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FresnelS and FresnelC

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19 Sep 2010 (Updated )

Efficient and accurate computation of the Fresnel sine and cosine integrals

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Description

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.

Acknowledgements

Jim Mc Elwaine, 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, and Fresnel Integral inspired this file.

MATLAB release MATLAB 7.10 (R2010a)
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Comments and Ratings (3)
02 May 2012 John D'Errico

New version submitted - thanks to Felipe.

02 May 2012 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.

12 Oct 2010 John Kot  
Updates
03 May 2012

Acknowledge two other files

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