numerical approximation
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1: Annie A. M. Cuyt
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►Her main research interest is in the area of numerical approximation theory and its applications to a diversity of problems in scientific computing.
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2: 3.4 Differentiation
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►With the choice (which is crucial when is large because of numerical cancellation) the integrand equals at the dominant points , and in combination with the factor in front of the integral sign this gives a rough approximation to .
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Laplacian
… ►Biharmonic Operator
…3: 18.40 Methods of Computation
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►The question is then: how is this possible given only , rather than itself? often converges to smooth results for off the real axis for at a distance greater than the pole spacing of the , this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to and evaluating these on the real axis in regions of higher pole density that those of the approximating function.
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4: 2.11 Remainder Terms; Stokes Phenomenon
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§2.11(i) Numerical Use of Asymptotic Expansions
… ►The rest of this section is devoted to general methods for increasing this accuracy. … ►§2.11(vi) Direct Numerical Transformations
… ►For example, extrapolated values may converge to an accurate value on one side of a Stokes line (§2.11(iv)), and converge to a quite inaccurate value on the other.5: Bibliography W
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Rational Chebyshev approximation.
Numer. Math. 10 (4), pp. 289–306.
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Error bounds for asymptotic approximations of special functions.
Ann. Numer. Math. 2 (1-4), pp. 181–197.
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6: 28.8 Asymptotic Expansions for Large
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►Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1).
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►Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1).
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7: Bibliography G
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Orthogonal Polynomials: Computation and Approximation.
Numerical Mathematics and Scientific Computation, Oxford University Press, New York.
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8: Mathematical Introduction
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►All of the special function chapters contain sections that describe available methods for computing the main functions in the chapter, and most also provide references to numerical tables of, and approximations for, these functions.
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►In referring to the numerical tables and approximations we use notation typified by , 8D or 8S.
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9: 35.10 Methods of Computation
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►For small values of the zonal polynomial expansion given by (35.8.1) can be summed numerically.
For large the asymptotic approximations referred to in §35.7(iv) are available.
►Other methods include numerical quadrature applied to double and multiple integral representations.
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►Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1).
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10: Mourad E. H. Ismail
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►Ismail has published numerous papers on special functions, orthogonal polynomials, approximation theory, combinatorics, asymptotics, and related topics.
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