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Numerical Evaluation of Special Functions
 In W. Gautschi (Ed.), AMS Proceedings of Symposia in Applied Mathematics 48
, 1994
"... . This document is an excerpt from the current hypertext version of an article that appeared in Walter Gautschi (ed.), Mathematics of Computation 19431993: A HalfCentury of Computational Mathematics, Proceedings of Symposia in Applied Mathematics 48, American Mathematical Society, Providence, ..."
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. This document is an excerpt from the current hypertext version of an article that appeared in Walter Gautschi (ed.), Mathematics of Computation 19431993: A HalfCentury of Computational Mathematics, Proceedings of Symposia in Applied Mathematics 48, American Mathematical Society, Providence, RI 02940, 1994. The symposium was held at the University of British Columbia August 913, 1993, in honor of the fiftieth anniversary of the journal Mathematics of Computation. The original abstract follows. Higher transcendental functions continue to play varied and important roles in investigations by engineers, mathematicians, scientists and statisticians. The purpose of this paper is to assist in locating useful approximations and software for the numerical generation of these functions, and to offer some suggestions for future developments in this field. 5.9. Mathieu, Lam'e, and Spheroidal Wave Functions. 5.9.1. Characteristic Values of Mathieu's Equation. Software Packages:...
The Asymptotic Chebyshev Coefficients ior Functions with Logarithmic Enclpoint Singularities: Mappings ad Singular Basis Functions
"... When a function is singular at the ends of its expansion interval, its Chebyshev coefficients a, converge very poorly. We analyze three numerical strategies for coping with such singularities of the form (1 + x) ~ log(1 f x), and in the process make some modest additions to the theory of Chebyshev e ..."
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When a function is singular at the ends of its expansion interval, its Chebyshev coefficients a, converge very poorly. We analyze three numerical strategies for coping with such singularities of the form (1 + x) ~ log(1 f x), and in the process make some modest additions to the theory of Chebyshev expansions. The first two numerical methods are the convergenceimproving changes of coordinate x = sin[ ( In/Z&] and x = tanh[ly/(l y”)‘/2]. We derive the asymptotic Chebyshev coefficients in the limit n + 00 for both mappings and for the original, untransformed Chebyshev series. For the original function, the asymptotic approximation for general R is augmented by the exact Chebyshev coefficients for integer k. Numerical tests show that the sine mapping is excellent for k 2 1, increasing the rate of convergence to b, = 0(1/n 4k+1). Although the tanh transfomation is guaranteed to be better for sufficiently large n, we offer both theoretical and numerical evidence to explain why the sine mapping is usually better in practice: “sufficiently large n ” is usually huge. Instead of mapping, one may use a third strategy: supplementing the Chebyshev polynomials with singular basis functions. Simple experiments how that this approach is also successful. 1. INTHBDIJCTION Solutions to differential equations often are singular at comers and endpoints. Several examples are discussed by Lund and Riley [13] and Lee, Schultz, and Boyd [El. Unfortunately, these boundary branch points seri