    if implementation == 'scalar':
        sin = math.sin
        cos = math.cos
        A = np.zeros([neig, neig], float)
        nlayers = len(zt)
        for layer in range(nlayers):
            a_slope = (ab[layer] - at[layer]) / (zb[layer] - zt[layer])
            b_slope = (bb[layer] - bt[layer]) / (zb[layer] - zt[layer])
            for i in range(neig):
                A[i, i] += (a_slope*b_slope*m[i]**(-3)*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])/4 -
                    a_slope*b_slope*m[i]**(-3)*cos(m[i]*zt[layer])*sin(m[i]*zt[layer])/4 -
                    a_slope*b_slope*m[i]**(-2)*sin(m[i]*zb[layer])**2*zt[layer]/2 +
                    a_slope*b_slope*m[i]**(-2)*zb[layer]*sin(m[i]*zb[layer])**2/4 -
                    a_slope*b_slope*m[i]**(-2)*zb[layer]*cos(m[i]*zb[layer])**2/4 +
                    a_slope*b_slope*m[i]**(-2)*zt[layer]*sin(m[i]*zt[layer])**2/4 +
                    a_slope*b_slope*m[i]**(-2)*zt[layer]*cos(m[i]*zt[layer])**2/4 -
                    a_slope*b_slope*m[i]**(-1)*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]**2/2
                    +
                    a_slope*b_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]
                    -
                    a_slope*b_slope*m[i]**(-1)*zb[layer]**2*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])/2
                    + a_slope*b_slope*zb[layer]*sin(m[i]*zb[layer])**2*zt[layer]**2/2 +
                    a_slope*b_slope*zb[layer]*cos(m[i]*zb[layer])**2*zt[layer]**2/2 -
                    a_slope*b_slope*zb[layer]**2*sin(m[i]*zb[layer])**2*zt[layer]/2 -
                    a_slope*b_slope*zb[layer]**2*cos(m[i]*zb[layer])**2*zt[layer]/2 +
                    a_slope*b_slope*zb[layer]**3*sin(m[i]*zb[layer])**2/6 +
                    a_slope*b_slope*zb[layer]**3*cos(m[i]*zb[layer])**2/6 -
                    a_slope*b_slope*zt[layer]**3*sin(m[i]*zt[layer])**2/6 -
                    a_slope*b_slope*zt[layer]**3*cos(m[i]*zt[layer])**2/6 +
                    a_slope*m[i]**(-2)*sin(m[i]*zb[layer])**2*bt[layer]/4 -
                    a_slope*m[i]**(-2)*sin(m[i]*zt[layer])**2*bt[layer]/4 +
                    a_slope*m[i]**(-1)*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]*bt[layer]/2
                    -
                    a_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])*bt[layer]/2
                    - a_slope*zb[layer]*sin(m[i]*zb[layer])**2*zt[layer]*bt[layer]/2 -
                    a_slope*zb[layer]*cos(m[i]*zb[layer])**2*zt[layer]*bt[layer]/2 +
                    a_slope*zb[layer]**2*sin(m[i]*zb[layer])**2*bt[layer]/4 +
                    a_slope*zb[layer]**2*cos(m[i]*zb[layer])**2*bt[layer]/4 +
                    a_slope*zt[layer]**2*sin(m[i]*zt[layer])**2*bt[layer]/4 +
                    a_slope*zt[layer]**2*cos(m[i]*zt[layer])**2*bt[layer]/4 +
                    b_slope*m[i]**(-2)*sin(m[i]*zb[layer])**2*at[layer]/4 -
                    b_slope*m[i]**(-2)*sin(m[i]*zt[layer])**2*at[layer]/4 +
                    b_slope*m[i]**(-1)*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]*at[layer]/2
                    -
                    b_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])*at[layer]/2
                    - b_slope*zb[layer]*sin(m[i]*zb[layer])**2*zt[layer]*at[layer]/2 -
                    b_slope*zb[layer]*cos(m[i]*zb[layer])**2*zt[layer]*at[layer]/2 +
                    b_slope*zb[layer]**2*sin(m[i]*zb[layer])**2*at[layer]/4 +
                    b_slope*zb[layer]**2*cos(m[i]*zb[layer])**2*at[layer]/4 +
                    b_slope*zt[layer]**2*sin(m[i]*zt[layer])**2*at[layer]/4 +
                    b_slope*zt[layer]**2*cos(m[i]*zt[layer])**2*at[layer]/4 -
                    m[i]**(-1)*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])*bt[layer]*at[layer]/2 +
                    m[i]**(-1)*cos(m[i]*zt[layer])*sin(m[i]*zt[layer])*bt[layer]*at[layer]/2 +
                    zb[layer]*sin(m[i]*zb[layer])**2*bt[layer]*at[layer]/2 +
                    zb[layer]*cos(m[i]*zb[layer])**2*bt[layer]*at[layer]/2 -
                    zt[layer]*sin(m[i]*zt[layer])**2*bt[layer]*at[layer]/2 -
                    zt[layer]*cos(m[i]*zt[layer])**2*bt[layer]*at[layer]/2)
            for i in range(neig-1):
                for j in range(i + 1, neig):
                    A[i, j] += (2*a_slope*b_slope*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer]) -
                        2*a_slope*b_slope*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zt[layer])*sin(m[j]*zt[layer]) -
                        2*a_slope*b_slope*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*sin(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer] +
                        2*a_slope*b_slope*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*sin(m[i]*zb[layer])*sin(m[j]*zb[layer]) -
                        a_slope*b_slope*m[i]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer]**2 +
                        2*a_slope*b_slope*m[i]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer] -
                        a_slope*b_slope*m[i]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]**2*cos(m[i]*zb[layer])*sin(m[j]*zb[layer]) -
                        6*a_slope*b_slope*m[j]*m[i]**2*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer]) +
                        6*a_slope*b_slope*m[j]*m[i]**2*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(m[j]*zt[layer])*sin(m[i]*zt[layer]) -
                        4*a_slope*b_slope*m[j]*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*cos(m[j]*zb[layer])*zt[layer] +
                        4*a_slope*b_slope*m[j]*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[i]*zb[layer])*cos(m[j]*zb[layer]) +
                        a_slope*b_slope*m[j]*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]**2 -
                        2*a_slope*b_slope*m[j]*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer] +
                        a_slope*b_slope*m[j]*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]**2*cos(m[j]*zb[layer])*sin(m[i]*zb[layer]) +
                        6*a_slope*b_slope*m[j]**2*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer]) -
                        6*a_slope*b_slope*m[j]**2*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(m[i]*zt[layer])*sin(m[j]*zt[layer]) +
                        2*a_slope*b_slope*m[j]**2*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer]**2 -
                        4*a_slope*b_slope*m[j]**2*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer] +
                        2*a_slope*b_slope*m[j]**2*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]**2*cos(m[i]*zb[layer])*sin(m[j]*zb[layer]) -
                        2*a_slope*b_slope*m[j]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer]) +
                        2*a_slope*b_slope*m[j]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zt[layer])*sin(m[i]*zt[layer]) +
                        4*a_slope*b_slope*m[j]**3*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*cos(m[j]*zb[layer])*zt[layer] -
                        4*a_slope*b_slope*m[j]**3*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[i]*zb[layer])*cos(m[j]*zb[layer]) -
                        2*a_slope*b_slope*m[j]**3*m[i]**2*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]**2 +
                        4*a_slope*b_slope*m[j]**3*m[i]**2*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer] -
                        2*a_slope*b_slope*m[j]**3*m[i]**2*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]**2*cos(m[j]*zb[layer])*sin(m[i]*zb[layer]) +
                        2*a_slope*b_slope*m[j]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*sin(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer] -
                        2*a_slope*b_slope*m[j]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*sin(m[i]*zb[layer])*sin(m[j]*zb[layer]) -
                        a_slope*b_slope*m[j]**4*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer]**2 +
                        2*a_slope*b_slope*m[j]**4*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer] -
                        a_slope*b_slope*m[j]**4*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]**2*cos(m[i]*zb[layer])*sin(m[j]*zb[layer]) +
                        a_slope*b_slope*m[j]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]**2 -
                        2*a_slope*b_slope*m[j]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer] +
                        a_slope*b_slope*m[j]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]**2*cos(m[j]*zb[layer])*sin(m[i]*zb[layer]) +
                        a_slope*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*sin(m[i]*zb[layer])*sin(m[j]*zb[layer])*bt[layer] -
                        a_slope*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*sin(m[i]*zt[layer])*sin(m[j]*zt[layer])*bt[layer] +
                        a_slope*m[i]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer]*bt[layer] -
                        a_slope*m[i]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*bt[layer] +
                        2*a_slope*m[j]*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*cos(m[j]*zb[layer])*bt[layer] -
                        2*a_slope*m[j]*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zt[layer])*cos(m[j]*zt[layer])*bt[layer] -
                        a_slope*m[j]*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]*bt[layer] +
                        a_slope*m[j]*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*bt[layer] -
                        2*a_slope*m[j]**2*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer]*bt[layer] +
                        2*a_slope*m[j]**2*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*bt[layer] -
                        2*a_slope*m[j]**3*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*cos(m[j]*zb[layer])*bt[layer] +
                        2*a_slope*m[j]**3*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zt[layer])*cos(m[j]*zt[layer])*bt[layer] +
                        2*a_slope*m[j]**3*m[i]**2*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]*bt[layer] -
                        2*a_slope*m[j]**3*m[i]**2*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*bt[layer] -
                        a_slope*m[j]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*sin(m[i]*zb[layer])*sin(m[j]*zb[layer])*bt[layer] +
                        a_slope*m[j]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*sin(m[i]*zt[layer])*sin(m[j]*zt[layer])*bt[layer] +
                        a_slope*m[j]**4*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer]*bt[layer] -
                        a_slope*m[j]**4*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*bt[layer] -
                        a_slope*m[j]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]*bt[layer] +
                        a_slope*m[j]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*bt[layer] +
                        b_slope*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*sin(m[i]*zb[layer])*sin(m[j]*zb[layer])*at[layer] -
                        b_slope*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*sin(m[i]*zt[layer])*sin(m[j]*zt[layer])*at[layer] +
                        b_slope*m[i]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer]*at[layer] -
                        b_slope*m[i]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*at[layer] +
                        2*b_slope*m[j]*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*cos(m[j]*zb[layer])*at[layer] -
                        2*b_slope*m[j]*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zt[layer])*cos(m[j]*zt[layer])*at[layer] -
                        b_slope*m[j]*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]*at[layer] +
                        b_slope*m[j]*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*at[layer] -
                        2*b_slope*m[j]**2*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer]*at[layer] +
                        2*b_slope*m[j]**2*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*at[layer] -
                        2*b_slope*m[j]**3*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*cos(m[j]*zb[layer])*at[layer] +
                        2*b_slope*m[j]**3*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zt[layer])*cos(m[j]*zt[layer])*at[layer] +
                        2*b_slope*m[j]**3*m[i]**2*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]*at[layer] -
                        2*b_slope*m[j]**3*m[i]**2*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*at[layer] -
                        b_slope*m[j]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*sin(m[i]*zb[layer])*sin(m[j]*zb[layer])*at[layer] +
                        b_slope*m[j]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*sin(m[i]*zt[layer])*sin(m[j]*zt[layer])*at[layer] +
                        b_slope*m[j]**4*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer]*at[layer] -
                        b_slope*m[j]**4*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*at[layer] -
                        b_slope*m[j]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]*at[layer] +
                        b_slope*m[j]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*zb[layer]*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*at[layer] -
                        m[i]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*bt[layer]*at[layer] +
                        m[i]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zt[layer])*sin(m[j]*zt[layer])*bt[layer]*at[layer] +
                        m[j]*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*bt[layer]*at[layer] -
                        m[j]*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zt[layer])*sin(m[i]*zt[layer])*bt[layer]*at[layer] +
                        2*m[j]**2*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*bt[layer]*at[layer] -
                        2*m[j]**2*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zt[layer])*sin(m[j]*zt[layer])*bt[layer]*at[layer] -
                        2*m[j]**3*m[i]**2*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*bt[layer]*at[layer] +
                        2*m[j]**3*m[i]**2*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zt[layer])*sin(m[i]*zt[layer])*bt[layer]*at[layer] -
                        m[j]**4*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*bt[layer]*at[layer] +
                        m[j]**4*m[i]*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[i]*zt[layer])*sin(m[j]*zt[layer])*bt[layer]*at[layer] +
                        m[j]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*bt[layer]*at[layer] -
                        m[j]**5*(m[i]**6 - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 -
                        m[j]**6)**(-1)*cos(m[j]*zt[layer])*sin(m[i]*zt[layer])*bt[layer]*at[layer])

        #A is symmetric
        for i in range(neig - 1):
            for j in range(i + 1, neig):
                A[j, i] = A[i, j]

    elif implementation == 'fortran':
        if MUST_TRY_FORTRAN:
            import geotecha.speccon.ext_integrals as ext_integ
            A = ext_integ.dim1sin_abf_linear(m, at, ab, bt, bb, zt, zb)
        else:
            try:
                import geotecha.speccon.ext_integrals as ext_integ
                A = ext_integ.dim1sin_abf_linear(m, at, ab, bt, bb, zt, zb)
            except ImportError:
                A = dim1sin_abf_linear(m, at, ab, bt, bb, zt, zb, implementation='vectorized')

    else:#default is 'vectorized' using numpy
        sin = np.sin
        cos = np.cos
        A = np.zeros([neig, neig], float)

        diag =  np.diag_indices(neig)
        triu = np.triu_indices(neig, k = 1)
        tril = (triu[1], triu[0])

        a_slope = (ab - at) / (zb - zt)
        b_slope = (bb - bt) / (zb - zt)

        mi = m[:, np.newaxis]
        A[diag] = np.sum(a_slope*b_slope*zb*sin(mi*zb)**2*zt**2/2 + a_slope*b_slope*zb*cos(mi*zb)**2*zt**2/2 -
            a_slope*b_slope*zb**2*sin(mi*zb)**2*zt/2 - a_slope*b_slope*zb**2*cos(mi*zb)**2*zt/2 +
            a_slope*b_slope*zb**3*sin(mi*zb)**2/6 + a_slope*b_slope*zb**3*cos(mi*zb)**2/6 -
            a_slope*b_slope*zt**3*sin(mi*zt)**2/6 - a_slope*b_slope*zt**3*cos(mi*zt)**2/6 -
            a_slope*b_slope*cos(mi*zb)*sin(mi*zb)*zt**2/(2*mi) +
            a_slope*b_slope*zb*cos(mi*zb)*sin(mi*zb)*zt/mi -
            a_slope*b_slope*zb**2*cos(mi*zb)*sin(mi*zb)/(2*mi) -
            a_slope*b_slope*sin(mi*zb)**2*zt/(2*mi**2) + a_slope*b_slope*zb*sin(mi*zb)**2/(4*mi**2)
            - a_slope*b_slope*zb*cos(mi*zb)**2/(4*mi**2) +
            a_slope*b_slope*zt*sin(mi*zt)**2/(4*mi**2) + a_slope*b_slope*zt*cos(mi*zt)**2/(4*mi**2)
            + a_slope*b_slope*cos(mi*zb)*sin(mi*zb)/(4*mi**3) -
            a_slope*b_slope*cos(mi*zt)*sin(mi*zt)/(4*mi**3) - a_slope*zb*sin(mi*zb)**2*zt*bt/2 -
            a_slope*zb*cos(mi*zb)**2*zt*bt/2 + a_slope*zb**2*sin(mi*zb)**2*bt/4 +
            a_slope*zb**2*cos(mi*zb)**2*bt/4 + a_slope*zt**2*sin(mi*zt)**2*bt/4 +
            a_slope*zt**2*cos(mi*zt)**2*bt/4 + a_slope*cos(mi*zb)*sin(mi*zb)*zt*bt/(2*mi) -
            a_slope*zb*cos(mi*zb)*sin(mi*zb)*bt/(2*mi) + a_slope*sin(mi*zb)**2*bt/(4*mi**2) -
            a_slope*sin(mi*zt)**2*bt/(4*mi**2) - b_slope*zb*sin(mi*zb)**2*zt*at/2 -
            b_slope*zb*cos(mi*zb)**2*zt*at/2 + b_slope*zb**2*sin(mi*zb)**2*at/4 +
            b_slope*zb**2*cos(mi*zb)**2*at/4 + b_slope*zt**2*sin(mi*zt)**2*at/4 +
            b_slope*zt**2*cos(mi*zt)**2*at/4 + b_slope*cos(mi*zb)*sin(mi*zb)*zt*at/(2*mi) -
            b_slope*zb*cos(mi*zb)*sin(mi*zb)*at/(2*mi) + b_slope*sin(mi*zb)**2*at/(4*mi**2) -
            b_slope*sin(mi*zt)**2*at/(4*mi**2) + zb*sin(mi*zb)**2*bt*at/2 + zb*cos(mi*zb)**2*bt*at/2
            - zt*sin(mi*zt)**2*bt*at/2 - zt*cos(mi*zt)**2*bt*at/2 -
            cos(mi*zb)*sin(mi*zb)*bt*at/(2*mi) + cos(mi*zt)*sin(mi*zt)*bt*at/(2*mi), axis=1)

