numerical approximation

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21: 29.20 Methods of Computation

… ►Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). …The approximations converge geometrically (§3.8(i)) to the eigenvalues and coefficients of Lamé functions as $n\to \mathrm{\infty }$. The numerical computations described in Jansen (1977) are based in part upon this method. ►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. …

22: 18.38 Mathematical Applications

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§18.38(i) Classical OP’s: Numerical Analysis

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23: 2.7 Differential Equations

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§2.7(iii) Liouville–Green (WKBJ) Approximation

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Liouville–Green Approximation Theorem

… ►By approximating … ►

§2.7(iv) Numerically Satisfactory Solutions

… ►This is characteristic of numerically satisfactory pairs. …

24: Bibliography B

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Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, Vol. 50, Birkhäuser Verlag, Basel.

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External Links:

ISBN 3-7643-1100-2, MathReview (A. Magnus), ZentralBlatt

Referenced by:

§3.11(iv).

Encodings:

BibTeX

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25: Bibliography L

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Some Differential Equations Satisfied by Hypergeometric Functions. In Approximation and Computation (West Lafayette, IN, 1993), Internat. Ser. Numer. Math., Vol. 119, pp. 371–381.

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External Links:

MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

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26: Roderick S. C. Wong

… ►Wong has published numerous papers in international journals and is serving on the Editorial Boards of several journals. He is the author of the book Asymptotic Approximations of Integrals, published by Academic Press in 1989 and reprinted by SIAM in its Classics in Applied Mathematics Series in 2001, and of Lecture Notes on Applied Analysis, published by World Scientific in 2010. … ►

2 Asymptotic Approximations

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27: 36.15 Methods of Computation

… ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi }$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …

28: Preface

… ►The authors will review the relevant published literature and produce approximately twice the number of formulas that were contained in the original Handbook. The DLMF will make full use of advanced communications and computational resources to present downloadable math data, manipulable graphs, tables of numerical values, and math-aware search. …

29: Bibliography O

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The Generalized Exponential Integral. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 497–510.

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External Links:

MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§2.11(iii), §8.19(viii).

Encodings:

BibTeX

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30: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

… ►Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation: … ►Because the method requires only one function evaluation per iteration, its numerical efficiency is ultimately higher than that of Newton’s method. … ►Initial approximations to the zeros can often be found from asymptotic or other approximations to $f(z)$, or by application of the phase principle or Rouché’s theorem; see §1.10(iv). … ►Corresponding numerical factors in this example for other zeros and other values of $j$ are obtained in Gautschi (1984, §4). …