approximation techniques

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1: 3.11 Approximation Techniques

§3.11 Approximation Techniques

… ►

§3.11(iv) Padé Approximations

… ►

§3.11(v) Least Squares Approximations

… ►

§3.11(vi) Splines

… ►See Knuth (1986, pp. 116-136).

2: 36.12 Uniform Approximation of Integrals

… ►This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes

\Psi_{K}(\mathbf{x};k)

in (36.2.10) away from

\mathbf{x}=\boldsymbol{{0}}

, in terms of canonical integrals

\Psi_{J}\left(\xi(\mathbf{x};k)\right)

for

J<K

. …

3: 33.23 Methods of Computation

… ►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

►WKBJ approximations (§2.7(iii)) for

\rho>\rho_{\operatorname{tp}}\left(\eta,\ell\right)

are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq. …Seaton (1984) estimates the accuracies of these approximations. ►Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for

F_{0}

and

G_{0}

in the region inside the turning point:

\rho<\rho_{\operatorname{tp}}\left(\eta,\ell\right)

4: Bibliography C

… ►

P. J. Cameron (1994) Combinatorics: Topics, Techniques, Algorithms. Cambridge University Press, Cambridge.

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External Links:: ISBN 0-521-45133-7; 0-521-45761-0, MathReview (E. J. F. Primrose), ZentralBlatt
Referenced by:: §26.13.
Encodings:: BibTeX

… ►

R. Cazenave (1969) Intégrales et Fonctions Elliptiques en Vue des Applications. Préface de Henri Villat. Publications Scientifiques et Techniques du Ministère de l’Air, No. 452, Centre de Documentation de l’Armement, Paris.

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External Links:: MathReview (B. Ć. Carlson)
Referenced by:: §19.11(i), §19.12, §19.14(i), §19.36(ii), §19.4(ii), §19.8(ii), §19.8(iii), Ch.19.
Encodings:: BibTeX

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E. W. Cheney (1982) 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:: BibTeX

… ►

J. A. Cochran and J. N. Hoffspiegel (1970) Numerical techniques for finding

\nu

-zeros of Hankel functions. Math. Comp. 24 (110), pp. 413–422.

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External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: §10.21(xiv).
Encodings:: BibTeX

… ►

W. J. Cody (1968) 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.
Encodings:: BibTeX

…

5: 9.16 Physical Applications

… ►The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood. …The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). … ►Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. … ►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. …

6: 25.10 Zeros

… ►

25.10.4

R(t)=(-1)^{m-1}\left(\frac{2\pi}{t}\right)^{1/4}\frac{\cos\left(t-(2m+1)\sqrt{% 2\pi t}-\frac{1}{8}\pi\right)}{\cos\left(\sqrt{2\pi t}\right)}+O\left(t^{-3/4}% \right).

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function and $m$ : nonnegative integer
Keywords:: asymptotic approximation
Source:: Titchmarsh (1986b, (4.17.5), p. 89)
Referenced by:: §25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii)
Permalink:: http://dlmf.nist.gov/25.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §25.10(ii), §25.10 and Ch.25

►Riemann also developed a technique for determining further terms. …

7: Bibliography R

… ►

A. Ralston (1965) 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:: BibTeX

… ►

M. Razaz and J. L. Schonfelder (1981) Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.

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External Links:: ISSN 0098-3500, Document
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

W. H. Reid (1972) Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.

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External Links:: MathReview (E. A. Boyd), ZentralBlatt
Referenced by:: (9.13.25), (9.13.26), (9.13.28), (9.13.29), (9.13.32), (9.13.33), (9.13.34), §9.13(ii), §9.13(ii).
Encodings:: BibTeX

►

W. H. Reid (1974a) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

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External Links:: MathReview (T. W. Kao), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

… ►

L. Robin (1959) Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome III. Collection Technique et Scientifique du C. N. E. T. Gauthier-Villars, Paris.

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External Links:: MathReview (R. Campbell), ZentralBlatt
Referenced by:: Ch.14.
Encodings:: BibTeX

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8: Bibliography F

… ►

G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:: ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)
Referenced by:: §18.32.
Encodings:: BibTeX

… ►

B. Friedman (1990) Principles and Techniques of Applied Mathematics. Dover, New York.

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Notes:: Reprint of First Edition, 1956, John-Wiley: New York
External Links:: ISBN 0-486-66444-9, MathReview Entry
Referenced by:: §1.16(ix), §1.17(ii), §1.17(ii), §1.17(i), §1.17(i), §1.17(iv), §1.18(ix), §1.18(iii), §1.18(vi), §1.18(i), §1.18(x).
Encodings:: BibTeX

…

9: Bibliography S

… ►

C. W. Schelin (1983) Calculator function approximation. Amer. Math. Monthly 90 (5), pp. 317–325.

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External Links:: ISSN 0002-9890, FullText, MathReview (J. Albrycht), ZentralBlatt
Referenced by:: §4.45(i).
Encodings:: BibTeX

… ►

T. Schmelzer and L. N. Trefethen (2007) 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

… ►

L. S. Schulman (1981) Techniques and Applications of Path Integration. John Wiley & Sons Inc., New York.

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Notes:: Reprinted by Dover in 2005.
External Links:: ISBN 0-471-76450-7, MathReview (Aubrey Truman), ZentralBlatt
Referenced by:: §20.13, §22.19(ii).
Encodings:: BibTeX

… ►

M. J. Seaton (1984) The accuracy of iterated JWBK approximations for Coulomb radial functions. Comput. Phys. Comm. 32 (2), pp. 115–119.

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External Links:: Document
Referenced by:: §33.23(vii).
Encodings:: BibTeX

… ►

A. H. Stroud (1971) Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, N.J..

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External Links:: ISBN 0130438936;978-0130438935, MathReview (R. D. Riess), ZentralBlatt
Referenced by:: §3.5(x).
Encodings:: BibTeX

…

10: 18.39 Applications in the Physical Sciences

… ►A relativistic treatment becoming necessary as

Z

becomes large as corrections to the non-relativistic Schrödinger picture are of approximate order

\left(\alpha Z\right)^{2}\sim\left(Z/137\right)^{2}

\alpha

being the dimensionless fine structure constant

{e}^{2}/(4\pi\varepsilon_{0}\hbar c)

, where

c

is the speed of light. … ►Shizgal (2015) gives a broad overview of techniques and applications of spectral and pseudo-spectral methods to problems arising in theoretical chemistry, chemical kinetics, transport theory, and astrophysics. … ►The technique to accomplish this follows the DVR idea, in which methods are based on finding tridiagonal representations of the co-ordinate,

x

. … ►While

s

in the basis of (18.39.44) is simply a variational parameter, care must be taken, or the relationship between the results of the matrix variational approximation and the Pollaczek polynomials is lost, although this has no effect on the use of the variational approximations Reinhardt (2021a, b). … ►This equivalent quadrature relationship, see Heller et al. (1973), Yamani and Reinhardt (1975), allows extraction of scattering information from the finite dimensional

L^{2}

functions of (18.39.53), provided that such information involves potentials, or projections onto

L^{2}

functions, exactly expressed, or well approximated, in the finite basis of (18.39.44). …