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FresnelS and FresnelC

version 1.1 (2.34 MB) by

Efficient and accurate computation of the Fresnel sine and cosine integrals

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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

>> fresnelC(1.38,0)
ans =
0.562975925772444

Verify the correctness of this value using quadgk.

>> FresnelCObj = @(t) cos(pi*t.^2/2);
>> quadgk(FresnelCObj,0,1.38,'abstol',1e-15')
ans =
0.562975925772444

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))
ans =
0.000193604455833333

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);
>> tic
>> FCpred = fresnelC(T);
>> toc
Elapsed time is 0.226884 seconds.

Comments and Ratings (5)

gzrollins

Very useful functions. Very well documented. First class work! Thank you!

John D'Errico

John D'Errico (view profile)

New version submitted - thanks to Felipe.

Felipe

Felipe (view profile)

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.

John Kot

Updates

1.1

Acknowledge two other files

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