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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 =
Verify the correctness of this value using quadgk.

>> FresnelCObj = @(t) cos(pi*t.^2/2);
>> quadgk(FresnelCObj,0,1.38,'abstol',1e-15')
ans =
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 =
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 (4)

John D'Errico

John D'Errico (view profile)

New version submitted - thanks to 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



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MATLAB 7.10 (R2010a)

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