NDLI: Talbot quadratures and rational approximations

Content Provider	Springer Nature Link
Author	Trefethen, L. N. Weideman, J. A. C. Schmelzer, T.
Copyright Year	2006
Abstract	Many computational problems can be solved with the aid of contour integrals containing e z in the integrand: examples include inverse Laplace transforms, special functions, functions of matrices and operators, parabolic PDEs, and reaction-diffusion equations. One approach to the numerical quadrature of such integrals is to apply the trapezoid rule on a Hankel contour defined by a suitable change of variables. Optimal parameters for three classes of such contours have recently been derived: (a) parabolas, (b) hyperbolas, and (c) cotangent contours, following Talbot in 1979. The convergence rates for these optimized quadrature formulas are very fast: roughly O(3-N ), where N is the number of sample points or function evaluations. On the other hand, convergence at a rate apparently about twice as fast, O(9.28903-N ), can be achieved by using a different approach: best supremum-norm rational approximants to e z for z∈(–∞,0], following Cody, Meinardus and Varga in 1969. (All these rates are doubled in the case of self-adjoint operators or real integrands.) It is shown that the quadrature formulas can be interpreted as rational approximations and the rational approximations as quadrature formulas, and the strengths and weaknesses of the different approaches are discussed in the light of these connections. A MATLAB function is provided for computing Cody–Meinardus–Varga approximants by the method of Carathéodory–Fejér approximation.
Starting Page	653
Ending Page	670
Page Count	18
File Format	PDF
ISSN	00063835
Journal	BIT Numerical Mathematics
Volume Number	46
Issue Number	3
e-ISSN	15729125
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2006-08-17
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	rational approximation Carathéodory–Fejér approximation quadrature Hankel contour inverse Laplace transform special functions trapezoid rule Talbot contour Mathematics Numeric Computing Computational Mathematics and Numerical Analysis
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computer Networks and Communications Computational Mathematics Software

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