function ilt = talbot_inversion_sym(f_s, t, M, P)
% ilt = talbot_inversion_sym(f_s, t, [M], [P])
%
% 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 is a symbolic implementation capable
% of much greater accuracy than talbot_inversion. For this reason, it takes
% an additional argument, P, the number of significant digits required by
% the calculation. In general, P should be about 0.6*M.
%
% 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 example_inversions.m for an example of stability [64]
% P: Optional precision of calculation in significant digits [32]
%
% Requires the Symbolic Toolbox(TM).
%
% 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.');
end
% Set M to 64 if user didn't specify an M.
if nargin < 3, M = 64; end
% Set P to the greater of the default from digits() or 0.6M.
if nargin < 4, P = max(floor(0.6*M), digits()); end
% Vectorized Talbot's algorithm
M = sym(M);
k = 1:(M-1); % Iteration index
% Calculate delta for every index.
delta = vpa([2*M/5, 2*pi/5 * k .* (cot(pi/M*k)+1i)], P);
% Calculate gamma for every index.
gamma = vpa([0.5*exp(delta(1)), ...
(1 + 1i*pi/M*k .* (1+cot(pi/M*k).^2) - 1i*cot(pi/M*k)) ...
.* exp(delta(2:end))], P);
% 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.
f_s_evals = arrayfun(f_s, delta_mesh./t_mesh, 'UniformOutput', false);
f_s_evals = vpa(reshape([f_s_evals{:}], size(delta_mesh)), P);
ilt = vpa(0.4./t .* sum(real(gamma_mesh .* f_s_evals), 2), P);
end