        mi = m[triu[0]][:, np.newaxis]
        mj = m[triu[1]][:, np.newaxis]
        A[triu] = np.sum(-a_slope*b_slope*mi**5*cos(mi*zb)*sin(mj*zb)*zt**2/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            2*a_slope*b_slope*mi**5*zb*cos(mi*zb)*sin(mj*zb)*zt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - a_slope*b_slope*mi**5*zb**2*cos(mi*zb)*sin(mj*zb)/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            a_slope*b_slope*mi**4*mj*cos(mj*zb)*sin(mi*zb)*zt**2/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*a_slope*b_slope*mi**4*mj*zb*cos(mj*zb)*sin(mi*zb)*zt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            a_slope*b_slope*mi**4*mj*zb**2*cos(mj*zb)*sin(mi*zb)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*a_slope*b_slope*mi**4*sin(mi*zb)*sin(mj*zb)*zt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            2*a_slope*b_slope*mi**4*zb*sin(mi*zb)*sin(mj*zb)/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4
            - mj**6) + 2*a_slope*b_slope*mi**3*mj**2*cos(mi*zb)*sin(mj*zb)*zt**2/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            4*a_slope*b_slope*mi**3*mj**2*zb*cos(mi*zb)*sin(mj*zb)*zt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) +
            2*a_slope*b_slope*mi**3*mj**2*zb**2*cos(mi*zb)*sin(mj*zb)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 4*a_slope*b_slope*mi**3*mj*cos(mi*zb)*cos(mj*zb)*zt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            4*a_slope*b_slope*mi**3*mj*zb*cos(mi*zb)*cos(mj*zb)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 2*a_slope*b_slope*mi**3*cos(mi*zb)*sin(mj*zb)/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            2*a_slope*b_slope*mi**3*cos(mi*zt)*sin(mj*zt)/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - 2*a_slope*b_slope*mi**2*mj**3*cos(mj*zb)*sin(mi*zb)*zt**2/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            4*a_slope*b_slope*mi**2*mj**3*zb*cos(mj*zb)*sin(mi*zb)*zt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) -
            2*a_slope*b_slope*mi**2*mj**3*zb**2*cos(mj*zb)*sin(mi*zb)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 6*a_slope*b_slope*mi**2*mj*cos(mj*zb)*sin(mi*zb)/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            6*a_slope*b_slope*mi**2*mj*cos(mj*zt)*sin(mi*zt)/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4
            - mj**6) - a_slope*b_slope*mi*mj**4*cos(mi*zb)*sin(mj*zb)*zt**2/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 2*a_slope*b_slope*mi*mj**4*zb*cos(mi*zb)*sin(mj*zb)*zt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            a_slope*b_slope*mi*mj**4*zb**2*cos(mi*zb)*sin(mj*zb)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 4*a_slope*b_slope*mi*mj**3*cos(mi*zb)*cos(mj*zb)*zt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            4*a_slope*b_slope*mi*mj**3*zb*cos(mi*zb)*cos(mj*zb)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 6*a_slope*b_slope*mi*mj**2*cos(mi*zb)*sin(mj*zb)/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            6*a_slope*b_slope*mi*mj**2*cos(mi*zt)*sin(mj*zt)/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4
            - mj**6) + a_slope*b_slope*mj**5*cos(mj*zb)*sin(mi*zb)*zt**2/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*a_slope*b_slope*mj**5*zb*cos(mj*zb)*sin(mi*zb)*zt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            a_slope*b_slope*mj**5*zb**2*cos(mj*zb)*sin(mi*zb)/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4
            - mj**6) + 2*a_slope*b_slope*mj**4*sin(mi*zb)*sin(mj*zb)*zt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*a_slope*b_slope*mj**4*zb*sin(mi*zb)*sin(mj*zb)/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            2*a_slope*b_slope*mj**3*cos(mj*zb)*sin(mi*zb)/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + 2*a_slope*b_slope*mj**3*cos(mj*zt)*sin(mi*zt)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + a_slope*mi**5*cos(mi*zb)*sin(mj*zb)*zt*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            a_slope*mi**5*zb*cos(mi*zb)*sin(mj*zb)*bt/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - a_slope*mi**4*mj*cos(mj*zb)*sin(mi*zb)*zt*bt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + a_slope*mi**4*mj*zb*cos(mj*zb)*sin(mi*zb)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) + a_slope*mi**4*sin(mi*zb)*sin(mj*zb)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) - a_slope*mi**4*sin(mi*zt)*sin(mj*zt)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            2*a_slope*mi**3*mj**2*cos(mi*zb)*sin(mj*zb)*zt*bt/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4
            - mj**6) + 2*a_slope*mi**3*mj**2*zb*cos(mi*zb)*sin(mj*zb)*bt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 2*a_slope*mi**3*mj*cos(mi*zb)*cos(mj*zb)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            2*a_slope*mi**3*mj*cos(mi*zt)*cos(mj*zt)*bt/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + 2*a_slope*mi**2*mj**3*cos(mj*zb)*sin(mi*zb)*zt*bt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*a_slope*mi**2*mj**3*zb*cos(mj*zb)*sin(mi*zb)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            a_slope*mi*mj**4*cos(mi*zb)*sin(mj*zb)*zt*bt/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - a_slope*mi*mj**4*zb*cos(mi*zb)*sin(mj*zb)*bt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*a_slope*mi*mj**3*cos(mi*zb)*cos(mj*zb)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            2*a_slope*mi*mj**3*cos(mi*zt)*cos(mj*zt)*bt/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - a_slope*mj**5*cos(mj*zb)*sin(mi*zb)*zt*bt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + a_slope*mj**5*zb*cos(mj*zb)*sin(mi*zb)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) - a_slope*mj**4*sin(mi*zb)*sin(mj*zb)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) + a_slope*mj**4*sin(mi*zt)*sin(mj*zt)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            b_slope*mi**5*cos(mi*zb)*sin(mj*zb)*zt*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - b_slope*mi**5*zb*cos(mi*zb)*sin(mj*zb)*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - b_slope*mi**4*mj*cos(mj*zb)*sin(mi*zb)*zt*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            b_slope*mi**4*mj*zb*cos(mj*zb)*sin(mi*zb)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + b_slope*mi**4*sin(mi*zb)*sin(mj*zb)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - b_slope*mi**4*sin(mi*zt)*sin(mj*zt)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - 2*b_slope*mi**3*mj**2*cos(mi*zb)*sin(mj*zb)*zt*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 2*b_slope*mi**3*mj**2*zb*cos(mi*zb)*sin(mj*zb)*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            2*b_slope*mi**3*mj*cos(mi*zb)*cos(mj*zb)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - 2*b_slope*mi**3*mj*cos(mi*zt)*cos(mj*zt)*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 2*b_slope*mi**2*mj**3*cos(mj*zb)*sin(mi*zb)*zt*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            2*b_slope*mi**2*mj**3*zb*cos(mj*zb)*sin(mi*zb)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4
            - mj**6) + b_slope*mi*mj**4*cos(mi*zb)*sin(mj*zb)*zt*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - b_slope*mi*mj**4*zb*cos(mi*zb)*sin(mj*zb)*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            2*b_slope*mi*mj**3*cos(mi*zb)*cos(mj*zb)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + 2*b_slope*mi*mj**3*cos(mi*zt)*cos(mj*zt)*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - b_slope*mj**5*cos(mj*zb)*sin(mi*zb)*zt*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            b_slope*mj**5*zb*cos(mj*zb)*sin(mi*zb)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - b_slope*mj**4*sin(mi*zb)*sin(mj*zb)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + b_slope*mj**4*sin(mi*zt)*sin(mj*zt)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - mi**5*cos(mi*zb)*sin(mj*zb)*bt*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + mi**5*cos(mi*zt)*sin(mj*zt)*bt*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + mi**4*mj*cos(mj*zb)*sin(mi*zb)*bt*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - mi**4*mj*cos(mj*zt)*sin(mi*zt)*bt*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + 2*mi**3*mj**2*cos(mi*zb)*sin(mj*zb)*bt*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*mi**3*mj**2*cos(mi*zt)*sin(mj*zt)*bt*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            2*mi**2*mj**3*cos(mj*zb)*sin(mi*zb)*bt*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + 2*mi**2*mj**3*cos(mj*zt)*sin(mi*zt)*bt*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - mi*mj**4*cos(mi*zb)*sin(mj*zb)*bt*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + mi*mj**4*cos(mi*zt)*sin(mj*zt)*bt*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + mj**5*cos(mj*zb)*sin(mi*zb)*bt*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - mj**5*cos(mj*zt)*sin(mi*zt)*bt*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6), axis=1)
        #A is symmetric
        A[tril] = A[triu]

    return A


def dim1sin_D_aDf_linear(m, at, ab, zt, zb, implementation='vectorized'):
    """Integration of sin(mi * z) * D[a(z) * D[sin(mj * z),z],z]
    between ztop and zbot where a(z) is piecewise linear functions of z.


    Calulation of integrals is performed at each element of a square symmetric
    matrix (size depends on size of `m`)

    Parameters
    ----------
    m : ``list`` of ``float``
        eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    at : ``list`` of ``float``
        Property at top of each layer.
    ab : ``list`` of ``float``
        Property at bottom of each layer.
    zt : ``list`` of ``float``
        Normalised depth or z-coordinate at top of each layer. `zt[0]` = 0
    zb : ``list`` of ``float``
        Normalised depth or z-coordinate at bottom of each layer. `zt[-1]` = 1
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A square symmetric matrix, size determined by size of `m`.

    See Also
    --------
    m_from_sin_mx : Used to generate 'm' input parameter.
    geotecha.speccon.integrals_generate_code.dim1sin_D_aDf_linear_implementations : Generation
        of code for this function.
    pdim1sin_D_aDf_linear : Same function with PolyLine
        inputs.
    geotecha.speccon.ext_integrals.dim1sin_D_aDf_linear : Equivalent fortran
        function.


    Notes
    -----
    The `dim1sin_D_aDf_linear` matrix, :math:`A` is given by:

    .. math:: \\mathbf{A}_{i,j}=\\int_{0}^1{\\frac{d}{dz}\\left({a\\left(z\\right)}\\frac{d\\phi_j}{dz}\\right)\\phi_i\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)` is a piecewise
    linear functions with respect to :math:`z`, that within a layer is defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.

    To make the above integration simpler we integate by parts to get:

    .. math:: \\mathbf{A}_{i,j}= \\left.\\phi_i{a\\left(z\\right)}\\frac{d\\phi_j}{dz}\\right|_{z=0}^{z=1} -\\int_{0}^1{{a\\left(z\\right)}\\frac{d\\phi_j}{dz}\\frac{d\\phi_i}{dz}\\,dz}

    In this case the sine basis functions means the left term in the above
    equation is zero, leaving us with

    .. math:: \\mathbf{A}_{i,j}= -\\int_{0}^1{{a\\left(z\\right)}\\frac{d\\phi_j}{dz}\\frac{d\\phi_i}{dz}\\,dz}


    """

    #import numpy as np #import this at module level
    #import math #import this at module level

    m = np.asarray(m)
    at = np.asarray(at)
    ab = np.asarray(ab)
    zt = np.asarray(zt)
    zb = np.asarray(zb)

    neig = len(m)

    if implementation == 'scalar':
        sin = math.sin
        cos = math.cos
        A = np.zeros([neig, neig], float)
        nlayers = len(zt)
        for layer in range(nlayers):
            a_slope = (ab[layer] - at[layer]) / (zb[layer] - zt[layer])
            for i in range(neig):
                A[i, i] += (m[i]**2*(-a_slope*m[i]**(-2)*sin(m[i]*zt[layer])**2/4 -
                    a_slope*zt[layer]**2*sin(m[i]*zt[layer])**2/4 -
                    a_slope*zt[layer]**2*cos(m[i]*zt[layer])**2/4 +
                    m[i]**(-1)*cos(m[i]*zt[layer])*sin(m[i]*zt[layer])*at[layer]/2 +
                    zt[layer]*sin(m[i]*zt[layer])**2*at[layer]/2 +
                    zt[layer]*cos(m[i]*zt[layer])**2*at[layer]/2) -
                    m[i]**2*(-a_slope*m[i]**(-2)*sin(m[i]*zb[layer])**2/4 -
                    a_slope*m[i]**(-1)*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])*zt[layer]/2 +
                    a_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])/2 -
                    a_slope*zb[layer]*sin(m[i]*zb[layer])**2*zt[layer]/2 -
                    a_slope*zb[layer]*cos(m[i]*zb[layer])**2*zt[layer]/2 +
                    a_slope*zb[layer]**2*sin(m[i]*zb[layer])**2/4 +
                    a_slope*zb[layer]**2*cos(m[i]*zb[layer])**2/4 +
                    m[i]**(-1)*cos(m[i]*zb[layer])*sin(m[i]*zb[layer])*at[layer]/2 +
                    zb[layer]*sin(m[i]*zb[layer])**2*at[layer]/2 +
                    zb[layer]*cos(m[i]*zb[layer])**2*at[layer]/2))
            for i in range(neig-1):
                for j in range(i + 1, neig):
                    A[i, j] += (m[j]*m[i]*(a_slope*m[i]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zt[layer])*cos(m[j]*zt[layer]) +
                        2*a_slope*m[j]*m[i]*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*sin(m[i]*zt[layer])*sin(m[j]*zt[layer]) +
                        a_slope*m[j]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zt[layer])*cos(m[j]*zt[layer]) + m[i]**3*(m[i]**4 -
                        2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[j]*zt[layer])*sin(m[i]*zt[layer])*at[layer] -
                        m[j]*m[i]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zt[layer])*sin(m[j]*zt[layer])*at[layer] -
                        m[j]**2*m[i]*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[j]*zt[layer])*sin(m[i]*zt[layer])*at[layer] +
                        m[j]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zt[layer])*sin(m[j]*zt[layer])*at[layer]) -
                        m[j]*m[i]*(a_slope*m[i]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zb[layer])*cos(m[j]*zb[layer]) -
                        a_slope*m[i]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer] +
                        a_slope*m[i]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*zb[layer]*cos(m[j]*zb[layer])*sin(m[i]*zb[layer]) +
                        2*a_slope*m[j]*m[i]*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*sin(m[i]*zb[layer])*sin(m[j]*zb[layer]) +
                        a_slope*m[j]*m[i]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer] -
                        a_slope*m[j]*m[i]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[j]*zb[layer]) +
                        a_slope*m[j]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zb[layer])*cos(m[j]*zb[layer]) +
                        a_slope*m[j]**2*m[i]*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*zt[layer] -
                        a_slope*m[j]**2*m[i]*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*zb[layer]*cos(m[j]*zb[layer])*sin(m[i]*zb[layer]) -
                        a_slope*m[j]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*zt[layer] +
                        a_slope*m[j]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*zb[layer]*cos(m[i]*zb[layer])*sin(m[j]*zb[layer]) +
                        m[i]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*at[layer] -
                        m[j]*m[i]**2*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*at[layer] -
                        m[j]**2*m[i]*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[j]*zb[layer])*sin(m[i]*zb[layer])*at[layer] +
                        m[j]**3*(m[i]**4 - 2*m[j]**2*m[i]**2 +
                        m[j]**4)**(-1)*cos(m[i]*zb[layer])*sin(m[j]*zb[layer])*at[layer]))

        #A is symmetric
        for i in range(neig - 1):
            for j in range(i + 1, neig):
                A[j, i] = A[i, j]

    elif implementation == 'fortran':
        if MUST_TRY_FORTRAN:
            import geotecha.speccon.ext_integrals as ext_integ
            A = ext_integ.dim1sin_d_adf_linear(m, at, ab, zt, zb)
        else:
            try:
                import geotecha.speccon.ext_integrals as ext_integ
                A = ext_integ.dim1sin_d_adf_linear(m, at, ab, zt, zb)
            except ImportError:
                A = dim1sin_D_aDf_linear(m, at, ab, zt, zb, implementation='vectorized')

    else:#default is 'vectorized' using numpy
        sin = np.sin
        cos = np.cos
        A = np.zeros([neig, neig], float)

        diag =  np.diag_indices(neig)
        triu = np.triu_indices(neig, k = 1)
        tril = (triu[1], triu[0])

        a_slope = (ab - at) / (zb - zt)

        mi = m[:, np.newaxis]
        A[diag] = np.sum(mi**2*(-a_slope*zt**2*sin(mi*zt)**2/4 - a_slope*zt**2*cos(mi*zt)**2/4 -
            a_slope*sin(mi*zt)**2/(4*mi**2) + zt*sin(mi*zt)**2*at/2 + zt*cos(mi*zt)**2*at/2 +
            cos(mi*zt)*sin(mi*zt)*at/(2*mi)) - mi**2*(-a_slope*zb*sin(mi*zb)**2*zt/2 -
            a_slope*zb*cos(mi*zb)**2*zt/2 + a_slope*zb**2*sin(mi*zb)**2/4 +
            a_slope*zb**2*cos(mi*zb)**2/4 - a_slope*cos(mi*zb)*sin(mi*zb)*zt/(2*mi) +
            a_slope*zb*cos(mi*zb)*sin(mi*zb)/(2*mi) - a_slope*sin(mi*zb)**2/(4*mi**2) +
            zb*sin(mi*zb)**2*at/2 + zb*cos(mi*zb)**2*at/2 + cos(mi*zb)*sin(mi*zb)*at/(2*mi)), axis=1)

        mi = m[triu[0]][:, np.newaxis]
        mj = m[triu[1]][:, np.newaxis]
        A[triu] = np.sum(mi*mj*(a_slope*mi**2*cos(mi*zt)*cos(mj*zt)/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            2*a_slope*mi*mj*sin(mi*zt)*sin(mj*zt)/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            a_slope*mj**2*cos(mi*zt)*cos(mj*zt)/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            mi**3*cos(mj*zt)*sin(mi*zt)*at/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            mi**2*mj*cos(mi*zt)*sin(mj*zt)*at/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            mi*mj**2*cos(mj*zt)*sin(mi*zt)*at/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            mj**3*cos(mi*zt)*sin(mj*zt)*at/(mi**4 - 2*mi**2*mj**2 + mj**4)) -
            mi*mj*(-a_slope*mi**3*cos(mj*zb)*sin(mi*zb)*zt/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            a_slope*mi**3*zb*cos(mj*zb)*sin(mi*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            a_slope*mi**2*mj*cos(mi*zb)*sin(mj*zb)*zt/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            a_slope*mi**2*mj*zb*cos(mi*zb)*sin(mj*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            a_slope*mi**2*cos(mi*zb)*cos(mj*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            a_slope*mi*mj**2*cos(mj*zb)*sin(mi*zb)*zt/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            a_slope*mi*mj**2*zb*cos(mj*zb)*sin(mi*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            2*a_slope*mi*mj*sin(mi*zb)*sin(mj*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            a_slope*mj**3*cos(mi*zb)*sin(mj*zb)*zt/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            a_slope*mj**3*zb*cos(mi*zb)*sin(mj*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            a_slope*mj**2*cos(mi*zb)*cos(mj*zb)/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            mi**3*cos(mj*zb)*sin(mi*zb)*at/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            mi**2*mj*cos(mi*zb)*sin(mj*zb)*at/(mi**4 - 2*mi**2*mj**2 + mj**4) -
            mi*mj**2*cos(mj*zb)*sin(mi*zb)*at/(mi**4 - 2*mi**2*mj**2 + mj**4) +
            mj**3*cos(mi*zb)*sin(mj*zb)*at/(mi**4 - 2*mi**2*mj**2 + mj**4)), axis=1)
        #A is symmetric
        A[tril] = A[triu]

    return A


def dim1sin_ab_linear(m, at, ab, bt, bb, zt, zb, implementation='vectorized'):
    """Integration of sin(mi * z) * a(z) * b(z)
    between ztop and zbot where a(z) and b(z) are piecewise linear functions of z.


    Calulation of integrals is performed at each element of a 1d array
    (size depends on size of `m`).

    Parameters
    ----------
    m : ``list`` of ``float``
        eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    at : ``list`` of ``float``
        Property at top of each layer.
    ab : ``list`` of ``float``
        Property at bottom of each layer.
    bt : ``list`` of ``float``
        2nd property at top of each layer.
    bb : ``list`` of ``float``
        2nd property at bottom of each layer.
    zt : ``list`` of ``float``
        Normalised depth or z-coordinate at top of each layer. `zt[0]` = 0
    zb : ``list`` of ``float``
        Normalised depth or z-coordinate at bottom of each layer. `zt[-1]` = 1
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A 1d array size determined by size of `m`. Treat as column
        vector

    See Also
    --------
    m_from_sin_mx : Used to generate 'm' input parameter.
    geotecha.speccon.integrals_generate_code.dim1sin_ab_linear_implementations : Generation
        of code for this function.
    pdim1sin_ab_linear : Same function with PolyLine
        inputs.
    geotecha.speccon.ext_integrals.dim1sin_ab_linear : Equivalent fortran
        function.


    Notes
    -----
    The `dim1sin_ab_linear` which should be treated as a column vector,
    :math:`A` is given by:

    .. math:: \\mathbf{A}_{i}=\\int_{0}^1{{a\\left(z\\right)}{b\\left(z\\right)}\\phi_i\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)` and :math:`b\\left(z\\right)` are piecewise
    linear functions with respect to :math:`z`, that within a layer are defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.


