Trefethen, Lloyd N. and Weideman, J. A. C. and Schmelzer, Thomas (2005) Talbot quadratures and rational approximations. Technical Report. Unspecified. (Submitted)

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Abstract
Many computational problems can be solved with the aid of contour integrals containing in the the integrand: examples include inverse Laplace transforms, special functions, functions of matrices and operators, parabolic PDEs, and reactiondiffusion 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 , where is the number of sample points or function evaluations. On the other hand, convergence at a rate apparently about twice as fast, , can be achieved by using a different approach: best supremumnorm rational approximants to for , following Cody, Meinardus and Varga in 1969. (All these rates are doubled in the case of selfadjoint 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 CodyMeinardusVarga approximants by the method of CarathèodoryFejèr approximation.
Item Type:  Technical Report (Technical Report) 

Subjects:  A  C > Approximations and expansions H  N > Numerical analysis 
Research Groups:  Numerical Analysis Group 
ID Code:  1133 
Deposited By:  Lotti Ekert 
Deposited On:  12 May 2011 07:36 
Last Modified:  29 May 2015 18:49 
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