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31: 9.19 Approximations

§9.19 Approximations

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§9.19(i) Approximations in Terms of Elementary Functions

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§9.19(ii) Expansions in Chebyshev Series

… ►The constants $a$ and $b$ are chosen numerically, with a view to equalizing the effort required for summing the series. … ►

§9.19(iii) Approximations in the Complex Plane

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32: Bibliography C

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Numerical computation of real or complex elliptic integrals. Numer. Algorithms 10 (1-2), pp. 13–26.

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External Links:

ISSN 1017-1398, MathReview, ZentralBlatt

Referenced by:

§19.36(i), §19.36(ii), §19.36(i), §19.36(i), §19.37(iv).

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Introduction to Approximation Theory. 2nd edition, Chelsea Publishing Co., New York.

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Notes:

Reprinted by AMS Chelsea Publishing, Providence, RI, 1998

External Links:

ISBN 0-8218-1374-9, MathReview, ZentralBlatt

Referenced by:

§18.38(i).

Encodings:

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A method for numerical calculation of Fourier integrals. Numer. Math. 8 (5), pp. 422–436.

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External Links:

ISSN 0029-599X, Document, MathReview (M. P. S. Madan), ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

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A method for numerical integration on an automatic copmputer. Numer. Math. 2 (4), pp. 197–205.

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External Links:

ISSN 0029-599X, Document, MathReview (A. H. Stroud), ZentralBlatt

Referenced by:

§3.5(iv).

Encodings:

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Chebyshev approximations for the Fresnel integrals. Math. Comp. 22 (102), pp. 450–453.

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Notes:

with Microfiche Supplement

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

3rd item.

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33: 15.19 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods. …However, since the growth near the singularities of the differential equation is algebraic rather than exponential, the resulting instabilities in the numerical integration might be tolerable in some cases. … ►Gauss quadrature approximations are discussed in Gautschi (2002b). … ►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the numerator parameters is a positive integer. …

34: Bibliography E

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The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.

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External Links:

ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§10.74(vii).

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Further results on McMahon’s asymptotic approximations. J. Phys. A 33 (36), pp. 6333–6341.

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External Links:

ISSN 0305-4470, Document, MathReview (R. G. Aĭrapetyan), ZentralBlatt

Referenced by:

§10.21(vi).

Encodings:

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The zeros of the complementary error function. Numer. Algorithms 49 (1-4), pp. 153–157.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§7.13(ii).

Encodings:

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35: 13.29 Methods of Computation

… ►For large values of the parameters $a$ and $b$ the approximations in §13.8 are available. … ►A comprehensive and powerful approach is to integrate the differential equations (13.2.1) and (13.14.1) by direct numerical methods. …

36: 30.16 Methods of Computation

… ►Approximations to eigenvalues can be improved by using the continued-fraction equations from §30.3(iii) and §30.8; see Bouwkamp (1947) and Meixner and Schäfke (1954, §3.93). … ►If $|{\gamma }^2|$ is large, then we can use the asymptotic expansions referred to in §30.9 to approximate ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$. ►If ${\lambda }_n^m\left({\gamma }^2\right)$ is known, then we can compute ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions $w(0)=1$, $w^{\prime }(0)=0$ if $n-m$ is even, or $w(0)=0$, $w^{\prime }(0)=1$ if $n-m$ is odd. …

37: 10.21 Zeros

… ►For the first zeros rounded numerical values of the coefficients are given by … ►The approximations that follow in §10.21(viii) do not suffer from this drawback. ►

§10.21(viii) Uniform Asymptotic Approximations for Large Order

… ►The latter reference includes numerical tables of the first few coefficients in the uniform asymptotic expansions. … ►This information includes asymptotic approximations analogous to those given in §§10.21(vi), 10.21(vii), and 10.21(x). …

38: Bibliography D

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Methods of Numerical Integration. 2nd edition, Computer Science and Applied Mathematics, Academic Press Inc., Orlando, FL.

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External Links:

ISBN 0-12-206360-0, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§3.5(iii), §3.5(i), §3.5(i), §3.5(ii), §3.5(iii), §3.5(iv), §3.5(v), §3.5(vi), §3.5(vii), §3.5(x).

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On the computation of Mathieu functions. J. Engrg. Math. 7, pp. 39–61.

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Notes:

Variable-precision Algol program, maximum precision 15D

External Links:

Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§28.36(ii), §28.36(iii).

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Texas Instruments, Inc..

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Notes:

Symbolic and extended-precision numerical computing system.

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

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Uniform asymptotic approximation of Mathieu functions. Methods Appl. Anal. 1 (2), pp. 143–168.

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External Links:

ISSN 1073-2772, Pdf, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§28.8(iv).

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Uniform asymptotic approximations for incomplete Riemann zeta functions. J. Comput. Appl. Math. 190 (1-2), pp. 339–353.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§8.22(ii).

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39: Tom H. Koornwinder

… ►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. … ►Currently he is on the editorial board for Constructive Approximation, and is editor for the volume on Multivariable Special Functions in the ongoing Askey–Bateman book project. …

40: 22.20 Methods of Computation

… ►for $n\ge 1$, where the square root is chosen so that $\mathrm{ph}{b}_n=\frac{1}{2}(\mathrm{ph}{a}_{n-1}+\mathrm{ph}{b}_{n-1})$, where $\mathrm{ph}{a}_{n-1}$ and $\mathrm{ph}{b}_{n-1}$ are chosen so that their difference is numerically less than $\pi$. … ►Next, compute ${\varphi }_N,{\varphi }_{N-1},\mathrm{\dots },{\varphi }_0$, where ►

22.20.3 $${\varphi }_N=2^N{a}_{N}x,$$

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Symbols:

$x$: real, $a_n$: numbers and ${\varphi }_N$: approximations

Referenced by:

§22.20(ii)

Permalink:

http://dlmf.nist.gov/22.20.E3

Encodings:

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See also:

Annotations for §22.20(ii), §22.20 and Ch.22

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22.20.4 $${\varphi }_{n-1}=\frac{1}{2}\left({\varphi }_n+\arcsin \left(\frac{c_n}{a_n}\sin {\varphi }_n\right)\right),$$

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Symbols:

$\arcsin z$: arcsine function, $\sin z$: sine function, $a_n$: numbers, $c_n$: numbers, $n$: positive and ${\varphi }_N$: approximations

Referenced by:

§22.20(ii)

Permalink:

http://dlmf.nist.gov/22.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.20(ii), §22.20 and Ch.22

… ►By application of the transformations given in §§22.7(i) and 22.7(ii), $k$ or $k^{\prime }$ can always be made sufficently small to enable the approximations given in §22.10(ii) to be applied. …