    """

    #import numpy as np #import this at module level
    #import math #import this at module level

    m = np.asarray(m)
    at = np.asarray(at)
    ab = np.asarray(ab)
    bt = np.asarray(bt)
    bb = np.asarray(bb)
    zt = np.asarray(zt)
    zb = np.asarray(zb)

    neig = len(m)

    if implementation == 'scalar':
        sin = math.sin
        cos = math.cos
        A = np.zeros(neig, float)

        nlayers = len(zt)
        for layer in range(nlayers):
            a_slope = (ab[layer] - at[layer]) / (zb[layer] - zt[layer])
            b_slope = (bb[layer] - bt[layer]) / (zb[layer] - zt[layer])
            for i in range(neig):
                A[i] += (2*a_slope*b_slope*m[i]**(-3)*cos(m[i]*zb[layer]) - 2*a_slope*b_slope*m[i]**(-3)*cos(m[i]*zt[layer])
                    - 2*a_slope*b_slope*m[i]**(-2)*sin(m[i]*zb[layer])*zt[layer] +
                    2*a_slope*b_slope*m[i]**(-2)*zb[layer]*sin(m[i]*zb[layer]) -
                    a_slope*b_slope*m[i]**(-1)*cos(m[i]*zb[layer])*zt[layer]**2 +
                    2*a_slope*b_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*zt[layer] -
                    a_slope*b_slope*m[i]**(-1)*zb[layer]**2*cos(m[i]*zb[layer]) +
                    a_slope*m[i]**(-2)*sin(m[i]*zb[layer])*bt[layer] -
                    a_slope*m[i]**(-2)*sin(m[i]*zt[layer])*bt[layer] +
                    a_slope*m[i]**(-1)*cos(m[i]*zb[layer])*zt[layer]*bt[layer] -
                    a_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*bt[layer] +
                    b_slope*m[i]**(-2)*sin(m[i]*zb[layer])*at[layer] -
                    b_slope*m[i]**(-2)*sin(m[i]*zt[layer])*at[layer] +
                    b_slope*m[i]**(-1)*cos(m[i]*zb[layer])*zt[layer]*at[layer] -
                    b_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*at[layer] -
                    m[i]**(-1)*cos(m[i]*zb[layer])*bt[layer]*at[layer] +
                    m[i]**(-1)*cos(m[i]*zt[layer])*bt[layer]*at[layer])

    elif implementation == 'fortran':
        if MUST_TRY_FORTRAN:
            import geotecha.speccon.ext_integrals as ext_integ
            A = ext_integ.dim1sin_ab_linear(m, at, ab, bt, bb, zt, zb)
        else:
            try:
                import geotecha.speccon.ext_integrals as ext_integ
                A = ext_integ.dim1sin_ab_linear(m, at, ab, bt, bb, zt, zb)
            except ImportError:
                A = dim1sin_ab_linear(m, at, ab, bt, bb, zt, zb, implementation='vectorized')

    else:#default is 'vectorized' using numpy
        sin = np.sin
        cos = np.cos
        A = np.zeros(neig, float)

        a_slope = (ab - at) / (zb - zt)
        b_slope = (bb - bt) / (zb - zt)
        mi = m[:, np.newaxis]
        A[:] = np.sum(-a_slope*b_slope*cos(mi*zb)*zt**2/mi + 2*a_slope*b_slope*zb*cos(mi*zb)*zt/mi -
            a_slope*b_slope*zb**2*cos(mi*zb)/mi - 2*a_slope*b_slope*sin(mi*zb)*zt/mi**2 +
            2*a_slope*b_slope*zb*sin(mi*zb)/mi**2 + 2*a_slope*b_slope*cos(mi*zb)/mi**3 -
            2*a_slope*b_slope*cos(mi*zt)/mi**3 + a_slope*cos(mi*zb)*zt*bt/mi -
            a_slope*zb*cos(mi*zb)*bt/mi + a_slope*sin(mi*zb)*bt/mi**2 - a_slope*sin(mi*zt)*bt/mi**2
            + b_slope*cos(mi*zb)*zt*at/mi - b_slope*zb*cos(mi*zb)*at/mi +
            b_slope*sin(mi*zb)*at/mi**2 - b_slope*sin(mi*zt)*at/mi**2 - cos(mi*zb)*bt*at/mi +
            cos(mi*zt)*bt*at/mi, axis=1)


    return A


def dim1sin_abc_linear(m, at, ab, bt, bb, ct, cb, zt, zb, implementation='vectorized'):
    """Integrations of sin(mi * z) * a(z) * b(z) * c(z)
    between ztop and zbot where a(z), b(z), c(z) are piecewise linear functions of z.

    Calulation of integrals is performed at each element of a 1d array
    (size depends on size of `m`).

    Parameters
    ----------
    m : ``list`` of ``float``
        eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    at : ``list`` of ``float``
        Property at top of each layer.
    ab : ``list`` of ``float``
        Property at bottom of each layer.
    bt : ``list`` of ``float``
        2nd property at top of each layer.
    bb : ``list`` of ``float``
        2nd property at bottom of each layer.
    ct : ``list`` of ``float``
        3rd property at top of each layer.
    cb : ``list`` of ``float``
        3rd property at bottom of each layer.
    zt : ``list`` of ``float``
        Normalised depth or z-coordinate at top of each layer. `zt[0]` = 0
    zb : ``list`` of ``float``
        Normalised depth or z-coordinate at bottom of each layer. `zt[-1]` = 1
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A 1d array size determined by size of `m`. Treat as column
        vector.

    See Also
    --------
    m_from_sin_mx : Used to generate 'm' input parameter.
    geotecha.speccon.integrals_generate_code.dim1sin_abc_linear_implementations : Generation
        of code for this function.
    pdim1sin_abc_linear : Same function with PolyLine
        inputs.
    geotecha.speccon.ext_integrals.dim1sin_abc_linear : Equivalent fortran
        function.


    Notes
    -----
    The `dim1sin_abc_linear` which should be treated as a column vector,
    :math:`A` is given by:

    .. math:: \\mathbf{A}_{i}=\\int_{0}^1{{a\\left(z\\right)}{b\\left(z\\right)}{c\\left(z\\right)}\\phi_i\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)`, :math:`b\\left(z\\right)`, and
    :math:`c\\left(z\\right)` are piecewise linear functions
    with respect to :math:`z`, that within a layer are defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.


    """

    #import numpy as np #import this at module level
    #import math #import this at module level

    m = np.asarray(m)
    at = np.asarray(at)
    ab = np.asarray(ab)
    bt = np.asarray(bt)
    bb = np.asarray(bb)
    ct = np.asarray(ct)
    cb = np.asarray(cb)
    zt = np.asarray(zt)
    zb = np.asarray(zb)

    neig = len(m)

    if implementation == 'scalar':
        sin = math.sin
        cos = math.cos
        A = np.zeros(neig, float)
        nlayers = len(zt)
        for layer in range(nlayers):
            a_slope = (ab[layer] - at[layer]) / (zb[layer] - zt[layer])
            b_slope = (bb[layer] - bt[layer]) / (zb[layer] - zt[layer])
            c_slope = (cb[layer] - ct[layer]) / (zb[layer] - zt[layer])
            for i in range(neig):
                A[i] += (-6*a_slope*b_slope*c_slope*m[i]**(-4)*sin(m[i]*zb[layer]) +
                    6*a_slope*b_slope*c_slope*m[i]**(-4)*sin(m[i]*zt[layer]) -
                    6*a_slope*b_slope*c_slope*m[i]**(-3)*cos(m[i]*zb[layer])*zt[layer] +
                    6*a_slope*b_slope*c_slope*m[i]**(-3)*zb[layer]*cos(m[i]*zb[layer]) +
                    3*a_slope*b_slope*c_slope*m[i]**(-2)*sin(m[i]*zb[layer])*zt[layer]**2 -
                    6*a_slope*b_slope*c_slope*m[i]**(-2)*zb[layer]*sin(m[i]*zb[layer])*zt[layer] +
                    3*a_slope*b_slope*c_slope*m[i]**(-2)*zb[layer]**2*sin(m[i]*zb[layer]) +
                    a_slope*b_slope*c_slope*m[i]**(-1)*cos(m[i]*zb[layer])*zt[layer]**3 -
                    3*a_slope*b_slope*c_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*zt[layer]**2
                    +
                    3*a_slope*b_slope*c_slope*m[i]**(-1)*zb[layer]**2*cos(m[i]*zb[layer])*zt[layer]
                    - a_slope*b_slope*c_slope*m[i]**(-1)*zb[layer]**3*cos(m[i]*zb[layer]) +
                    2*a_slope*b_slope*m[i]**(-3)*cos(m[i]*zb[layer])*ct[layer] -
                    2*a_slope*b_slope*m[i]**(-3)*cos(m[i]*zt[layer])*ct[layer] -
                    2*a_slope*b_slope*m[i]**(-2)*sin(m[i]*zb[layer])*zt[layer]*ct[layer] +
                    2*a_slope*b_slope*m[i]**(-2)*zb[layer]*sin(m[i]*zb[layer])*ct[layer] -
                    a_slope*b_slope*m[i]**(-1)*cos(m[i]*zb[layer])*zt[layer]**2*ct[layer] +
                    2*a_slope*b_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*zt[layer]*ct[layer] -
                    a_slope*b_slope*m[i]**(-1)*zb[layer]**2*cos(m[i]*zb[layer])*ct[layer] +
                    2*a_slope*c_slope*m[i]**(-3)*cos(m[i]*zb[layer])*bt[layer] -
                    2*a_slope*c_slope*m[i]**(-3)*cos(m[i]*zt[layer])*bt[layer] -
                    2*a_slope*c_slope*m[i]**(-2)*sin(m[i]*zb[layer])*zt[layer]*bt[layer] +
                    2*a_slope*c_slope*m[i]**(-2)*zb[layer]*sin(m[i]*zb[layer])*bt[layer] -
                    a_slope*c_slope*m[i]**(-1)*cos(m[i]*zb[layer])*zt[layer]**2*bt[layer] +
                    2*a_slope*c_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*zt[layer]*bt[layer] -
                    a_slope*c_slope*m[i]**(-1)*zb[layer]**2*cos(m[i]*zb[layer])*bt[layer] +
                    a_slope*m[i]**(-2)*sin(m[i]*zb[layer])*ct[layer]*bt[layer] -
                    a_slope*m[i]**(-2)*sin(m[i]*zt[layer])*ct[layer]*bt[layer] +
                    a_slope*m[i]**(-1)*cos(m[i]*zb[layer])*zt[layer]*ct[layer]*bt[layer] -
                    a_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*ct[layer]*bt[layer] +
                    2*b_slope*c_slope*m[i]**(-3)*cos(m[i]*zb[layer])*at[layer] -
                    2*b_slope*c_slope*m[i]**(-3)*cos(m[i]*zt[layer])*at[layer] -
                    2*b_slope*c_slope*m[i]**(-2)*sin(m[i]*zb[layer])*zt[layer]*at[layer] +
                    2*b_slope*c_slope*m[i]**(-2)*zb[layer]*sin(m[i]*zb[layer])*at[layer] -
                    b_slope*c_slope*m[i]**(-1)*cos(m[i]*zb[layer])*zt[layer]**2*at[layer] +
                    2*b_slope*c_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*zt[layer]*at[layer] -
                    b_slope*c_slope*m[i]**(-1)*zb[layer]**2*cos(m[i]*zb[layer])*at[layer] +
                    b_slope*m[i]**(-2)*sin(m[i]*zb[layer])*ct[layer]*at[layer] -
                    b_slope*m[i]**(-2)*sin(m[i]*zt[layer])*ct[layer]*at[layer] +
                    b_slope*m[i]**(-1)*cos(m[i]*zb[layer])*zt[layer]*ct[layer]*at[layer] -
                    b_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*ct[layer]*at[layer] +
                    c_slope*m[i]**(-2)*sin(m[i]*zb[layer])*bt[layer]*at[layer] -
                    c_slope*m[i]**(-2)*sin(m[i]*zt[layer])*bt[layer]*at[layer] +
                    c_slope*m[i]**(-1)*cos(m[i]*zb[layer])*zt[layer]*bt[layer]*at[layer] -
                    c_slope*m[i]**(-1)*zb[layer]*cos(m[i]*zb[layer])*bt[layer]*at[layer] -
                    m[i]**(-1)*cos(m[i]*zb[layer])*ct[layer]*bt[layer]*at[layer] +
                    m[i]**(-1)*cos(m[i]*zt[layer])*ct[layer]*bt[layer]*at[layer])

    elif implementation == 'fortran':
        if MUST_TRY_FORTRAN:
            import geotecha.speccon.ext_integrals as ext_integ
            A = ext_integ.dim1sin_abc_linear(m, at, ab, bt, bb,  ct, cb, zt, zb)
        else:
            try:
                import geotecha.speccon.ext_integrals as ext_integ
                A = ext_integ.dim1sin_abc_linear(m, at, ab, bt, bb,  ct, cb, zt, zb)
            except ImportError:
                A = dim1sin_abc_linear(m, at, ab, bt, bb,  ct, cb, zt, zb, implementation='vectorized')

    else:#default is 'vectorized' using numpy
        sin = np.sin
        cos = np.cos
        A = np.zeros(neig, float)


        a_slope = (ab - at) / (zb - zt)
        b_slope = (bb - bt) / (zb - zt)
        c_slope = (cb - ct) / (zb - zt)

        mi = m[:, np.newaxis]
        A[:] = np.sum(a_slope*b_slope*c_slope*cos(mi*zb)*zt**3/mi - 3*a_slope*b_slope*c_slope*zb*cos(mi*zb)*zt**2/mi +
            3*a_slope*b_slope*c_slope*zb**2*cos(mi*zb)*zt/mi -
            a_slope*b_slope*c_slope*zb**3*cos(mi*zb)/mi +
            3*a_slope*b_slope*c_slope*sin(mi*zb)*zt**2/mi**2 -
            6*a_slope*b_slope*c_slope*zb*sin(mi*zb)*zt/mi**2 +
            3*a_slope*b_slope*c_slope*zb**2*sin(mi*zb)/mi**2 -
            6*a_slope*b_slope*c_slope*cos(mi*zb)*zt/mi**3 +
            6*a_slope*b_slope*c_slope*zb*cos(mi*zb)/mi**3 -
            6*a_slope*b_slope*c_slope*sin(mi*zb)/mi**4 + 6*a_slope*b_slope*c_slope*sin(mi*zt)/mi**4
            - a_slope*b_slope*cos(mi*zb)*zt**2*ct/mi + 2*a_slope*b_slope*zb*cos(mi*zb)*zt*ct/mi -
            a_slope*b_slope*zb**2*cos(mi*zb)*ct/mi - 2*a_slope*b_slope*sin(mi*zb)*zt*ct/mi**2 +
            2*a_slope*b_slope*zb*sin(mi*zb)*ct/mi**2 + 2*a_slope*b_slope*cos(mi*zb)*ct/mi**3 -
            2*a_slope*b_slope*cos(mi*zt)*ct/mi**3 - a_slope*c_slope*cos(mi*zb)*zt**2*bt/mi +
            2*a_slope*c_slope*zb*cos(mi*zb)*zt*bt/mi - a_slope*c_slope*zb**2*cos(mi*zb)*bt/mi -
            2*a_slope*c_slope*sin(mi*zb)*zt*bt/mi**2 + 2*a_slope*c_slope*zb*sin(mi*zb)*bt/mi**2 +
            2*a_slope*c_slope*cos(mi*zb)*bt/mi**3 - 2*a_slope*c_slope*cos(mi*zt)*bt/mi**3 +
            a_slope*cos(mi*zb)*zt*ct*bt/mi - a_slope*zb*cos(mi*zb)*ct*bt/mi +
            a_slope*sin(mi*zb)*ct*bt/mi**2 - a_slope*sin(mi*zt)*ct*bt/mi**2 -
            b_slope*c_slope*cos(mi*zb)*zt**2*at/mi + 2*b_slope*c_slope*zb*cos(mi*zb)*zt*at/mi -
            b_slope*c_slope*zb**2*cos(mi*zb)*at/mi - 2*b_slope*c_slope*sin(mi*zb)*zt*at/mi**2 +
            2*b_slope*c_slope*zb*sin(mi*zb)*at/mi**2 + 2*b_slope*c_slope*cos(mi*zb)*at/mi**3 -
            2*b_slope*c_slope*cos(mi*zt)*at/mi**3 + b_slope*cos(mi*zb)*zt*ct*at/mi -
            b_slope*zb*cos(mi*zb)*ct*at/mi + b_slope*sin(mi*zb)*ct*at/mi**2 -
            b_slope*sin(mi*zt)*ct*at/mi**2 + c_slope*cos(mi*zb)*zt*bt*at/mi -
            c_slope*zb*cos(mi*zb)*bt*at/mi + c_slope*sin(mi*zb)*bt*at/mi**2 -
            c_slope*sin(mi*zt)*bt*at/mi**2 - cos(mi*zb)*ct*bt*at/mi + cos(mi*zt)*ct*bt*at/mi, axis=1)


    return A


def dim1sin_D_aDb_linear(m, at, ab, bt, bb, zt, zb, implementation='vectorized'):
    """Integrations of `sin(mi * z) * D[a(z) * D[b(z), z], z]`
    between ztop and zbot where a(z) is a piecewise linear function of z,
    and b(z) is a linear function of z.

    Calulation of integrals is performed at each element of a 1d array
    (size depends on size of `m`).

    .. warning::
        The functions produced are set up to accept the b(z) input as
        piecewise linear, i.e. zt, zb, bt, bb etc. It is up to the user to
        ensure that the bt and bb are such that they define a continuous
        linear function. eg. to define b(z)=z+1 then use
        zt=[0,0.4], zb=[0.4, 1], bt=[1,1.4], bb=[1.4,2].


    Parameters
    ----------
    m : ``list`` of ``float``
        eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    at : ``list`` of ``float``
        Property at top of each layer.
    ab : ``list`` of ``float``
        Property at bottom of each layer.
    bt : ``list`` of ``float``
        2nd property at top of each layer.
    bb : ``list`` of ``float``
        2nd property at bottom of each layer.
    zt : ``list`` of ``float``
        Normalised depth or z-coordinate at top of each layer. `zt[0]` = 0
    zb : ``list`` of ``float``
        Normalised depth or z-coordinate at bottom of each layer. `zt[-1]` = 1
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A 1d array size determined by size of `m`. Treat as column
        vector

    See Also
    --------
    m_from_sin_mx : Used to generate 'm' input parameter.
    geotecha.speccon.integrals_generate_code.dim1sin_D_aDb_linear_implementations : Generation
        of code for this function.
    pdim1sin_D_aDb_linear : Same function with PolyLine
        inputs.
    geotecha.speccon.ext_integrals.dim1sin_D_aDb_linear : Equivalent fortran
        function.


    Notes
    -----
    The `dim1sin_D_aDb_linear` which should be treated as a column vector,
    :math:`A` is given by:

    .. math:: \\mathbf{A}_{i}=\\int_{0}^1{\\frac{d}{dz}\\left({a\\left(z\\right)}\\frac{d}{dz}{b\\left(z\\right)}\\right)\\phi_i\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)` is a piecewise
    linear functions with respect to :math:`z`, that within a layer is defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.

