numerical approximation

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11: 3.11 Approximation Techniques

§3.11 Approximation Techniques

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§3.11(i) Minimax Polynomial Approximations

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§3.11(iv) Padé Approximations

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12: 35.9 Applications

… ►The asymptotic approximations of §35.7(iv) are applied in numerous statistical contexts in Butler and Wood (2002). …

13: 9.12 Scorer Functions

… ►For the above properties and further results, including the distribution of complex zeros, asymptotic approximations for the numerically large real or complex zeros, and numerical tables see Gil et al. (2003c). …

14: Bibliography R

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Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.

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External Links:

ISSN 0029-599X, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§3.11(i).

Encodings:

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15: Bibliography P

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Evaluation of Fermi-Dirac Integral. In Nonlinear Numerical Methods and Rational Approximation (Wilrijk, 1987), A. Cuyt (Ed.), Mathematics and Its Applications, Vol. 43, pp. 435–444.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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16: Bibliography S

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Computing the gamma function using contour integrals and rational approximations. SIAM J. Numer. Anal. 45 (2), pp. 558–571.

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External Links:

ISSN 0036-1429, Document, MathReview (Amparo Gil), ZentralBlatt

Referenced by:

§5.21, §5.23(iii).

Encodings:

BibTeX

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17: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

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Approximation of Functions: Theory and Numerical Methods. Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag, New York.

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Notes:

Expanded translation from the German edition. Translated by Larry L. Schumaker

External Links:

MathReview, ZentralBlatt

Referenced by:

§3.11(i), §3.11(iii).

Encodings:

BibTeX

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18: Bibliography T

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Computational aspects of incomplete gamma functions with large complex parameters. In Approximation and Computation. A Festschrift in Honor of Walter Gautschi, R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 551–562.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.11(iii), §8.20(ii), §8.25(i), §8.25(iii), 2nd item, 3rd item.

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Numerical and asymptotic aspects of parabolic cylinder functions. J. Comput. Appl. Math. 121 (1-2), pp. 221–246.

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Notes:

Numerical analysis in the 20th century, Vol. I, Approximation theory

External Links:

ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt

Referenced by:

§12.10(vii), §12.10(vi), §12.10(vi), §12.18.

Encodings:

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Numerical evaluation of exponential integral: Theis well function approximation. Journal of Hydrology 205 (1-2), pp. 38–51.

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External Links:

Document

Referenced by:

§6.18(i).

Encodings:

BibTeX

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19: 4.45 Methods of Computation

… ►Initial approximations are obtainable, for example, from the power series (4.13.6) (with $t\ge 0$) when $x$ is close to $-1/\mathrm{e}$, from the asymptotic expansion (4.13.10) when $x$ is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of $x$. …

20: 33.23 Methods of Computation

… ►When numerical values of the Coulomb functions are available for some radii, their values for other radii may be obtained by direct numerical integration of equations (33.2.1) or (33.14.1), provided that the integration is carried out in a stable direction (§3.7). … ►

§33.23(vi) Other Numerical Methods

… ►Noble (2004) obtains double-precision accuracy for $W_{-\eta ,\mu }\left(2\rho \right)$ for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). ►

§33.23(vii) WKBJ Approximations

… ►Seaton (1984) estimates the accuracies of these approximations. …