Concerns about symbolic expressions and quadgk accuracy (Matlab and Mathematica differences)
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From: Josh on 13 Aug 2010 16:07 Hi everyone, I've been comparing numerical integration results between Matlab and Mathematica (quadgk and NIntegrate options). I have given both the same symbolic expression that has variables t and tau and is highly oscillatory. So far only Mathematica has been able to produce the correct result. However Matlab's result is just noticeably there but is much noiser. Unfortunately I can't post plots of the two results. The integration is performed by picking a t value and evaluating "FullFunction" using feval(FullFunction, t=t_numer(j), tau) where "tau" is kept symbolic. This result is re-cast as an anonymous function in tau as "FullFunction_tStep = @(tau) feval(FullFunction,t_numer(j),tau)". Then this result, "FullFunction_tStep" is passed to quadgk using quadgk(FullFunction_tStep,a,b). The problem originates from one of three possible sources: (1) the partial numerical and symbolic feval step, (2) re-cast the result of #1 into anonymous function in tau and/or (3) the quadgk() function. Is there another way to handle the feval and re-casting steps to carry out the integration? Thank you very much in advance. MATLAB CODE: a = 0; b = Inf; N2 = 4096 t_numer = linspace(-pi,pi,N2); AbsTols = 1e-4; %1e-6; RelTols = 1e-4; %1e-12; MaxIntervals = 60000; % long function FullFunction = @(t,tau) (i.*((pi./(tau/2)).^(3/2)).*(i*((1/(p))^(3/4))*(((E0*c./(w0*tau)).*(cos(w0*t)......... %pick t-step, eval t as t_numer, leave tau symbolic, then integrate over full tau for j=1:length(t_numer) FullFunction_tStep = @(tau) feval(FullFunction,t_numer(j),tau); F =quadgk(FullFunction_tStep,a,b,'RelTol',RelTols,... 'AbsTol',AbsTols,'MaxIntervalCount',MaxIntervals); result(j) = F + conj(F); fprintf('Completed t step # %u out of %u\n',j,N2); end |