    :math:`b\\left(z\\right)` is a linear function of :math:`z` defined by

    .. math:: b\\left(z\\right) = b_t+\\left({b_b-b_t}\\right)z

    with :math:`t` and :math:`b` subscripts now representing 'top' and
    'bottom' of the profile respectively.

    Using the product rule for differentiation the above integral can be split
    into:

    .. math:: \\mathbf{A}_{i}=\\int_{0}^1{\\frac{da\\left(z\\right)}{dz}\\frac{db\\left(z\\right)}{dz}\\phi_i\\,dz} +
                              \\int_{0}^1{a\\left(z\\right)\\frac{d^2b\\left(z\\right)}{dz^2}\\phi_i\\,dz}

    The right hand term is zero because :math:`b\\left(z\\right)` is a
    continuous linear function so it's second derivative is zero.  The
    first derivative of :math:`b\\left(z\\right)` is a constant so the
    left term can be integrated by parts to give:

    .. math:: \\mathbf{A}_{i}=\\frac{db\\left(z\\right)}{dz}\\left(
                \\left.\\phi_i{a\\left(z\\right)}\\right|_{z=0}^{z=1} -
                -\\int_{0}^1{{a\\left(z\\right)}\\frac{d\\phi_i}{dz}\\,dz}
                \\right)


    """


        #import numpy as np #import this at module level
    #import math #import this at module level

    m = np.asarray(m)
    at = np.asarray(at)
    ab = np.asarray(ab)
    bt = np.asarray(bt)
    bb = np.asarray(bb)
    zt = np.asarray(zt)
    zb = np.asarray(zb)

    neig = len(m)

    if implementation == 'scalar':
        sin = math.sin
        cos = math.cos
        A = np.zeros(neig, float)
        nlayers = len(zt)
        for layer in range(nlayers):
            a_slope = (ab[layer] - at[layer]) / (zb[layer] - zt[layer])
            b_slope = (bb[layer] - bt[layer]) / (zb[layer] - zt[layer])
            for i in range(neig):
                A[i] += (b_slope*m[i]*(a_slope*m[i]**(-2)*cos(zt[layer]*m[i]) + at[layer]*m[i]**(-1)*sin(zt[layer]*m[i])) -
                    b_slope*m[i]*(a_slope*m[i]**(-2)*cos(zb[layer]*m[i]) +
                    a_slope*zb[layer]*m[i]**(-1)*sin(zb[layer]*m[i]) -
                    a_slope*zt[layer]*m[i]**(-1)*sin(zb[layer]*m[i]) +
                    at[layer]*m[i]**(-1)*sin(zb[layer]*m[i])))

        for i in range(neig):
            A[i] += (-(zb[0] - zt[0])**(-1)*(bb[0] - bt[0])*at[0]*sin(zt[0]*m[i]) + (zb[nlayers - 1] - zt[nlayers -
                1])**(-1)*(bb[nlayers - 1] - bt[nlayers - 1])*ab[nlayers - 1]*sin(zb[nlayers -
                1]*m[i]))
    elif implementation == 'fortran':
        if MUST_TRY_FORTRAN:
            import geotecha.speccon.ext_integrals as ext_integ
            A = ext_integ.dim1sin_d_adb_linear(m, at, ab, bt, bb, zt, zb)
        else:
            try:
                import geotecha.speccon.ext_integrals as ext_integ
                A = ext_integ.dim1sin_d_adb_linear(m, at, ab, bt, bb, zt, zb)
            except ImportError:
                A = dim1sin_D_aDb_linear(m, at, ab, bt, bb, zt, zb, implementation='vectorized')

    else:#default is 'vectorized' using numpy
        sin = np.sin
        cos = np.cos
        A = np.zeros(neig, float)

        a_slope = (ab - at) / (zb - zt)
        b_slope = (bb - bt) / (zb - zt)
        mi = m[:, np.newaxis]
        A[:] = np.sum(b_slope*mi*(a_slope*cos(mi*zt)/mi**2 + sin(mi*zt)*at/mi) - b_slope*mi*(-a_slope*sin(mi*zb)*zt/mi +
            a_slope*zb*sin(mi*zb)/mi + a_slope*cos(mi*zb)/mi**2 + sin(mi*zb)*at/mi), axis=1)
        mi = m
        A[:]+= (-(zb[0] - zt[0])**(-1)*sin(mi*zt[0])*(bb[0] - bt[0])*at[0] + sin(mi*zb[-1])*(zb[-1] -
            zt[-1])**(-1)*(bb[-1] - bt[-1])*ab[-1])
    return A


def Eload_linear(loadtim, loadmag, eigs, tvals, dT=1.0, implementation='vectorized'):
    """Integration of load(tau) * exp(dT * eig * (t-tau)) between [0, t], where
    load(tau) is piecewise linear.

    Performs integrations involving a piecewise linear load.  A 2d array of
    dimensions A[len(tvals), len(eigs)]
    is produced where the 'i'th row of A contains the diagonal elements of the
    spectral 'E' matrix calculated for the time value tvals[i]. i.e. rows of
    this matrix will be assembled into the diagonal matrix 'E' elsewhere.

    Parameters
    ----------
    loadtim : 1d numpy.ndarray
        List of times describing load application.
    loadmag : 1d numpy.ndarray
        List of load magnitudes.
    eigs : 1d numpy.ndarray
        List of eigenvalues.
    tvals : 1d numpy.ndarray`
        List of time values to calculate integral at.
    dT : ``float``, optional
        Time factor multiple (Default dT=1.0).
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.


    Returns
    -------
    A : numpy.ndarray
        A 2d array of dimesnions A[len(tvals), len(eigs)] (note dtype=complex.
        The 'i'th row of A is the diagonal elements of the spectral 'E' matrix
        calculated for the time tvals[i].


    Notes
    -----
    Assuming the load are formulated as the product of separate time and depth
    dependant functions:

    .. math:: \\sigma\\left({Z,t}\\right)=\\sigma\\left({Z}\\right)\\sigma\\left({t}\\right)

    the solution to the consolidation equation using the spectral method has
    the form:

    .. math:: u\\left(Z,t\\right)=\\mathbf{\\Phi v E}\\left(\\mathbf{\\Gamma v}\\right)^{-1}\\mathbf{\\theta}

    The matrix :math:`E` is a time dependent diagonal matrix due to time
    dependant loadings.  The version of :math:`E` calculated here in
    `Eload_linear` is from loading terms in the governing equation that are NOT
    differentiated wrt :math:`t`.
    The diagonal elements of :math:`E` are given by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{0}^t{{\\sigma\\left(\\tau\\right)}{\exp\\left({(dT\\left(t-\\tau\\right)\\lambda_i}\\right)}\\,d\\tau}

    where

     :math:`\\lambda_i` is the `ith` eigenvalue of the problem,
     :math:`dT` is a time factor for numerical convienience,
     :math:`\\sigma\left(\\tau\\right)` is the time dependant portion of the loading function.

    When the time dependant loading term :math:`\\sigma\\left(\\tau\\right)` is
    piecewise in time. The contribution of each load segment is found by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{t_s}^{t_f}{{\\sigma\\left(\\tau\\right)}\\exp\\left({dT\\left(t-\\tau\\right)*\\lambda_i}\\right)\\,d\\tau}

    where

    .. math:: t_s = \\min\\left(t,t_{increment\\:start}\\right)

    .. math:: t_f = \\min\\left(t,t_{increment\\:end}\\right)

    (note that this function,`Eload_linear`, rather than use :math:`t_s` and
    :math:`t_f`,
    explicitly finds increments that the current time falls in, falls after,
    and falls before and treates each case on it's own.)

    Each :math:`t` value of interest requires a separate diagonal matrix
    :math:`E`.  To use space more efficiently and to facilitate numpy
    broadcasting when using the results of the function, the diagonal elements
    of :math:`E` for each time value `t` value are stored in the rows of
    array :math:`A` returned by `Eload_linear`.  Thus:

    .. math:: \\mathbf{A}=\\left(\\begin{matrix}E_{0,0}(t_0)&E_{1,1}(t_0)& \cdots & E_{neig-1,neig-1}(t_0)\\\ E_{0,0}(t_1)&E_{1,1}(t_1)& \\cdots & E_{neig-1,neig-1}(t_1)\\\ \\vdots&\\vdots&\\ddots&\\vdots \\\ E_{0,0}(t_m)&E_{1,1}(t_m)& \cdots & E_{neig-1,neig-1}(t_m)\\end{matrix}\\right)


    See Also
    --------
    pEload_linear : Same function with PolyLine inputs.
    geotecha.speccon.ext_integrals.eload_linear : Equivalent fortran function.
    geotecha.speccon.integrals_generate_code.Eload_linear_implementations : Generation
        of code for this function.
    Eload_coslinear : Similar function with additional cosine term.
    EDload_linear : Similar function but the time dependent
        loading function is differentiated with respect to time.


    """

    loadtim = np.asarray(loadtim)
    loadmag = np.asarray(loadmag)
    eigs = np.asarray(eigs)
    tvals = np.asarray(tvals)

    if implementation == 'scalar':
        sin = np.sin
        cos = np.cos
        exp = np.exp

        A = np.zeros([len(tvals), len(eigs)], dtype=complex)

        (ramps_less_than_t, constants_less_than_t, steps_less_than_t,
            ramps_containing_t, constants_containing_t) = segment_containing_also_segments_less_than_xi(loadtim, loadmag, tvals, steps_or_equal_to = True)

        for i, t in enumerate(tvals):
            for k in constants_containing_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += (-exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT*eig) + loadmag[k]/(dT*eig))
            for k in constants_less_than_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += (exp(-dT*eig*(t - loadtim[k + 1]))*loadmag[k]/(dT*eig) - exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig))
            for k in ramps_containing_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += ((-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*loadmag[k + 1] +
                        dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*loadmag[k] -
                        dT*eig*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k] + dT*eig*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k])**(-1)*loadmag[k + 1] + (-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k + 1] - (-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k])/(dT*eig) -
                        (-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*loadmag[k + 1] + dT*eig*t*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] -
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] - dT*eig*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k] + dT*eig*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] - (-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k])/(dT*eig))
            for k in ramps_less_than_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += (-(-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] + dT*eig*t*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] -
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] - dT*eig*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k] + dT*eig*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] - (-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k])/(dT*eig)
                        + (-dT*eig*t*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k + 1] + dT*eig*t*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k] + dT*eig*(t - loadtim[k +
                        1])*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k + 1] - dT*eig*(t - loadtim[k +
                        1])*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k] + dT*eig*exp(-dT*eig*(t - loadtim[k +
                        1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*loadmag[k
                        + 1] - dT*eig*loadtim[k + 1]*exp(-dT*eig*(t - loadtim[k +
                        1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*loadmag[k] +
                        exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k + 1] - exp(-dT*eig*(t - loadtim[k +
                        1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*loadmag[k])/(dT*eig))
    elif implementation == 'fortran':
        #note than all fortran subroutines are lowercase.
        if MUST_TRY_FORTRAN:
            from geotecha.speccon.ext_integrals import eload_linear as fn
        else:
            try:
                from geotecha.speccon.ext_integrals import eload_linear as fn
            except ImportError:
                fn = Eload_linear
        A = fn(loadtim, loadmag, eigs, tvals, dT)
    else:#default is 'vectorized' using numpy
        sin = np.sin
        cos = np.cos
        exp = np.exp

        A = np.zeros([len(tvals), len(eigs)],dtype=complex)

        (ramps_less_than_t, constants_less_than_t, steps_less_than_t,
            ramps_containing_t, constants_containing_t) = segment_containing_also_segments_less_than_xi(loadtim, loadmag, tvals, steps_or_equal_to = True)

        eig = eigs[:, None]
        for i, t in enumerate(tvals):
            k = constants_containing_t[i]
            if len(k):
                A[i, :] += np.sum(-exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT*eig) + loadmag[k]/(dT*eig), axis=1)

            k = constants_less_than_t[i]
            if len(k):
                A[i, :] += np.sum(exp(-dT*eig*(t - loadtim[k + 1]))*loadmag[k]/(dT*eig) - exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig), axis=1)

            k = ramps_containing_t[i]
            if len(k):
                A[i, :] += np.sum((-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*loadmag[k + 1] +
                        dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*loadmag[k] -
                        dT*eig*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k] + dT*eig*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k])**(-1)*loadmag[k + 1] + (-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k + 1] - (-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k])/(dT*eig) -
                        (-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*loadmag[k + 1] + dT*eig*t*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] -
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] - dT*eig*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k] + dT*eig*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] - (-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k])/(dT*eig), axis=1)

            k = ramps_less_than_t[i]
            if len(k):
                A[i, :] += np.sum(-(-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] + dT*eig*t*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] -
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] - dT*eig*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k] + dT*eig*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] - (-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k])/(dT*eig)
                        + (-dT*eig*t*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k + 1] + dT*eig*t*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k] + dT*eig*(t - loadtim[k +
                        1])*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k + 1] - dT*eig*(t - loadtim[k +
                        1])*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k] + dT*eig*exp(-dT*eig*(t - loadtim[k +
                        1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*loadmag[k
                        + 1] - dT*eig*loadtim[k + 1]*exp(-dT*eig*(t - loadtim[k +
                        1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*loadmag[k] +
                        exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k])**(-1)*loadmag[k + 1] - exp(-dT*eig*(t - loadtim[k +
                        1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k])**(-1)*loadmag[k])/(dT*eig), axis=1)
    return A


def EDload_linear(loadtim, loadmag, eigs, tvals, dT=1.0, implementation='vectorized'):
    """Integration of D[load(tau), tau] * exp(dT * eig * (t-tau)) between
    [0, t], where load(tau) is piecewise linear.

    Performs integrations involving a piecewise linear load.  A 2d array of
    dimensions A[len(tvals), len(eigs)]
    is produced where the 'i'th row of A contains the diagonal elements of the
    spectral 'E' matrix calculated for the time value tvals[i]. i.e. rows of
    this matrix will be assembled into the diagonal matrix 'E' elsewhere.

    Parameters
    ----------
    loadtim : 1d numpy.ndarray
        List of times describing load application.
    loadmag : 1d numpy.ndarray
        List of load magnitudes.
    eigs : 1d numpy.ndarray
        List of eigenvalues.
    tvals : 1d numpy.ndarray`
        List of time values to calculate integral at.
    dT : ``float``, optional
        Time factor multiple (Default dT=1.0).
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.


    Returns
    -------
    A : numpy.ndarray
        A 2d array of dimesnions A[len(tvals), len(eigs)].
        The 'i'th row of A is the diagonal elements of the spectral 'E' matrix
        calculated for the time tvals[i].


    Notes
    -----
    Assuming the load are formulated as the product of separate time and depth
    dependant functions:

    .. math:: \\sigma\\left({Z,t}\\right)=\\sigma\\left({Z}\\right)\\sigma\\left({t}\\right)

    the solution to the consolidation equation using the spectral method has
    the form:

    .. math:: u\\left(Z,t\\right)=\\mathbf{\\Phi v E}\\left(\\mathbf{\\Gamma v}\\right)^{-1}\\mathbf{\\theta}

    The matrix :math:`E` is a time dependent diagonal matrix due to time
    dependant loadings.  The version of :math:`E` calculated here in
    `EDload_linear` is from loading terms in the governing equation that are NOT
    differentiated wrt :math:`t`.
    The diagonal elements of :math:`E` are given by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{0}^t{\\frac{d{\\sigma\\left(\\tau\\right)}}{d\\tau}{\\exp\\left({(dT\\left(t-\\tau\\right)\\lambda_i}\\right)}\\,d\\tau}

    where

     :math:`\\lambda_i` is the `ith` eigenvalue of the problem,
     :math:`dT` is a time factor for numerical convienience,
     :math:`\\sigma\left(\\tau\\right)` is the time dependant portion of the loading function.

    When the time dependant loading term :math:`\\sigma\\left(\\tau\\right)` is
    piecewise in time. The contribution of each load segment is found by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{t_s}^{t_f}{\\frac{d{\\sigma\\left(\\tau\\right)}}{d\\tau}\\exp\\left({dT\\left(t-\\tau\\right)*\\lambda_i}\\right)\\,d\\tau}

    where

    .. math:: t_s = \\min\\left(t,t_{increment\\:start}\\right)

    .. math:: t_f = \\min\\left(t,t_{increment\\:end}\\right)

    (note that this function,`EDload_linear`, rather than use :math:`t_s` and
    :math:`t_f`,
    explicitly finds increments that the current time falls in, falls after,
    and falls before and treates each case on it's own.)

    Each :math:`t` value of interest requires a separate diagonal matrix
    :math:`E`.  To use space more efficiently and to facilitate numpy
    broadcasting when using the results of the function, the diagonal elements
    of :math:`E` for each time value `t` value are stored in the rows of
    array :math:`A` returned by `EDload_linear`.  Thus:

    .. math:: \\mathbf{A}=\\left(\\begin{matrix}E_{0,0}(t_0)&E_{1,1}(t_0)& \cdots & E_{neig-1,neig-1}(t_0)\\\ E_{0,0}(t_1)&E_{1,1}(t_1)& \\cdots & E_{neig-1,neig-1}(t_1)\\\ \\vdots&\\vdots&\\ddots&\\vdots \\\ E_{0,0}(t_m)&E_{1,1}(t_m)& \cdots & E_{neig-1,neig-1}(t_m)\\end{matrix}\\right)


    See Also
    --------
    pEDload_linear : Same function with PolyLine inputs.
    geotecha.speccon.ext_integrals.edload_linear : Equivalent fortran function.
    geotecha.speccon.integrals_generate_code.EDload_linear_implementations : Generation
        of code for this function.
    EDload_coslinear : Similar function with additional cosine term.
    Eload_linear : Similar function but the time dependent loading function
        is not differentiated with repect to time.

