laplace

Class summary

Talbot(f[, n, shift, vectorized])

Numerical inverse Laplace transform, Talbot method

Function summary

cot(phi)	Reciprocal of tan function.
csc(phi)	Reciprocal of sin function.

Module listing

Numerical inverse Laplace transform

class geotecha.mathematics.laplace.Talbot(f, n=24, shift=0.0, vectorized=True)

Bases: object

Numerical inverse Laplace transform, Talbot method

Parameters:

f : function or method

Function to perform inverse Laplace transform on. Function should be vectorised (but doesn’t have to be).

n : even int, optional

Number of integration points. Nf n is even it will be rounded up to nearest even number. Default n=24.

shift : float, optional

Shift contour to the right in case there is a pole on the positive real axis. Default shift=0.0.

vectorized : True/False, optional

If True then f accepts vector inputs and numpy broadcasting will be used. Otherwise function evaluation will occur in loops. Default vectorized=True.

See also

geotecha.mathematics.mp_laplace.Talbot

higher precision numerical inverse Laplace transform.

Notes

Talbot suggested that the Bromwich line be deformed into a contour that begins and ends in the left half plane, i.e., z infinity at both ends. Due to the exponential factor the integrand decays rapidly on such a contour. In such situations the trapezoidal rule converge extraordinarily rapidly.

Shift contour to the right in case there is a pole on the positive real axis : Note the contour will not be optimal since it was originally devoloped for function with singularities on the negative real axis For example take F(s) = 1/(s-1), it has a pole at s = 1, the contour needs to be shifted with one unit, i.e shift = 1.

References

Code adapted (vectorised, args added) from [1] and [2] (including much of the text taken verbatim). Algorithm from [3]:

[1]	(1, 2) Created by Fernando Damian Nieuwveldt, 25 October 2009, fdnieuwveldt@gmail.com, http://code.activestate.com/recipes/576934-numerical-inversion-of-the-laplace-transform-using/

[2]	(1, 2) Adapted to mpmath and classes by Dieter Kadelka, 27 October 2009, Dieter.Kadelka@kit.edu, http://code.activestate.com/recipes/578799-numerical-inversion-of-the-laplace-transform-with-/

[3]	(1, 2) L.N.Trefethen, J.A.C.Weideman, and T.Schmelzer. Talbot quadratures and rational approximations. BIT. Numerical Mathematics, 46(3):653 670, 2006.

Methods

__call__(t[, args])

Numerical inverse laplace transform of F at various time t.

geotecha.mathematics.laplace.cot(phi)

Reciprocal of tan function.

Parameters:

phi : float of 1d array of float

Point to evaluate at.

Returns:

out : float or 1d array of float

cot(phi)

geotecha.mathematics.laplace.csc(phi)

Reciprocal of sin function.

Parameters:

phi : float of 1d array of float

Point to evaluate at.

Returns:

out : float or 1d array of float

csc(phi)