I noticed the many codes on the FEX to compute the Fresnel integrals for real arguments, and it left me wondering how I might try solving this problem in MATLAB for both high accuracy and high efficiency.
The approach I took yields a maximum error of roughly 1e-14 as far as I could get reasonable values to compare it to. (The screenshot shows the predicted error for a sampling of points.)
I've supplied functions for both the Fresnel sine and cosine integrals, as well as a .pdf file that explains the approach I took.
Evaluate the Fresnel cosine integral C(x) at x = 1.38
Verify the correctness of this value using quadgk.
>> FresnelCObj = @(t) cos(pi*t.^2/2);
Now, how fast is fresnelC? Using Steve Eddins timeit code to yield an accurate estimate of the time required, we see that it is reasonably fast for scalar input.
>> timeit(@() fresnelC(1.38))
More importantly, these functions are properly vectorized. So 1 million evaluations are easy to do, and are much faster than 1 million times the time taken for one evaluation.
>> T = rand(1000000,1);
>> FCpred = fresnelC(T);
Elapsed time is 0.226884 seconds.
Very useful functions. Very well documented. First class work! Thank you!
New version submitted - thanks to Felipe.
Hi John. Thanks for citing ("acknowledging") related submissions. There are two new ones, that came after yours: 33577 and 34134. You might want to cite these, too. That way folks will find your submission in all cases. I'm trying to kill duplicates. I'm assuming yours is superior -- both in accuracy and speed, not to mention readability.
Acknowledge two other files
Inspired by: Jim McElwaine, Fresnel Sine and Cosine Integrals, Computation of Special Functions, Fresnel integrals, Fresnel Cosine and Sine Integral Function, Fresnel integral, Fresnel integrals, Cornu spiral_Fresnel Integral, Complex Erf (Error Function), Fresnel Integrals, Fresnel Integral
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