    """
    loadtim = np.asarray(loadtim)
    loadmag = np.asarray(loadmag)
    eigs = np.asarray(eigs)
    tvals = np.asarray(tvals)

    if implementation == 'scalar':
        sin = math.sin
        cos = math.cos
        exp = math.exp

        A = np.zeros([len(tvals), len(eigs)])

        (ramps_less_than_t, constants_less_than_t, steps_less_than_t,
            ramps_containing_t, constants_containing_t) = segment_containing_also_segments_less_than_xi(loadtim, loadmag, tvals, steps_or_equal_to = True)

        for i, t in enumerate(tvals):
            for k in steps_less_than_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += (exp(-dT*eig*(t - loadtim[k]))*(loadmag[k + 1] - loadmag[k]))
            for k in ramps_containing_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += ((loadtim[k + 1] - loadtim[k])**(-1)*(loadmag[k + 1] - loadmag[k])/(dT*eig) - (loadtim[k + 1] -
                        loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*(loadmag[k + 1] -
                        loadmag[k])/(dT*eig))
            for k in ramps_less_than_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += (-(loadtim[k + 1] - loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*(loadmag[k + 1] -
                        loadmag[k])/(dT*eig) + exp(-dT*eig*(t - loadtim[k + 1]))*(loadtim[k + 1] -
                        loadtim[k])**(-1)*(loadmag[k + 1] - loadmag[k])/(dT*eig))

    elif implementation == 'fortran':
        #note than all fortran subroutines are lowercase.
        if MUST_TRY_FORTRAN:
            from geotecha.speccon.ext_integrals import edload_linear as fn
        else:
            try:
                from geotecha.speccon.ext_integrals import edload_linear as fn
            except ImportError:
                fn = EDload_linear
        A = fn(loadtim, loadmag, eigs, tvals, dT)
    else:#default is 'vectorized' using numpy
        sin = np.sin
        cos = np.cos
        exp = np.exp

        A = np.zeros([len(tvals), len(eigs)])

        (ramps_less_than_t, constants_less_than_t, steps_less_than_t,
            ramps_containing_t, constants_containing_t) = segment_containing_also_segments_less_than_xi(loadtim, loadmag, tvals, steps_or_equal_to = True)

        eig = eigs[:, None]


        for i, t in enumerate(tvals):
            k = steps_less_than_t[i]
            if len(k):
                A[i, :] += np.sum(exp(-dT*eig*(t - loadtim[k]))*(loadmag[k + 1] - loadmag[k]), axis=1)
            k = ramps_containing_t[i]
            if len(k):
                A[i, :] += np.sum((loadtim[k + 1] - loadtim[k])**(-1)*(loadmag[k + 1] - loadmag[k])/(dT*eig) - (loadtim[k + 1] -
                        loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*(loadmag[k + 1] -
                        loadmag[k])/(dT*eig), axis=1)
            k = ramps_less_than_t[i]
            if len(k):
                A[i, :] += np.sum(-(loadtim[k + 1] - loadtim[k])**(-1)*exp(-dT*eig*(t - loadtim[k]))*(loadmag[k + 1] -
                        loadmag[k])/(dT*eig) + exp(-dT*eig*(t - loadtim[k + 1]))*(loadtim[k + 1] -
                        loadtim[k])**(-1)*(loadmag[k + 1] - loadmag[k])/(dT*eig), axis=1)
    return A



def Eload_coslinear(loadtim, loadmag, omega, phase, eigs, tvals, dT=1.0, implementation='vectorized'):
    """Integration of cos(omega*tau+phase)*load(tau) * exp(dT * eig * (t-tau)) between [0, t], where
    load(tau) is piecewise linear.

    Performs integrations involving a piecewise linear load.  A 2d array of
    dimensions A[len(tvals), len(eigs)]
    is produced where the 'i'th row of A contains the diagonal elements of the
    spectral 'E' matrix calculated for the time value tvals[i]. i.e. rows of
    this matrix will be assembled into the diagonal matrix 'E' elsewhere.

    Parameters
    ----------
    loadtim : 1d numpy.ndarray
        List of times describing load application.
    loadmag : 1d numpy.ndarray
        List of load magnitudes.
    omega, phase : float
        Parameters that describe a cyclic load cos(omega * t + phase).
    eigs : 1d numpy.ndarray
        List of eigenvalues.
    tvals : 1d numpy.ndarray`
        List of time values to calculate integral at.
    dT : ``float``, optional
        Time factor multiple (Default dT=1.0).
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.


    Returns
    -------
    A : numpy.ndarray
        A 2d array of dimesnions A[len(tvals), len(eigs)].
        The 'i'th row of A is the diagonal elements of the spectral 'E' matrix
        calculated for the time tvals[i].


    Notes
    -----
    Assuming the load are formulated as the product of separate time and depth
    dependant functions:

    .. math:: \\sigma\\left({Z,t}\\right)=\\sigma\\left({Z}\\right)\\sigma\\left({t}\\right)

    the solution to the consolidation equation using the spectral method has
    the form:

    .. math:: u\\left(Z,t\\right)=\\mathbf{\\Phi v E}\\left(\\mathbf{\\Gamma v}\\right)^{-1}\\mathbf{\\theta}

    The matrix :math:`E` is a time dependent diagonal matrix due to time
    dependant loadings.  The version of :math:`E` calculated here in
    `Eload_coslinear` is from loading terms in the governing equation that are NOT
    differentiated wrt :math:`t`.
    The diagonal elements of :math:`E` are given by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{0}^t{{\\cos\\left(\\omega\\tau+\\textrm{phase}\\right)}{\\sigma\\left(\\tau\\right)}{\exp\\left({(dT\\left(t-\\tau\\right)\\lambda_i}\\right)}\\,d\\tau}

    where

     :math:`\\lambda_i` is the `ith` eigenvalue of the problem,
     :math:`dT` is a time factor for numerical convienience,
     :math:`\\sigma\left(\\tau\\right)` is the time dependant portion of the loading function.

    When the time dependant loading term :math:`\\sigma\\left(\\tau\\right)` is
    piecewise in time. The contribution of each load segment is found by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{t_s}^{t_f}{{\\cos\\left(\\omega\\tau+\\textrm{phase}\\right)}{\\sigma\\left(\\tau\\right)}\\exp\\left({dT\\left(t-\\tau\\right)*\\lambda_i}\\right)\\,d\\tau}

    where

    .. math:: t_s = \\min\\left(t,t_{increment\\:start}\\right)

    .. math:: t_f = \\min\\left(t,t_{increment\\:end}\\right)

    (note that this function,`Eload_coslinear`, rather than use :math:`t_s` and
    :math:`t_f`,
    explicitly finds increments that the current time falls in, falls after,
    and falls before and treates each case on it's own.)

    Each :math:`t` value of interest requires a separate diagonal matrix
    :math:`E`.  To use space more efficiently and to facilitate numpy
    broadcasting when using the results of the function, the diagonal elements
    of :math:`E` for each time value `t` value are stored in the rows of
    array :math:`A` returned by `Eload_coslinear`.  Thus:

    .. math:: \\mathbf{A}=\\left(\\begin{matrix}E_{0,0}(t_0)&E_{1,1}(t_0)& \cdots & E_{neig-1,neig-1}(t_0)\\\ E_{0,0}(t_1)&E_{1,1}(t_1)& \\cdots & E_{neig-1,neig-1}(t_1)\\\ \\vdots&\\vdots&\\ddots&\\vdots \\\ E_{0,0}(t_m)&E_{1,1}(t_m)& \cdots & E_{neig-1,neig-1}(t_m)\\end{matrix}\\right)

    See Also
    --------
    pEload_coslinear : Same function with PolyLine inputs.
    geotecha.speccon.ext_integrals.eload_coslinear : Equivalent fortran function.
    geotecha.speccon.integrals_generate_code.Eload_coslinear_implementations : Generation
        of code for this function.
    Eload_linear : Similar function with no cosine term.
    EDload_coslinear : Similar function but the combined
        loading function is differentiated with respect to time.


    """
    loadtim = np.asarray(loadtim)
    loadmag = np.asarray(loadmag)
    eigs = np.asarray(eigs)
    tvals = np.asarray(tvals)

    if implementation == 'scalar':
        sin = math.sin
        cos = math.cos
        exp = math.exp

        A = np.zeros([len(tvals), len(eigs)])

        (ramps_less_than_t, constants_less_than_t, steps_less_than_t,
            ramps_containing_t, constants_containing_t) = segment_containing_also_segments_less_than_xi(loadtim, loadmag, tvals, steps_or_equal_to = True)

        for i, t in enumerate(tvals):
            for k in constants_containing_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += ((cos(omega*t + phase)/(1 + omega**2/(dT**2*eig**2)) + omega*sin(omega*t + phase)/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig) - (cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))/(1 +
                        omega**2/(dT**2*eig**2)) + omega*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t -
                        omega*(t - loadtim[k]) + phase)/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig))
            for k in constants_less_than_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += ((cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k + 1]))/(1 +
                        omega**2/(dT**2*eig**2)) + omega*exp(-dT*eig*(t - loadtim[k +
                        1]))*sin(omega*t - omega*(t - loadtim[k + 1]) + phase)/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig) - (cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))/(1 +
                        omega**2/(dT**2*eig**2)) + omega*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t -
                        omega*(t - loadtim[k]) + phase)/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig))
            for k in ramps_containing_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += ((-dT*eig*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        + dT*eig*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - dT*eig*cos(omega*t +
                        phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + dT*eig*cos(omega*t +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] -
                        omega*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] +
                        omega*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - omega*sin(omega*t +
                        phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + omega*sin(omega*t +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] + cos(omega*t +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        - cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] -
                        omega**2*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**2*cos(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**2*cos(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        2*omega*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        2*omega*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**3*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**3*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*sin(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**3*sin(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**2*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**2*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2))/(dT*eig)
                        - (-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] +
                        dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] - dT*eig*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        dT*eig*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        dT*eig*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] -
                        omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1]
                        + omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1] - omega*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        omega*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]
                        - (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT*eig) +
                        omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1]/(dT*eig) - omega**2*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) - omega**2*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) + omega**2*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - 2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) - omega**3*t*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) - omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) - omega**3*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**2*eig**2))/(dT*eig))
            for k in ramps_less_than_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += (-(-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] +
                        dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] - dT*eig*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        dT*eig*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        dT*eig*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] -
                        omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1]
                        + omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1] - omega*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        omega*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]
                        - (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT*eig) +
                        omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1]/(dT*eig) - omega**2*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) - omega**2*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) + omega**2*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - 2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) - omega**3*t*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) - omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) - omega**3*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**2*eig**2))/(dT*eig) + (-dT*eig*t*cos(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        + dT*eig*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + dT*eig*(t - loadtim[k
                        + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - dT*eig*(t -
                        loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + dT*eig*cos(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - dT*eig*loadtim[k
                        + 1]*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] -
                        omega*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] +
                        omega*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + omega*(t - loadtim[k
                        + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k +
                        1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - omega*(t -
                        loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + omega*exp(-dT*eig*(t
                        - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - omega*loadtim[k +
                        1]*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k +
                        1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + cos(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        - cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] -
                        omega**2*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) + omega**2*(t
                        - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        omega**2*(t - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**2*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        omega**2*loadtim[k + 1]*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        2*omega*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        2*omega*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**3*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**3*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**3*(t - loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t
                        - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*(t - loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t
                        - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**3*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*loadtim[k + 1]*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**2*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**2*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2))/(dT*eig))
    elif implementation == 'fortran':
        #note than all fortran subroutines are lowercase.
        if MUST_TRY_FORTRAN:
            from geotecha.speccon.ext_integrals import eload_coslinear as fn
        else:
            try:
                from geotecha.speccon.ext_integrals import eload_coslinear as fn
            except ImportError:
                fn = Eload_coslinear
        A = fn(loadtim, loadmag, omega, phase, eigs, tvals, dT)
    else:#default is 'vectorized' using numpy
        sin = np.sin
        cos = np.cos
        exp = np.exp

        A = np.zeros([len(tvals), len(eigs)])

        (ramps_less_than_t, constants_less_than_t, steps_less_than_t,
            ramps_containing_t, constants_containing_t) = segment_containing_also_segments_less_than_xi(loadtim, loadmag, tvals, steps_or_equal_to = True)

        eig = eigs[:, None]
        for i, t in enumerate(tvals):
            k = constants_containing_t[i]
            if len(k):
                A[i, :] += np.sum((cos(omega*t + phase)/(1 + omega**2/(dT**2*eig**2)) + omega*sin(omega*t + phase)/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig) - (cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))/(1 +
                        omega**2/(dT**2*eig**2)) + omega*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t -
                        omega*(t - loadtim[k]) + phase)/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig), axis=1)

            k = constants_less_than_t[i]
            if len(k):
                A[i, :] += np.sum((cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k + 1]))/(1 +
                        omega**2/(dT**2*eig**2)) + omega*exp(-dT*eig*(t - loadtim[k +
                        1]))*sin(omega*t - omega*(t - loadtim[k + 1]) + phase)/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig) - (cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))/(1 +
                        omega**2/(dT**2*eig**2)) + omega*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t -
                        omega*(t - loadtim[k]) + phase)/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig), axis=1)

            k = ramps_containing_t[i]
            if len(k):
                A[i, :] += np.sum((-dT*eig*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        + dT*eig*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - dT*eig*cos(omega*t +
                        phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + dT*eig*cos(omega*t +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] -
                        omega*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] +
                        omega*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - omega*sin(omega*t +
                        phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + omega*sin(omega*t +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] + cos(omega*t +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        - cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] -
                        omega**2*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**2*cos(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**2*cos(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        2*omega*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        2*omega*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**3*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**3*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*sin(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**3*sin(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**2*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**2*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2))/(dT*eig)
                        - (-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] +
                        dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] - dT*eig*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        dT*eig*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        dT*eig*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] -
                        omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1]
                        + omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1] - omega*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        omega*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]
                        - (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT*eig) +
                        omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1]/(dT*eig) - omega**2*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) - omega**2*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) + omega**2*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - 2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) - omega**3*t*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) - omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) - omega**3*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**2*eig**2))/(dT*eig), axis=1)

            k = ramps_less_than_t[i]
            if len(k):
                A[i, :] += np.sum(-(-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] +
                        dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] - dT*eig*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        dT*eig*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        dT*eig*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] -
                        omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1]
                        + omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1] - omega*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        omega*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]
                        - (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT*eig) +
                        omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1]/(dT*eig) - omega**2*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) - omega**2*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) + omega**2*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - 2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) - omega**3*t*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) - omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) - omega**3*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**2*eig**2))/(dT*eig) + (-dT*eig*t*cos(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        + dT*eig*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + dT*eig*(t - loadtim[k
                        + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - dT*eig*(t -
                        loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + dT*eig*cos(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - dT*eig*loadtim[k
                        + 1]*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] -
                        omega*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] +
                        omega*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + omega*(t - loadtim[k
                        + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k +
                        1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - omega*(t -
                        loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + omega*exp(-dT*eig*(t
                        - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - omega*loadtim[k +
                        1]*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k +
                        1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + cos(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        - cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] -
                        omega**2*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) + omega**2*(t
                        - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        omega**2*(t - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**2*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        omega**2*loadtim[k + 1]*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        2*omega*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        2*omega*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**3*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**3*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**3*(t - loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t
                        - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*(t - loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t
                        - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**3*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*loadtim[k + 1]*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**2*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**2*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2))/(dT*eig), axis=1)
    return A


def EDload_coslinear(loadtim, loadmag, omega, phase,eigs, tvals, dT=1.0, implementation='vectorized'):
    """Integration of D[cos(omega*tau+phase)*load(tau), tau] * exp(dT * eig * (t-tau)) between [0, t], where
    load(tau) is piecewise linear.

    Performs integrations involving a piecewise linear load.  A 2d array of
    dimensions A[len(tvals), len(eigs)]
    is produced where the 'i'th row of A contains the diagonal elements of the
    spectral 'E' matrix calculated for the time value tvals[i]. i.e. rows of
    this matrix will be assembled into the diagonal matrix 'E' elsewhere.

    Parameters
    ----------
    loadtim : 1d numpy.ndarray
        List of times describing load application.
    loadmag : 1d numpy.ndarray
        List of load magnitudes.
    omega, phase : float
        Parameters that describe a cyclic load cos(omega * t + phase).
    eigs : 1d numpy.ndarray
        List of eigenvalues.
    tvals : 1d numpy.ndarray`
        List of time values to calculate integral at.
    dT : ``float``, optional
        Time factor multiple (Default dT=1.0).
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.


    Returns
    -------
    A : numpy.ndarray
        A 2d array of dimesnions A[len(tvals), len(eigs)].
        The 'i'th row of A is the diagonal elements of the spectral 'E' matrix
        calculated for the time tvals[i].


    Notes
    -----
    Assuming the load are formulated as the product of separate time and depth
    dependant functions:

    .. math:: \\sigma\\left({Z,t}\\right)=\\sigma\\left({Z}\\right)\\sigma\\left({t}\\right)

    the solution to the consolidation equation using the spectral method has
    the form:

    .. math:: u\\left(Z,t\\right)=\\mathbf{\\Phi v E}\\left(\\mathbf{\\Gamma v}\\right)^{-1}\\mathbf{\\theta}

    The matrix :math:`E` is a time dependent diagonal matrix due to time
    dependant loadings.  The version of :math:`E` calculated here in
    `EDload_coslinear` is from loading terms in the governing equation that are NOT
    differentiated wrt :math:`t`.
    The diagonal elements of :math:`E` are given by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{0}^t{\\frac{d{{\\cos\\left(\\omega\\tau+\\textrm{phase}\\right)}\\sigma\\left(\\tau\\right)}}{d\\tau}{\\exp\\left({(dT\\left(t-\\tau\\right)\\lambda_i}\\right)}\\,d\\tau}

    where

     :math:`\\lambda_i` is the `ith` eigenvalue of the problem,
     :math:`dT` is a time factor for numerical convienience,
     :math:`\\sigma\left(\\tau\\right)` is the time dependant portion of the loading function.

    When the time dependant loading term :math:`\\sigma\\left(\\tau\\right)` is
    piecewise in time. The contribution of each load segment is found by:

    .. math:: \\mathbf{E}_{i,i}=\\int_{t_s}^{t_f}{\\frac{d{{\\cos\\left(\\omega\\tau+\\textrm{phase}\\right)}\\sigma\\left(\\tau\\right)}}{d\\tau}\\exp\\left({dT\\left(t-\\tau\\right)*\\lambda_i}\\right)\\,d\\tau}

    where

    .. math:: t_s = \\min\\left(t,t_{increment\\:start}\\right)

    .. math:: t_f = \\min\\left(t,t_{increment\\:end}\\right)

    (note that this function,`EDload_coslinear`, rather than use :math:`t_s` and
    :math:`t_f`,
    explicitly finds increments that the current time falls in, falls after,
    and falls before and treates each case on it's own.)

    Each :math:`t` value of interest requires a separate diagonal matrix
    :math:`E`.  To use space more efficiently and to facilitate numpy
    broadcasting when using the results of the function, the diagonal elements
    of :math:`E` for each time value `t` value are stored in the rows of
    array :math:`A` returned by `EDload_coslinear`.  Thus:

    .. math:: \\mathbf{A}=\\left(\\begin{matrix}E_{0,0}(t_0)&E_{1,1}(t_0)& \cdots & E_{neig-1,neig-1}(t_0)\\\ E_{0,0}(t_1)&E_{1,1}(t_1)& \\cdots & E_{neig-1,neig-1}(t_1)\\\ \\vdots&\\vdots&\\ddots&\\vdots \\\ E_{0,0}(t_m)&E_{1,1}(t_m)& \cdots & E_{neig-1,neig-1}(t_m)\\end{matrix}\\right)

    See Also
    --------
    pEDload_coslinear : Same function with PolyLine inputs.
    geotecha.speccon.ext_integrals.edload_coslinear : Equivalent fortran function.
    geotecha.speccon.integrals_generate_code.EDload_coslinear_implementations : Generation
        of code for this function.
    EDload_linear : Similar function with no cosine term.
    Eload_coslinear : Similar function but the combined
        loading function is not differentiated with respect to time.


