Numerical Inverse Laplace Transform
04 Jan 2013
Numerical approximation of the inverse Laplace transform for use with any function defined in "s".
|talbot_inversion(f_s, t, M)
function ilt = talbot_inversion(f_s, t, M)
% ilt = talbot_inversion(f_s, t, [M])
% Returns an approximation to the inverse Laplace transform of function
% handle f_s evaluated at each value in t (1xn) using Talbot's method as
% summarized in the source below.
% This implementation is very coarse; use talbot_inversion_sym for better
% precision. Further, please see example_inversions.m for discussion.
% f_s: Handle to function of s
% t: Times at which to evaluate the inverse Laplace transformation of f_s
% M: Optional, number of terms to sum for each t (64 is a good guess);
% highly oscillatory functions require higher M, but this can grow
% unstable; see test_talbot.m for an example of stability.
% Abate, Joseph, and Ward Whitt. "A Unified Framework for Numerically
% Inverting Laplace Transforms." INFORMS Journal of Computing, vol. 18.4
% (2006): 408-421. Print.
% The paper is also online: http://www.columbia.edu/~ww2040/allpapers.html.
% Tucker McClure
% Copyright 2012, The MathWorks, Inc.
% Make sure t is n-by-1.
if size(t, 1) == 1
t = t';
elseif size(t, 2) > 1
error('Input times, t, must be a vector.');
% Set M to 64 if user didn't specify an M.
if nargin < 3
M = 64;
% Vectorized Talbot's algorithm
k = 1:(M-1); % Iteration index
% Calculate delta for every index.
delta = zeros(1, M);
delta(1) = 2*M/5;
delta(2:end) = 2*pi/5 * k .* (cot(pi/M*k)+1i);
% Calculate gamma for every index.
gamma = zeros(1, M);
gamma(1) = 0.5*exp(delta(1));
gamma(2:end) = (1 + 1i*pi/M*k.*(1+cot(pi/M*k).^2)-1i*cot(pi/M*k))...
% Make a mesh so we can do this entire calculation across all k for all
% given times without a single loop (it's faster this way).
[delta_mesh, t_mesh] = meshgrid(delta, t);
gamma_mesh = meshgrid(gamma, t);
% Finally, calculate the inverse Laplace transform for each given time.
ilt = 0.4./t .* sum(real( gamma_mesh ...
.* arrayfun(f_s, delta_mesh./t_mesh)), 2);