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Computing Fresnel integrals via modified trapezium rules

Alazah, M., Chandler-Wilde, S. N. and La Porte, S. (2014) Computing Fresnel integrals via modified trapezium rules. Numerische Mathematik, 128 (4). pp. 635-661. ISSN 0029-599X

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To link to this item DOI: 10.1007/s00211-014-0627-z

Abstract/Summary

In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis. Our starting point is a method for computation of the error function of complex argument due to Matta and Reichel (J Math Phys 34:298–307, 1956) and Hunter and Regan (Math Comp 26:539–541, 1972). We construct approximations which we prove are exponentially convergent as a function of N , the number of quadrature points, obtaining explicit error bounds which show that accuracies of 10−15 uniformly on the real line are achieved with N=12 , this confirmed by computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement.

Item Type:Article
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:36434
Publisher:Springer

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