    """
    loadtim = np.asarray(loadtim)
    loadmag = np.asarray(loadmag)
    eigs = np.asarray(eigs)
    tvals = np.asarray(tvals)

    if implementation == 'scalar':
        sin = math.sin
        cos = math.cos
        exp = math.exp

        A = np.zeros([len(tvals), len(eigs)])

        (ramps_less_than_t, constants_less_than_t, steps_less_than_t,
            ramps_containing_t, constants_containing_t) = segment_containing_also_segments_less_than_xi(loadtim, loadmag, tvals, steps_or_equal_to = True)

        for i, t in enumerate(tvals):
            for k in steps_less_than_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += (cos(omega*loadtim[k] + phase)*exp(-dT*eig*(t - loadtim[k]))*(loadmag[k + 1] - loadmag[k]))
            for k in ramps_containing_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += (omega*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        - omega*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] -
                        omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1]
                        + omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*sin(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - omega*sin(omega*t +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] +
                        omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1] - omega*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        omega*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] -
                        cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] + cos(omega*t +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]
                        - (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega**2*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) -
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT*eig) -
                        omega**2*cos(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**2*cos(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1]/(dT*eig) + omega**2*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) + omega**2*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) - omega**2*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) -
                        2*omega*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        2*omega*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - 2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) + omega**3*t*sin(omega*t +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k +
                        1]/(dT**2*eig**2) - omega**3*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1]
                        + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*sin(omega*t + phase)*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*sin(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k]/(dT**2*eig**2) - omega**3*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**2*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**2*eig**2)
                        - omega**4*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**3*eig**3) +
                        omega**4*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**3*eig**3) +
                        omega**4*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**3*eig**3) - omega**4*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**3*eig**3)
                        - omega**4*cos(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**3*eig**3) +
                        omega**4*cos(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**3*eig**3) -
                        omega**4*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1]/(dT**3*eig**3) + omega**4*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**3*eig**3) + omega**4*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**3*eig**3) -
                        omega**4*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**3*eig**3))
            for k in ramps_less_than_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += (-omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] -
                        omega*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - omega*(t - loadtim[k
                        + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k +
                        1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] + omega*(t -
                        loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] +
                        omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1] - omega*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        omega*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k +
                        1]) + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - omega*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*loadtim[k + 1]*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] +
                        omega*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]
                        - (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k
                        + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        + cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] +
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) -
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT*eig) -
                        omega**2*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) + omega**2*(t
                        - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        omega**2*(t - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1]/(dT*eig) + omega**2*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) + omega**2*cos(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT*eig) -
                        omega**2*loadtim[k + 1]*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**2*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - 2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) - 2*omega*exp(-dT*eig*(t - loadtim[k +
                        1]))*sin(omega*t - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        2*omega*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*t*exp(-dT*eig*(t - loadtim[k +
                        1]))*sin(omega*t - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*(t - loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t
                        - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**3*(t - loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t
                        - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k]/(dT**2*eig**2) - omega**3*exp(-dT*eig*(t -
                        loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*loadtim[k + 1]*exp(-dT*eig*(t -
                        loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**3*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**2*eig**2)
                        + omega**2*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**2*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**4*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**3*eig**3) - omega**4*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**3*eig**3)
                        - omega**4*t*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**3*eig**3) +
                        omega**4*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**3*eig**3) +
                        omega**4*(t - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**3*eig**3) -
                        omega**4*(t - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**3*eig**3) -
                        omega**4*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1]/(dT**3*eig**3) + omega**4*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**3*eig**3) + omega**4*cos(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**3*eig**3)
                        - omega**4*loadtim[k + 1]*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**3*eig**3) -
                        omega**4*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**3*eig**3))
            for k in constants_containing_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += (-omega*(sin(omega*t + phase)/(1 + omega**2/(dT**2*eig**2)) - omega*cos(omega*t + phase)/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig) + omega*(exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)/(1 +
                        omega**2/(dT**2*eig**2)) - omega*cos(omega*t - omega*(t - loadtim[k]) +
                        phase)*exp(-dT*eig*(t - loadtim[k]))/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig))
            for k in constants_less_than_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += (-omega*(exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) + phase)/(1 +
                        omega**2/(dT**2*eig**2)) - omega*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig) + omega*(exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)/(1 +
                        omega**2/(dT**2*eig**2)) - omega*cos(omega*t - omega*(t - loadtim[k]) +
                        phase)*exp(-dT*eig*(t - loadtim[k]))/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig))

    elif implementation == 'fortran':
        #note than all fortran subroutines are lowercase.
        if MUST_TRY_FORTRAN:
            from geotecha.speccon.ext_integrals import edload_coslinear as fn
        else:
            try:
                from geotecha.speccon.ext_integrals import edload_coslinear as fn
            except ImportError:
                fn = EDload_coslinear
        A = fn(loadtim, loadmag, omega, phase, eigs, tvals, dT)
    else:#default is 'vectorized' using numpy
        sin = np.sin
        cos = np.cos
        exp = np.exp

        A = np.zeros([len(tvals), len(eigs)])

        (ramps_less_than_t, constants_less_than_t, steps_less_than_t,
            ramps_containing_t, constants_containing_t) = segment_containing_also_segments_less_than_xi(loadtim, loadmag, tvals, steps_or_equal_to = True)

        eig = eigs[:, None]


        for i, t in enumerate(tvals):
            k = steps_less_than_t[i]
            if len(k):
                A[i, :] += np.sum(cos(omega*loadtim[k] + phase)*exp(-dT*eig*(t - loadtim[k]))*(loadmag[k + 1] - loadmag[k]), axis=1)
            k = ramps_containing_t[i]
            if len(k):
                A[i, :] += np.sum(omega*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        - omega*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] -
                        omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1]
                        + omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*sin(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - omega*sin(omega*t +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] +
                        omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1] - omega*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        omega*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] -
                        cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] + cos(omega*t +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]
                        - (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega**2*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) -
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT*eig) -
                        omega**2*cos(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**2*cos(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1]/(dT*eig) + omega**2*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) + omega**2*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) - omega**2*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) -
                        2*omega*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        2*omega*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - 2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) + omega**3*t*sin(omega*t +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k +
                        1]/(dT**2*eig**2) - omega**3*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1]
                        + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*sin(omega*t + phase)*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*sin(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k]/(dT**2*eig**2) - omega**3*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*loadtim[k]*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**2*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**2*eig**2)
                        - omega**4*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**3*eig**3) +
                        omega**4*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**3*eig**3) +
                        omega**4*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**3*eig**3) - omega**4*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**3*eig**3)
                        - omega**4*cos(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**3*eig**3) +
                        omega**4*cos(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**3*eig**3) -
                        omega**4*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1]/(dT**3*eig**3) + omega**4*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**3*eig**3) + omega**4*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**3*eig**3) -
                        omega**4*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**3*eig**3), axis=1)
            k = ramps_less_than_t[i]
            if len(k):
                A[i, :] += np.sum(-omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] -
                        omega*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - omega*(t - loadtim[k
                        + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k +
                        1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] + omega*(t -
                        loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] +
                        omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1] - omega*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        omega*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k +
                        1]) + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - omega*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        omega*loadtim[k + 1]*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] +
                        omega*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]
                        - (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k
                        + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        + cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] +
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) -
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT*eig) -
                        omega**2*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) + omega**2*(t
                        - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        omega**2*(t - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1]/(dT*eig) + omega**2*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) + omega**2*cos(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT*eig) -
                        omega**2*loadtim[k + 1]*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**2*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - 2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) - 2*omega*exp(-dT*eig*(t - loadtim[k +
                        1]))*sin(omega*t - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        2*omega*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k
                        + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*t*exp(-dT*eig*(t - loadtim[k +
                        1]))*sin(omega*t - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*(t - loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t
                        - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**3*(t - loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t
                        - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k]/(dT**2*eig**2) - omega**3*exp(-dT*eig*(t -
                        loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2) + omega**3*loadtim[k + 1]*exp(-dT*eig*(t -
                        loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**3*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**2*eig**2)
                        + omega**2*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**2*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**4*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**3*eig**3) - omega**4*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**3*eig**3)
                        - omega**4*t*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**3*eig**3) +
                        omega**4*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**3*eig**3) +
                        omega**4*(t - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**3*eig**3) -
                        omega**4*(t - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**3*eig**3) -
                        omega**4*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1]/(dT**3*eig**3) + omega**4*(-dT*eig*loadtim[k +
                        1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**3*eig**3) + omega**4*cos(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**3*eig**3)
                        - omega**4*loadtim[k + 1]*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**3*eig**3) -
                        omega**4*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**3*eig**3), axis=1)
            k = constants_containing_t[i]
            if len(k):
                A[i,:] += np.sum(-omega*(sin(omega*t + phase)/(1 + omega**2/(dT**2*eig**2)) - omega*cos(omega*t + phase)/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig) + omega*(exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)/(1 +
                        omega**2/(dT**2*eig**2)) - omega*cos(omega*t - omega*(t - loadtim[k]) +
                        phase)*exp(-dT*eig*(t - loadtim[k]))/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig), axis=1)
            k = constants_less_than_t[i]
            if len(k):
                A[i,:] += np.sum(-omega*(exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) + phase)/(1 +
                        omega**2/(dT**2*eig**2)) - omega*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig) + omega*(exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)/(1 +
                        omega**2/(dT**2*eig**2)) - omega*cos(omega*t - omega*(t - loadtim[k]) +
                        phase)*exp(-dT*eig*(t - loadtim[k]))/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig), axis=1)
    return A


def Eload_sinlinear(loadtim, loadmag, omega, phase, eigs, tvals, dT=1.0, implementation='vectorized'):

    loadtim = np.asarray(loadtim)
    loadmag = np.asarray(loadmag)
    eigs = np.asarray(eigs)
    tvals = np.asarray(tvals)

    if implementation == 'scalar':
        sin = np.sin
        cos = np.cos
        exp = np.exp
        #note that math module doesn't like complex numbers

        A = np.zeros([len(tvals), len(eigs)], dtype=complex)

        (ramps_less_than_t, constants_less_than_t, steps_less_than_t,
            ramps_containing_t, constants_containing_t) = segment_containing_also_segments_less_than_xi(loadtim, loadmag, tvals, steps_or_equal_to = True)

        for i, t in enumerate(tvals):
            for k in constants_containing_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += ((sin(omega*t + phase)/(1 + omega**2/(dT**2*eig**2)) - omega*cos(omega*t + phase)/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig) - (exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)/(1 +
                        omega**2/(dT**2*eig**2)) - omega*cos(omega*t - omega*(t - loadtim[k]) +
                        phase)*exp(-dT*eig*(t - loadtim[k]))/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig))
            for k in constants_less_than_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += ((exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) + phase)/(1 +
                        omega**2/(dT**2*eig**2)) - omega*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig) - (exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)/(1 +
                        omega**2/(dT**2*eig**2)) - omega*cos(omega*t - omega*(t - loadtim[k]) +
                        phase)*exp(-dT*eig*(t - loadtim[k]))/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig))
            for k in ramps_containing_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += ((-dT*eig*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        + dT*eig*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - dT*eig*sin(omega*t +
                        phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + dT*eig*sin(omega*t +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] +
                        omega*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] -
                        omega*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + omega*cos(omega*t +
                        phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - omega*cos(omega*t +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] + sin(omega*t +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        - sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] -
                        omega**2*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**2*sin(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**2*sin(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        2*omega*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        2*omega*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**3*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**3*cos(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*cos(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**2*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**2*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2))/(dT*eig)
                        - (-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1] - dT*eig*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        dT*eig*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        dT*eig*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]
                        - omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] + omega*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        omega*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1]
                        - (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) + omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) + omega**2*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) - omega**2*loadtim[k + 1]*(-dT*eig*loadtim[k + 1]
                        + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) + omega**2*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - 2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) + omega**3*t*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) - omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**2*eig**2)
                        - omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) + omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**2*eig**2) + omega**3*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**2*eig**2) -
                        omega**3*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2))/(dT*eig))
            for k in ramps_less_than_t[i]:
                for j, eig in enumerate(eigs):
                    A[i,j] += (-(-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1] - dT*eig*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        dT*eig*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        dT*eig*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]
                        - omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] + omega*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        omega*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1]
                        - (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) + omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) + omega**2*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) - omega**2*loadtim[k + 1]*(-dT*eig*loadtim[k + 1]
                        + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) + omega**2*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - 2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) + omega**3*t*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) - omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**2*eig**2)
                        - omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) + omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**2*eig**2) + omega**3*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**2*eig**2) -
                        omega**3*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2))/(dT*eig) + (-dT*eig*t*exp(-dT*eig*(t -
                        loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        + dT*eig*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + dT*eig*(t - loadtim[k
                        + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k +
                        1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - dT*eig*(t -
                        loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + dT*eig*exp(-dT*eig*(t
                        - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - dT*eig*loadtim[k
                        + 1]*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k +
                        1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + omega*t*cos(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        - omega*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - omega*(t - loadtim[k
                        + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] + omega*(t -
                        loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - omega*cos(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] + omega*loadtim[k +
                        1]*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + exp(-dT*eig*(t -
                        loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        - exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1])
                        + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] -
                        omega**2*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) + omega**2*(t
                        - loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        omega**2*(t - loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t
                        - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**2*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        omega**2*loadtim[k + 1]*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        2*omega*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        2*omega*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**3*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*(t - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**3*(t - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**3*loadtim[k + 1]*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**2*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**2*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2))/(dT*eig))
    elif implementation == 'fortran':
        #note than all fortran subroutines are lowercase.
        if MUST_TRY_FORTRAN:
            from geotecha.speccon.ext_integrals import eload_sinlinear as fn
        else:
            try:
                from geotecha.speccon.ext_integrals import eload_sinlinear as fn
            except ImportError:
                fn = Eload_sinlinear
        A = fn(loadtim, loadmag, omega, phase, eigs, tvals, dT)
    else:#default is 'vectorized' using numpy
        sin = np.sin
        cos = np.cos
        exp = np.exp

        A = np.zeros([len(tvals), len(eigs)], dtype=complex)

        (ramps_less_than_t, constants_less_than_t, steps_less_than_t,
            ramps_containing_t, constants_containing_t) = segment_containing_also_segments_less_than_xi(loadtim, loadmag, tvals, steps_or_equal_to = True)

        eig = eigs[:, None]
        for i, t in enumerate(tvals):
            k = constants_containing_t[i]
            if len(k):
                A[i, :] += np.sum((sin(omega*t + phase)/(1 + omega**2/(dT**2*eig**2)) - omega*cos(omega*t + phase)/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig) - (exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)/(1 +
                        omega**2/(dT**2*eig**2)) - omega*cos(omega*t - omega*(t - loadtim[k]) +
                        phase)*exp(-dT*eig*(t - loadtim[k]))/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig), axis=1)

            k = constants_less_than_t[i]
            if len(k):
                A[i, :] += np.sum((exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) + phase)/(1 +
                        omega**2/(dT**2*eig**2)) - omega*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig) - (exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)/(1 +
                        omega**2/(dT**2*eig**2)) - omega*cos(omega*t - omega*(t - loadtim[k]) +
                        phase)*exp(-dT*eig*(t - loadtim[k]))/(dT*eig*(1 +
                        omega**2/(dT**2*eig**2))))*loadmag[k]/(dT*eig), axis=1)

            k = ramps_containing_t[i]
            if len(k):
                A[i, :] += np.sum((-dT*eig*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        + dT*eig*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - dT*eig*sin(omega*t +
                        phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + dT*eig*sin(omega*t +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] +
                        omega*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] -
                        omega*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + omega*cos(omega*t +
                        phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - omega*cos(omega*t +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] + sin(omega*t +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        - sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] -
                        omega**2*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        omega**2*sin(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**2*sin(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        2*omega*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        2*omega*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**3*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*t*cos(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k]
                        - 2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) +
                        omega**3*cos(omega*t + phase)*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*cos(omega*t + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**2*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**2*sin(omega*t + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2))/(dT*eig)
                        - (-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1] - dT*eig*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        dT*eig*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        dT*eig*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]
                        - omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] + omega*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        omega*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1]
                        - (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) + omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) + omega**2*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) - omega**2*loadtim[k + 1]*(-dT*eig*loadtim[k + 1]
                        + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) + omega**2*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - 2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) + omega**3*t*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) - omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**2*eig**2)
                        - omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) + omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**2*eig**2) + omega**3*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**2*eig**2) -
                        omega**3*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2))/(dT*eig), axis=1)

            k = ramps_less_than_t[i]
            if len(k):
                A[i, :] += np.sum(-(-dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        dT*eig*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        dT*eig*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*exp(-dT*eig*(t - loadtim[k]))*sin(omega*t - omega*(t -
                        loadtim[k]) + phase)*loadmag[k + 1] - dT*eig*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        dT*eig*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] +
                        dT*eig*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1] +
                        omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]
                        - omega*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t -
                        loadtim[k])*cos(omega*t - omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k + 1] + omega*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] +
                        omega*loadtim[k + 1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k] -
                        omega*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1] +
                        (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k + 1]
                        - (-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t
                        - loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k] -
                        omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) + omega**2*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) + omega**2*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) - omega**2*loadtim[k + 1]*(-dT*eig*loadtim[k + 1]
                        + dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT*eig) + omega**2*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT*eig) - 2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k + 1]/(dT*eig) +
                        2*omega*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT*eig) + omega**3*t*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) - omega**3*t*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k]/(dT**2*eig**2)
                        - omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) + omega**3*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*(t - loadtim[k])*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**2*eig**2) + omega**3*loadtim[k +
                        1]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t -
                        omega*(t - loadtim[k]) + phase)*exp(-dT*eig*(t -
                        loadtim[k]))*loadmag[k]/(dT**2*eig**2) -
                        omega**3*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*cos(omega*t - omega*(t -
                        loadtim[k]) + phase)*exp(-dT*eig*(t - loadtim[k]))*loadmag[k +
                        1]/(dT**2*eig**2) - omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) + phase)*loadmag[k +
                        1]/(dT**2*eig**2) + omega**2*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*exp(-dT*eig*(t -
                        loadtim[k]))*sin(omega*t - omega*(t - loadtim[k]) +
                        phase)*loadmag[k]/(dT**2*eig**2))/(dT*eig) + (-dT*eig*t*exp(-dT*eig*(t -
                        loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        + dT*eig*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + dT*eig*(t - loadtim[k
                        + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k +
                        1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - dT*eig*(t -
                        loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + dT*eig*exp(-dT*eig*(t
                        - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] - dT*eig*loadtim[k
                        + 1]*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k +
                        1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + omega*t*cos(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        - omega*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - omega*(t - loadtim[k
                        + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] + omega*(t -
                        loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] - omega*cos(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t - loadtim[k +
                        1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1] + omega*loadtim[k +
                        1]*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] + exp(-dT*eig*(t -
                        loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k +
                        1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]
                        - exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t - loadtim[k + 1])
                        + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] - 2*omega**2*loadtim[k
                        + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k +
                        1]/(dT**3*eig**3) + omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k] -
                        omega**2*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        omega**2*t*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) + omega**2*(t
                        - loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        omega**2*(t - loadtim[k + 1])*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t
                        - omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**2*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*loadtim[k]*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) -
                        omega**2*loadtim[k + 1]*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t -
                        omega*(t - loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) -
                        2*omega*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT*eig) +
                        2*omega*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT*eig) +
                        omega**3*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) -
                        omega**3*t*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t
                        - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*(t - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**3*(t - loadtim[k + 1])*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**3*cos(omega*t - omega*(t - loadtim[k + 1]) + phase)*exp(-dT*eig*(t -
                        loadtim[k + 1]))*loadtim[k]*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**3*loadtim[k + 1]*cos(omega*t - omega*(t - loadtim[k + 1]) +
                        phase)*exp(-dT*eig*(t - loadtim[k + 1]))*(-dT*eig*loadtim[k + 1] +
                        dT*eig*loadtim[k] - 2*omega**2*loadtim[k + 1]/(dT*eig) +
                        2*omega**2*loadtim[k]/(dT*eig) - omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2) -
                        omega**2*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k + 1]/(dT**2*eig**2) +
                        omega**2*exp(-dT*eig*(t - loadtim[k + 1]))*sin(omega*t - omega*(t -
                        loadtim[k + 1]) + phase)*(-dT*eig*loadtim[k + 1] + dT*eig*loadtim[k] -
                        2*omega**2*loadtim[k + 1]/(dT*eig) + 2*omega**2*loadtim[k]/(dT*eig) -
                        omega**4*loadtim[k + 1]/(dT**3*eig**3) +
                        omega**4*loadtim[k]/(dT**3*eig**3))**(-1)*loadmag[k]/(dT**2*eig**2))/(dT*eig), axis=1)
    return A

def pdim1sin_DD_abDDf_linear(m, a, b, **kwargs):
    """Integration of sin(mi * z) * D[a(z) * b(z) D[sin(mj * z),z,2],z,2]
    between ztop and zbot where a(z) & b(z) is piecewise linear functions of z.

    Wrapper for dim1sin_DD_abDDf_linear to allow PolyLine inputs.


    Calulation of integrals is performed at each element of a square symmetric
    matrix (size depends on size of `m`).

    Parameters
    ----------
    m : ``list`` of ``float``
        eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    a, b : PolyLine object
        PolyLine defining piecewise linear relationship.
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A square symmetric matrix, size determined by size of `m`.

    See Also
    -------
    dim1sin_DD_abDDf_linear : Wrapped function.


    Notes
    -----
    The `dim1sin_DD_abDDf_linear` matrix, :math:`A` is given by:

    .. math:: \\mathbf{A}_{i,j}=\\int_{0}^1{\\frac{d^2}{dz^2}\\left({a\\left(z\\right)}{b\\left(z\\right)}\\frac{d^2\\phi_j}{dz^2}\\right)\\phi_i\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)` and :math:`b\\left(z\\right)` are piecewise
    linear functions with respect to :math:`z`, that within a layer is defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.

    To make the above integration simpler we integate by parts to get:

    .. math:: \\mathbf{A}_{i,j}= \\left.{\\frac{d}{dz}\\left({a\\left(z\\right)}{b\\left(z\\right)}\\frac{d^2\\phi_j}{dz^2}\\right)\\phi_i}\\right|_{z=0}^{z=1}
                    - \\left.{{a\\left(z\\right)}{b\\left(z\\right)}\\frac{d^2\\phi_j}{dz^2}\\frac{d\\phi_i}{dz}}\\right|_{z=0}^{z=1}
               +\\int_{0}^1{{a\\left(z\\right)}{b\\left(z\\right)}\\frac{d^2\\phi_j}{dz^2}\\frac{d^2\\phi_i}{dz^2}\\,dz}

    In this case the sine basis functions means the end point terms in the above
    equation are zero, leaving us with

    .. math:: \\mathbf{A}_{i,j}= \\int_{0}^1{{a\\left(z\\right)}{b\\left(z\\right)}\\frac{d^2\\phi_j}{dz^2}\\frac{d^2\\phi_i}{dz^2}\\,dz}


    """

    a, b = pwise.polyline_make_x_common(a,b)
    return dim1sin_DD_abDDf_linear(m, a.y1, a.y2, b.y1, b.y2, a.x1, a.x2, **kwargs)


def dim1sin_DD_abDDf_linear(m, at, ab, bt, bb,  zt, zb, implementation='vectorized'):
    """Integration of sin(mi * z) * D[a(z) * b(z) D[sin(mj * z),z,2],z,2]
    between ztop and zbot where a(z) & b(z) is piecewise linear functions of z.

    Calulation of integrals is performed at each element of a square symmetric
    matrix (size depends on size of `m`).

    Parameters
    ----------
    m : ``list`` of ``float``
        eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx
    at : ``list`` of ``float``
        Property at top of each layer.
    ab : ``list`` of ``float``
        Property at bottom of each layer.
    bt : ``list`` of ``float``
        2nd property at top of each layer.
    bb : ``list`` of ``float``
        2nd property at bottom of each layer.
    zt : ``list`` of ``float``
        Normalised depth or z-coordinate at top of each layer. `zt[0]` = 0
    zb : ``list`` of ``float``
        Normalised depth or z-coordinate at bottom of each layer. `zt[-1]` = 1
    implementation : ['vectorized', 'scalar', 'fortran'], optional
        Functional implementation: 'scalar' = python loops (slow),
        'fortran' = fortran code (fastest), 'vectorized' = numpy(fast).
        Default implementation='vectorized'.  If fortran extention module
        cannot be imported then 'vectorized' version will be used.
        If anything other than 'fortran' or 'scalar' is used then
        default vectorized version will be used.

    Returns
    -------
    A : numpy.ndarray
        A square symmetric matrix, size determined by size of `m`.

    See Also
    --------
    m_from_sin_mx : Used to generate 'm' input parameter.
    geotecha.integrals_generate_code.dim1sin_DD_abDDf_linear_implementations : Generation
        of code for this function.
    pdim1sin_DD_abDDf_linear : Same function with PolyLine
        inputs.
    geotecha.speccon.ext_integrals.dim1sin_dd_abddf_linear : Equivalent fortran
        function.

    Notes
    -----
    The `dim1sin_DD_abDDf_linear` matrix, :math:`A` is given by:

    .. math:: \\mathbf{A}_{i,j}=\\int_{0}^1{\\frac{d^2}{dz^2}\\left({a\\left(z\\right)}{b\\left(z\\right)}\\frac{d^2\\phi_j}{dz^2}\\right)\\phi_i\\,dz}

    where the basis function :math:`\\phi_i` is given by:

    .. math:: \\phi_i\\left(z\\right)=\\sin\\left({m_i}z\\right)

    and :math:`a\\left(z\\right)` and :math:`b\\left(z\\right)` are piecewise
    linear functions with respect to :math:`z`, that within a layer is defined by:

    .. math:: a\\left(z\\right) = a_t+\\frac{a_b-a_t}{z_b-z_t}\\left(z-z_t\\right)

    with :math:`t` and :math:`b` subscripts representing 'top' and 'bottom' of
    each layer respectively.

    To make the above integration simpler we integate by parts to get:

    .. math:: \\mathbf{A}_{i,j}= \\left.{\\frac{d}{dz}\\left({a\\left(z\\right)}{b\\left(z\\right)}\\frac{d^2\\phi_j}{dz^2}\\right)\\phi_i}\\right|_{z=0}^{z=1}
                    - \\left.{{a\\left(z\\right)}{b\\left(z\\right)}\\frac{d^2\\phi_j}{dz^2}\\frac{d\\phi_i}{dz}}\\right|_{z=0}^{z=1}
               +\\int_{0}^1{{a\\left(z\\right)}{b\\left(z\\right)}\\frac{d^2\\phi_j}{dz^2}\\frac{d^2\\phi_i}{dz^2}\\,dz}

    In this case the sine basis functions means the end point terms in the above
    equation are zero, leaving us with

    .. math:: \\mathbf{A}_{i,j}= \\int_{0}^1{{a\\left(z\\right)}{b\\left(z\\right)}\\frac{d^2\\phi_j}{dz^2}\\frac{d^2\\phi_i}{dz^2}\\,dz}


    """


    #import numpy as np #import this at module level
    #import math #import this at module level

    m = np.asarray(m)
    at = np.asarray(at)
    ab = np.asarray(ab)
    bt = np.asarray(bt)
    bb = np.asarray(bb)
    zt = np.asarray(zt)
    zb = np.asarray(zb)

    neig = len(m)

    if implementation == 'scalar':
        sin = math.sin
        cos = math.cos
        A = np.zeros([neig, neig], float)
        nlayers = len(zt)
        for layer in range(nlayers):
            a_slope = (ab[layer] - at[layer]) / (zb[layer] - zt[layer])
            b_slope = (bb[layer] - bt[layer]) / (zb[layer] - zt[layer])
            for i in range(neig):
                A[i, i] += (-m[i]**4*(a_slope*b_slope*m[i]**(-3)*cos(zt[layer]*m[i])*sin(zt[layer]*m[i])/4 +
                    a_slope*b_slope*zt[layer]*m[i]**(-2)*sin(zt[layer]*m[i])**2/4 +
                    a_slope*b_slope*zt[layer]*m[i]**(-2)*cos(zt[layer]*m[i])**2/4 +
                    a_slope*b_slope*zt[layer]**3*sin(zt[layer]*m[i])**2/6 +
                    a_slope*b_slope*zt[layer]**3*cos(zt[layer]*m[i])**2/6 -
                    a_slope*bt[layer]*m[i]**(-2)*cos(zt[layer]*m[i])**2/4 -
                    a_slope*zt[layer]**2*bt[layer]*sin(zt[layer]*m[i])**2/4 -
                    a_slope*zt[layer]**2*bt[layer]*cos(zt[layer]*m[i])**2/4 -
                    b_slope*at[layer]*m[i]**(-2)*cos(zt[layer]*m[i])**2/4 -
                    b_slope*zt[layer]**2*at[layer]*sin(zt[layer]*m[i])**2/4 -
                    b_slope*zt[layer]**2*at[layer]*cos(zt[layer]*m[i])**2/4 -
                    bt[layer]*at[layer]*m[i]**(-1)*cos(zt[layer]*m[i])*sin(zt[layer]*m[i])/2 +
                    zt[layer]*bt[layer]*at[layer]*sin(zt[layer]*m[i])**2/2 +
                    zt[layer]*bt[layer]*at[layer]*cos(zt[layer]*m[i])**2/2) +
                    m[i]**4*(a_slope*b_slope*m[i]**(-3)*cos(zb[layer]*m[i])*sin(zb[layer]*m[i])/4 +
                    a_slope*b_slope*zb[layer]*m[i]**(-2)*sin(zb[layer]*m[i])**2/4 -
                    a_slope*b_slope*zb[layer]*m[i]**(-2)*cos(zb[layer]*m[i])**2/4 +
                    a_slope*b_slope*zb[layer]*zt[layer]*m[i]**(-1)*cos(zb[layer]*m[i])*sin(zb[layer]*m[i])
                    + a_slope*b_slope*zb[layer]*zt[layer]**2*sin(zb[layer]*m[i])**2/2 +
                    a_slope*b_slope*zb[layer]*zt[layer]**2*cos(zb[layer]*m[i])**2/2 -
                    a_slope*b_slope*zb[layer]**2*m[i]**(-1)*cos(zb[layer]*m[i])*sin(zb[layer]*m[i])/2
                    - a_slope*b_slope*zb[layer]**2*zt[layer]*sin(zb[layer]*m[i])**2/2 -
                    a_slope*b_slope*zb[layer]**2*zt[layer]*cos(zb[layer]*m[i])**2/2 +
                    a_slope*b_slope*zb[layer]**3*sin(zb[layer]*m[i])**2/6 +
                    a_slope*b_slope*zb[layer]**3*cos(zb[layer]*m[i])**2/6 +
                    a_slope*b_slope*zt[layer]*m[i]**(-2)*cos(zb[layer]*m[i])**2/2 -
                    a_slope*b_slope*zt[layer]**2*m[i]**(-1)*cos(zb[layer]*m[i])*sin(zb[layer]*m[i])/2
                    - a_slope*bt[layer]*m[i]**(-2)*cos(zb[layer]*m[i])**2/4 -
                    a_slope*zb[layer]*bt[layer]*m[i]**(-1)*cos(zb[layer]*m[i])*sin(zb[layer]*m[i])/2
                    - a_slope*zb[layer]*zt[layer]*bt[layer]*sin(zb[layer]*m[i])**2/2 -
                    a_slope*zb[layer]*zt[layer]*bt[layer]*cos(zb[layer]*m[i])**2/2 +
                    a_slope*zb[layer]**2*bt[layer]*sin(zb[layer]*m[i])**2/4 +
                    a_slope*zb[layer]**2*bt[layer]*cos(zb[layer]*m[i])**2/4 +
                    a_slope*zt[layer]*bt[layer]*m[i]**(-1)*cos(zb[layer]*m[i])*sin(zb[layer]*m[i])/2
                    - b_slope*at[layer]*m[i]**(-2)*cos(zb[layer]*m[i])**2/4 -
                    b_slope*zb[layer]*at[layer]*m[i]**(-1)*cos(zb[layer]*m[i])*sin(zb[layer]*m[i])/2
                    - b_slope*zb[layer]*zt[layer]*at[layer]*sin(zb[layer]*m[i])**2/2 -
                    b_slope*zb[layer]*zt[layer]*at[layer]*cos(zb[layer]*m[i])**2/2 +
                    b_slope*zb[layer]**2*at[layer]*sin(zb[layer]*m[i])**2/4 +
                    b_slope*zb[layer]**2*at[layer]*cos(zb[layer]*m[i])**2/4 +
                    b_slope*zt[layer]*at[layer]*m[i]**(-1)*cos(zb[layer]*m[i])*sin(zb[layer]*m[i])/2
                    - bt[layer]*at[layer]*m[i]**(-1)*cos(zb[layer]*m[i])*sin(zb[layer]*m[i])/2 +
                    zb[layer]*bt[layer]*at[layer]*sin(zb[layer]*m[i])**2/2 +
                    zb[layer]*bt[layer]*at[layer]*cos(zb[layer]*m[i])**2/2))
            for i in range(neig-1):
                for j in range(i + 1, neig):
                    A[i, j] += (-m[j]**2*m[i]**2*(2*a_slope*b_slope*sin(zt[layer]*m[j])*m[i]**3*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zt[layer]*m[i]) -
                        6*a_slope*b_slope*m[j]*cos(zt[layer]*m[j])*m[i]**2*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zt[layer]*m[i]) +
                        6*a_slope*b_slope*m[j]**2*sin(zt[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zt[layer]*m[i]) -
                        2*a_slope*b_slope*m[j]**3*cos(zt[layer]*m[j])*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zt[layer]*m[i]) +
                        a_slope*bt[layer]*sin(zt[layer]*m[j])*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zt[layer]*m[i]) +
                        2*a_slope*bt[layer]*m[j]*cos(zt[layer]*m[j])*m[i]**3*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zt[layer]*m[i]) -
                        2*a_slope*bt[layer]*m[j]**3*cos(zt[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zt[layer]*m[i]) -
                        a_slope*bt[layer]*m[j]**4*sin(zt[layer]*m[j])*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zt[layer]*m[i]) +
                        b_slope*at[layer]*sin(zt[layer]*m[j])*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zt[layer]*m[i]) +
                        2*b_slope*at[layer]*m[j]*cos(zt[layer]*m[j])*m[i]**3*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zt[layer]*m[i]) -
                        2*b_slope*at[layer]*m[j]**3*cos(zt[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zt[layer]*m[i]) -
                        b_slope*at[layer]*m[j]**4*sin(zt[layer]*m[j])*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zt[layer]*m[i]) -
                        bt[layer]*at[layer]*sin(zt[layer]*m[j])*m[i]**5*(m[i]**6 - 3*m[j]**2*m[i]**4
                        + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zt[layer]*m[i]) +
                        bt[layer]*at[layer]*m[j]*cos(zt[layer]*m[j])*m[i]**4*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zt[layer]*m[i]) +
                        2*bt[layer]*at[layer]*m[j]**2*sin(zt[layer]*m[j])*m[i]**3*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zt[layer]*m[i]) -
                        2*bt[layer]*at[layer]*m[j]**3*cos(zt[layer]*m[j])*m[i]**2*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zt[layer]*m[i]) -
                        bt[layer]*at[layer]*m[j]**4*sin(zt[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zt[layer]*m[i]) +
                        bt[layer]*at[layer]*m[j]**5*cos(zt[layer]*m[j])*(m[i]**6 - 3*m[j]**2*m[i]**4
                        + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zt[layer]*m[i])) +
                        m[j]**2*m[i]**2*(2*a_slope*b_slope*sin(zb[layer]*m[j])*m[i]**3*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        6*a_slope*b_slope*m[j]*cos(zb[layer]*m[j])*m[i]**2*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        6*a_slope*b_slope*m[j]**2*sin(zb[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        2*a_slope*b_slope*m[j]**3*cos(zb[layer]*m[j])*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        2*a_slope*b_slope*zb[layer]*sin(zb[layer]*m[j])*m[i]**4*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        4*a_slope*b_slope*zb[layer]*m[j]*cos(zb[layer]*m[j])*m[i]**3*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        4*a_slope*b_slope*zb[layer]*m[j]**3*cos(zb[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        2*a_slope*b_slope*zb[layer]*m[j]**4*sin(zb[layer]*m[j])*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        2*a_slope*b_slope*zb[layer]*zt[layer]*sin(zb[layer]*m[j])*m[i]**5*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        2*a_slope*b_slope*zb[layer]*zt[layer]*m[j]*cos(zb[layer]*m[j])*m[i]**4*(m[i]**6
                        - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i])
                        -
                        4*a_slope*b_slope*zb[layer]*zt[layer]*m[j]**2*sin(zb[layer]*m[j])*m[i]**3*(m[i]**6
                        - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i])
                        +
                        4*a_slope*b_slope*zb[layer]*zt[layer]*m[j]**3*cos(zb[layer]*m[j])*m[i]**2*(m[i]**6
                        - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i])
                        +
                        2*a_slope*b_slope*zb[layer]*zt[layer]*m[j]**4*sin(zb[layer]*m[j])*m[i]*(m[i]**6
                        - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i])
                        - 2*a_slope*b_slope*zb[layer]*zt[layer]*m[j]**5*cos(zb[layer]*m[j])*(m[i]**6
                        - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i])
                        - a_slope*b_slope*zb[layer]**2*sin(zb[layer]*m[j])*m[i]**5*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        a_slope*b_slope*zb[layer]**2*m[j]*cos(zb[layer]*m[j])*m[i]**4*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        2*a_slope*b_slope*zb[layer]**2*m[j]**2*sin(zb[layer]*m[j])*m[i]**3*(m[i]**6
                        - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i])
                        -
                        2*a_slope*b_slope*zb[layer]**2*m[j]**3*cos(zb[layer]*m[j])*m[i]**2*(m[i]**6
                        - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i])
                        - a_slope*b_slope*zb[layer]**2*m[j]**4*sin(zb[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        a_slope*b_slope*zb[layer]**2*m[j]**5*cos(zb[layer]*m[j])*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) -
                        2*a_slope*b_slope*zt[layer]*sin(zb[layer]*m[j])*m[i]**4*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) -
                        4*a_slope*b_slope*zt[layer]*m[j]*cos(zb[layer]*m[j])*m[i]**3*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        4*a_slope*b_slope*zt[layer]*m[j]**3*cos(zb[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        2*a_slope*b_slope*zt[layer]*m[j]**4*sin(zb[layer]*m[j])*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) -
                        a_slope*b_slope*zt[layer]**2*sin(zb[layer]*m[j])*m[i]**5*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        a_slope*b_slope*zt[layer]**2*m[j]*cos(zb[layer]*m[j])*m[i]**4*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        2*a_slope*b_slope*zt[layer]**2*m[j]**2*sin(zb[layer]*m[j])*m[i]**3*(m[i]**6
                        - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i])
                        -
                        2*a_slope*b_slope*zt[layer]**2*m[j]**3*cos(zb[layer]*m[j])*m[i]**2*(m[i]**6
                        - 3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i])
                        - a_slope*b_slope*zt[layer]**2*m[j]**4*sin(zb[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        a_slope*b_slope*zt[layer]**2*m[j]**5*cos(zb[layer]*m[j])*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        a_slope*bt[layer]*sin(zb[layer]*m[j])*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        2*a_slope*bt[layer]*m[j]*cos(zb[layer]*m[j])*m[i]**3*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        2*a_slope*bt[layer]*m[j]**3*cos(zb[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        a_slope*bt[layer]*m[j]**4*sin(zb[layer]*m[j])*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) -
                        a_slope*zb[layer]*bt[layer]*sin(zb[layer]*m[j])*m[i]**5*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        a_slope*zb[layer]*bt[layer]*m[j]*cos(zb[layer]*m[j])*m[i]**4*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        2*a_slope*zb[layer]*bt[layer]*m[j]**2*sin(zb[layer]*m[j])*m[i]**3*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        2*a_slope*zb[layer]*bt[layer]*m[j]**3*cos(zb[layer]*m[j])*m[i]**2*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) -
                        a_slope*zb[layer]*bt[layer]*m[j]**4*sin(zb[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        a_slope*zb[layer]*bt[layer]*m[j]**5*cos(zb[layer]*m[j])*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        a_slope*zt[layer]*bt[layer]*sin(zb[layer]*m[j])*m[i]**5*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        a_slope*zt[layer]*bt[layer]*m[j]*cos(zb[layer]*m[j])*m[i]**4*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) -
                        2*a_slope*zt[layer]*bt[layer]*m[j]**2*sin(zb[layer]*m[j])*m[i]**3*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        2*a_slope*zt[layer]*bt[layer]*m[j]**3*cos(zb[layer]*m[j])*m[i]**2*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        a_slope*zt[layer]*bt[layer]*m[j]**4*sin(zb[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        a_slope*zt[layer]*bt[layer]*m[j]**5*cos(zb[layer]*m[j])*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        b_slope*at[layer]*sin(zb[layer]*m[j])*m[i]**4*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        2*b_slope*at[layer]*m[j]*cos(zb[layer]*m[j])*m[i]**3*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        2*b_slope*at[layer]*m[j]**3*cos(zb[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        b_slope*at[layer]*m[j]**4*sin(zb[layer]*m[j])*(m[i]**6 - 3*m[j]**2*m[i]**4 +
                        3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) -
                        b_slope*zb[layer]*at[layer]*sin(zb[layer]*m[j])*m[i]**5*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        b_slope*zb[layer]*at[layer]*m[j]*cos(zb[layer]*m[j])*m[i]**4*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        2*b_slope*zb[layer]*at[layer]*m[j]**2*sin(zb[layer]*m[j])*m[i]**3*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        2*b_slope*zb[layer]*at[layer]*m[j]**3*cos(zb[layer]*m[j])*m[i]**2*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) -
                        b_slope*zb[layer]*at[layer]*m[j]**4*sin(zb[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        b_slope*zb[layer]*at[layer]*m[j]**5*cos(zb[layer]*m[j])*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        b_slope*zt[layer]*at[layer]*sin(zb[layer]*m[j])*m[i]**5*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        b_slope*zt[layer]*at[layer]*m[j]*cos(zb[layer]*m[j])*m[i]**4*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) -
                        2*b_slope*zt[layer]*at[layer]*m[j]**2*sin(zb[layer]*m[j])*m[i]**3*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        2*b_slope*zt[layer]*at[layer]*m[j]**3*cos(zb[layer]*m[j])*m[i]**2*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        b_slope*zt[layer]*at[layer]*m[j]**4*sin(zb[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        b_slope*zt[layer]*at[layer]*m[j]**5*cos(zb[layer]*m[j])*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) -
                        bt[layer]*at[layer]*sin(zb[layer]*m[j])*m[i]**5*(m[i]**6 - 3*m[j]**2*m[i]**4
                        + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        bt[layer]*at[layer]*m[j]*cos(zb[layer]*m[j])*m[i]**4*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) +
                        2*bt[layer]*at[layer]*m[j]**2*sin(zb[layer]*m[j])*m[i]**3*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) -
                        2*bt[layer]*at[layer]*m[j]**3*cos(zb[layer]*m[j])*m[i]**2*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i]) -
                        bt[layer]*at[layer]*m[j]**4*sin(zb[layer]*m[j])*m[i]*(m[i]**6 -
                        3*m[j]**2*m[i]**4 + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*cos(zb[layer]*m[i]) +
                        bt[layer]*at[layer]*m[j]**5*cos(zb[layer]*m[j])*(m[i]**6 - 3*m[j]**2*m[i]**4
                        + 3*m[j]**4*m[i]**2 - m[j]**6)**(-1)*sin(zb[layer]*m[i])))

        #A is symmetric
        for i in range(neig - 1):
            for j in range(i + 1, neig):
                A[j, i] = A[i, j]

    elif implementation == 'fortran':
        if MUST_TRY_FORTRAN:
            import geotecha.speccon.ext_integrals as ext_integ
            A = ext_integ.dim1sin_dd_abddf_linear(m, at, ab, bt, bb, zt, zb)
        else:
            try:
                import geotecha.speccon.ext_integrals as ext_integ
                A = ext_integ.dim1sin_dd_abddf_linear(m, at, ab, bt, bb, zt, zb)
            except ImportError:
                A = dim1sin_DD_abDDf_linear(m, at, ab, bt, bb, zt, zb, implementation='vectorized')

    else:#default is 'vectorized' using numpy
        sin = np.sin
        cos = np.cos
        A = np.zeros([neig, neig], float)

        diag =  np.diag_indices(neig)
        triu = np.triu_indices(neig, k = 1)
        tril = (triu[1], triu[0])

        a_slope = (ab - at) / (zb - zt)
        b_slope = (bb - bt) / (zb - zt)

        mi = m[:, np.newaxis]
        A[diag] = np.sum(-mi**4*(a_slope*b_slope*zt**3*sin(mi*zt)**2/6 + a_slope*b_slope*zt**3*cos(mi*zt)**2/6 +
            a_slope*b_slope*zt*sin(mi*zt)**2/(4*mi**2) + a_slope*b_slope*zt*cos(mi*zt)**2/(4*mi**2)
            + a_slope*b_slope*cos(mi*zt)*sin(mi*zt)/(4*mi**3) - a_slope*zt**2*sin(mi*zt)**2*bt/4 -
            a_slope*zt**2*cos(mi*zt)**2*bt/4 - a_slope*cos(mi*zt)**2*bt/(4*mi**2) -
            b_slope*zt**2*sin(mi*zt)**2*at/4 - b_slope*zt**2*cos(mi*zt)**2*at/4 -
            b_slope*cos(mi*zt)**2*at/(4*mi**2) + zt*sin(mi*zt)**2*bt*at/2 + zt*cos(mi*zt)**2*bt*at/2
            - cos(mi*zt)*sin(mi*zt)*bt*at/(2*mi)) + mi**4*(a_slope*b_slope*zb*sin(mi*zb)**2*zt**2/2
            + a_slope*b_slope*zb*cos(mi*zb)**2*zt**2/2 - a_slope*b_slope*zb**2*sin(mi*zb)**2*zt/2 -
            a_slope*b_slope*zb**2*cos(mi*zb)**2*zt/2 + a_slope*b_slope*zb**3*sin(mi*zb)**2/6 +
            a_slope*b_slope*zb**3*cos(mi*zb)**2/6 -
            a_slope*b_slope*cos(mi*zb)*sin(mi*zb)*zt**2/(2*mi) +
            a_slope*b_slope*zb*cos(mi*zb)*sin(mi*zb)*zt/mi -
            a_slope*b_slope*zb**2*cos(mi*zb)*sin(mi*zb)/(2*mi) +
            a_slope*b_slope*cos(mi*zb)**2*zt/(2*mi**2) + a_slope*b_slope*zb*sin(mi*zb)**2/(4*mi**2)
            - a_slope*b_slope*zb*cos(mi*zb)**2/(4*mi**2) +
            a_slope*b_slope*cos(mi*zb)*sin(mi*zb)/(4*mi**3) - a_slope*zb*sin(mi*zb)**2*zt*bt/2 -
            a_slope*zb*cos(mi*zb)**2*zt*bt/2 + a_slope*zb**2*sin(mi*zb)**2*bt/4 +
            a_slope*zb**2*cos(mi*zb)**2*bt/4 + a_slope*cos(mi*zb)*sin(mi*zb)*zt*bt/(2*mi) -
            a_slope*zb*cos(mi*zb)*sin(mi*zb)*bt/(2*mi) - a_slope*cos(mi*zb)**2*bt/(4*mi**2) -
            b_slope*zb*sin(mi*zb)**2*zt*at/2 - b_slope*zb*cos(mi*zb)**2*zt*at/2 +
            b_slope*zb**2*sin(mi*zb)**2*at/4 + b_slope*zb**2*cos(mi*zb)**2*at/4 +
            b_slope*cos(mi*zb)*sin(mi*zb)*zt*at/(2*mi) - b_slope*zb*cos(mi*zb)*sin(mi*zb)*at/(2*mi)
            - b_slope*cos(mi*zb)**2*at/(4*mi**2) + zb*sin(mi*zb)**2*bt*at/2 +
            zb*cos(mi*zb)**2*bt*at/2 - cos(mi*zb)*sin(mi*zb)*bt*at/(2*mi)), axis=1)

        mi = m[triu[0]][:, np.newaxis]
        mj = m[triu[1]][:, np.newaxis]
        A[triu] = np.sum(-mi**2*mj**2*(2*a_slope*b_slope*mi**3*cos(mi*zt)*sin(mj*zt)/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - 6*a_slope*b_slope*mi**2*mj*cos(mj*zt)*sin(mi*zt)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 6*a_slope*b_slope*mi*mj**2*cos(mi*zt)*sin(mj*zt)/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            2*a_slope*b_slope*mj**3*cos(mj*zt)*sin(mi*zt)/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + a_slope*mi**4*sin(mi*zt)*sin(mj*zt)*bt/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + 2*a_slope*mi**3*mj*cos(mi*zt)*cos(mj*zt)*bt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*a_slope*mi*mj**3*cos(mi*zt)*cos(mj*zt)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) - a_slope*mj**4*sin(mi*zt)*sin(mj*zt)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) + b_slope*mi**4*sin(mi*zt)*sin(mj*zt)*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            2*b_slope*mi**3*mj*cos(mi*zt)*cos(mj*zt)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - 2*b_slope*mi*mj**3*cos(mi*zt)*cos(mj*zt)*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - b_slope*mj**4*sin(mi*zt)*sin(mj*zt)*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - mi**5*cos(mi*zt)*sin(mj*zt)*bt*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + mi**4*mj*cos(mj*zt)*sin(mi*zt)*bt*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 2*mi**3*mj**2*cos(mi*zt)*sin(mj*zt)*bt*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            2*mi**2*mj**3*cos(mj*zt)*sin(mi*zt)*bt*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - mi*mj**4*cos(mi*zt)*sin(mj*zt)*bt*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + mj**5*cos(mj*zt)*sin(mi*zt)*bt*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6)) + mi**2*mj**2*(-a_slope*b_slope*mi**5*cos(mi*zb)*sin(mj*zb)*zt**2/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            2*a_slope*b_slope*mi**5*zb*cos(mi*zb)*sin(mj*zb)*zt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - a_slope*b_slope*mi**5*zb**2*cos(mi*zb)*sin(mj*zb)/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            a_slope*b_slope*mi**4*mj*cos(mj*zb)*sin(mi*zb)*zt**2/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*a_slope*b_slope*mi**4*mj*zb*cos(mj*zb)*sin(mi*zb)*zt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            a_slope*b_slope*mi**4*mj*zb**2*cos(mj*zb)*sin(mi*zb)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*a_slope*b_slope*mi**4*sin(mi*zb)*sin(mj*zb)*zt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            2*a_slope*b_slope*mi**4*zb*sin(mi*zb)*sin(mj*zb)/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4
            - mj**6) + 2*a_slope*b_slope*mi**3*mj**2*cos(mi*zb)*sin(mj*zb)*zt**2/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            4*a_slope*b_slope*mi**3*mj**2*zb*cos(mi*zb)*sin(mj*zb)*zt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) +
            2*a_slope*b_slope*mi**3*mj**2*zb**2*cos(mi*zb)*sin(mj*zb)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 4*a_slope*b_slope*mi**3*mj*cos(mi*zb)*cos(mj*zb)*zt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            4*a_slope*b_slope*mi**3*mj*zb*cos(mi*zb)*cos(mj*zb)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 2*a_slope*b_slope*mi**3*cos(mi*zb)*sin(mj*zb)/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            2*a_slope*b_slope*mi**2*mj**3*cos(mj*zb)*sin(mi*zb)*zt**2/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) +
            4*a_slope*b_slope*mi**2*mj**3*zb*cos(mj*zb)*sin(mi*zb)*zt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) -
            2*a_slope*b_slope*mi**2*mj**3*zb**2*cos(mj*zb)*sin(mi*zb)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 6*a_slope*b_slope*mi**2*mj*cos(mj*zb)*sin(mi*zb)/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            a_slope*b_slope*mi*mj**4*cos(mi*zb)*sin(mj*zb)*zt**2/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 2*a_slope*b_slope*mi*mj**4*zb*cos(mi*zb)*sin(mj*zb)*zt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            a_slope*b_slope*mi*mj**4*zb**2*cos(mi*zb)*sin(mj*zb)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 4*a_slope*b_slope*mi*mj**3*cos(mi*zb)*cos(mj*zb)*zt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            4*a_slope*b_slope*mi*mj**3*zb*cos(mi*zb)*cos(mj*zb)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 6*a_slope*b_slope*mi*mj**2*cos(mi*zb)*sin(mj*zb)/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            a_slope*b_slope*mj**5*cos(mj*zb)*sin(mi*zb)*zt**2/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4
            - mj**6) - 2*a_slope*b_slope*mj**5*zb*cos(mj*zb)*sin(mi*zb)*zt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + a_slope*b_slope*mj**5*zb**2*cos(mj*zb)*sin(mi*zb)/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            2*a_slope*b_slope*mj**4*sin(mi*zb)*sin(mj*zb)*zt/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4
            - mj**6) - 2*a_slope*b_slope*mj**4*zb*sin(mi*zb)*sin(mj*zb)/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*a_slope*b_slope*mj**3*cos(mj*zb)*sin(mi*zb)/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            a_slope*mi**5*cos(mi*zb)*sin(mj*zb)*zt*bt/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - a_slope*mi**5*zb*cos(mi*zb)*sin(mj*zb)*bt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - a_slope*mi**4*mj*cos(mj*zb)*sin(mi*zb)*zt*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            a_slope*mi**4*mj*zb*cos(mj*zb)*sin(mi*zb)*bt/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + a_slope*mi**4*sin(mi*zb)*sin(mj*zb)*bt/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - 2*a_slope*mi**3*mj**2*cos(mi*zb)*sin(mj*zb)*zt*bt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 2*a_slope*mi**3*mj**2*zb*cos(mi*zb)*sin(mj*zb)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            2*a_slope*mi**3*mj*cos(mi*zb)*cos(mj*zb)*bt/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + 2*a_slope*mi**2*mj**3*cos(mj*zb)*sin(mi*zb)*zt*bt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*a_slope*mi**2*mj**3*zb*cos(mj*zb)*sin(mi*zb)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            a_slope*mi*mj**4*cos(mi*zb)*sin(mj*zb)*zt*bt/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - a_slope*mi*mj**4*zb*cos(mi*zb)*sin(mj*zb)*bt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*a_slope*mi*mj**3*cos(mi*zb)*cos(mj*zb)*bt/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            a_slope*mj**5*cos(mj*zb)*sin(mi*zb)*zt*bt/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + a_slope*mj**5*zb*cos(mj*zb)*sin(mi*zb)*bt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - a_slope*mj**4*sin(mi*zb)*sin(mj*zb)*bt/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + b_slope*mi**5*cos(mi*zb)*sin(mj*zb)*zt*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            b_slope*mi**5*zb*cos(mi*zb)*sin(mj*zb)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - b_slope*mi**4*mj*cos(mj*zb)*sin(mi*zb)*zt*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + b_slope*mi**4*mj*zb*cos(mj*zb)*sin(mi*zb)*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) + b_slope*mi**4*sin(mi*zb)*sin(mj*zb)*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            2*b_slope*mi**3*mj**2*cos(mi*zb)*sin(mj*zb)*zt*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4
            - mj**6) + 2*b_slope*mi**3*mj**2*zb*cos(mi*zb)*sin(mj*zb)*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + 2*b_slope*mi**3*mj*cos(mi*zb)*cos(mj*zb)*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            2*b_slope*mi**2*mj**3*cos(mj*zb)*sin(mi*zb)*zt*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4
            - mj**6) - 2*b_slope*mi**2*mj**3*zb*cos(mj*zb)*sin(mi*zb)*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) + b_slope*mi*mj**4*cos(mi*zb)*sin(mj*zb)*zt*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) -
            b_slope*mi*mj**4*zb*cos(mi*zb)*sin(mj*zb)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - 2*b_slope*mi*mj**3*cos(mi*zb)*cos(mj*zb)*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - b_slope*mj**5*cos(mj*zb)*sin(mi*zb)*zt*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) +
            b_slope*mj**5*zb*cos(mj*zb)*sin(mi*zb)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - b_slope*mj**4*sin(mi*zb)*sin(mj*zb)*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) - mi**5*cos(mi*zb)*sin(mj*zb)*bt*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + mi**4*mj*cos(mj*zb)*sin(mi*zb)*bt*at/(mi**6 - 3*mi**4*mj**2 + 3*mi**2*mj**4 -
            mj**6) + 2*mi**3*mj**2*cos(mi*zb)*sin(mj*zb)*bt*at/(mi**6 - 3*mi**4*mj**2 +
            3*mi**2*mj**4 - mj**6) - 2*mi**2*mj**3*cos(mj*zb)*sin(mi*zb)*bt*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) - mi*mj**4*cos(mi*zb)*sin(mj*zb)*bt*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6) + mj**5*cos(mj*zb)*sin(mi*zb)*bt*at/(mi**6 -
            3*mi**4*mj**2 + 3*mi**2*mj**4 - mj**6)), axis=1)
        #A is symmetric
        A[tril] = A[triu]

    return A

if __name__ == '__main__':
    import nose
    nose.runmodule(argv=['nose', '--verbosity=3', '--with-doctest'])
#    nose.runmodule(argv=['nose', '--verbosity=3'])