numerical inversion

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11: 22.19 Physical Applications

… ►Numerous other physical or engineering applications involving Jacobian elliptic functions, and their inverses, to problems of classical dynamics, electrostatics, and hydrodynamics appear in Bowman (1953, Chapters VII and VIII) and Lawden (1989, Chapter 5). …

12: Bibliography X

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Prolate spheroidal wavefunctions, quadrature and interpolation. Inverse Problems 17 (4), pp. 805–838.

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Notes:

Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000)

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ISSN 0266-5611, Document, MathReview (Thomas Sauer), ZentralBlatt

Referenced by:

§30.13(v).

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Variable precision computation of elementary functions. J. Numer. Methods Comput. Appl. 15 (3), pp. 161–171 (Chinese).

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Notes:

English translation in Chinese J. Numer. Math. Appl. 17 (1995), no. 1, pp. 13–24

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ISSN 1000-3266, MathReview, ZentralBlatt

Referenced by:

§4.48(iii).

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13: 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 $\pi$. … ►

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

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

►and the inverse sine has its principal value (§4.23(ii)). … ►

§22.20(v) Inverse Functions

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14: Peter A. Clarkson

… ►Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations. …His well-known book Solitons, Nonlinear Evolution Equations and Inverse Scattering (with M. …

15: 1.2 Elementary Algebra

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The Inverse

►If det($𝐀$) $\ne$ $0$, $𝐀$ has a unique inverse, ${𝐀}^{-1}$, such that … ►has a unique solution, $𝐛={𝐀}^{-1}𝐜$. …Numerical methods and issues for solution of (1.2.61) appear in §§3.2(i) to 3.2(iii). … ►Numerical methods and issues for solution of (1.2.72) appear in §§3.2(iv) to 3.2(vii). …

16: 2.9 Difference Equations

… ►As in the case of differential equations (§§2.7(iii), 2.7(iv)) recessive solutions are unique and dominant solutions are not; furthermore, one member of a numerically satisfactory pair has to be recessive. … ►For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a). …

17: 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).

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A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.

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Notes:

Extended-precision Fortran, maximum accuracy 12D.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

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Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.

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External Links:

Document

Referenced by:

§25.18(i).

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A survey of numerical methods for the computation of Bessel function integrals. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 249–265.

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Notes:

International conference on special functions: theory and computation (Turin, 1984)

External Links:

ISSN 0373-1243, MathReview, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

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Numerical Solution of Sturm-Liouville Problems. Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York.

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External Links:

ISBN 0-19-853415-9, MathReview (L. Greenberg), ZentralBlatt

Referenced by:

§3.7(iv).

Encodings:

BibTeX

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18: 10.41 Asymptotic Expansions for Large Order

… ►For numerical tables of $\eta =\eta (z)$ and the coefficients $U_k(p)$, $V_k(p)$, see Olver (1962, pp. 43–51). … ►For expansions in inverse factorial series see Dunster et al. (1993). …

19: 18.39 Applications in the Physical Sciences

… ►Table 18.39.1 lists typical non-classical weight functions, many related to the non-classical Freud weights of §18.32, and §32.15, all of which require numerical computation of the recursion coefficients (i. … ►

18.39.50 $$w^{\mathrm{CP}}(x)=\frac{(l+1+\frac{2Z}{s})}{\pi \mathrm{\Gamma }(2l+2)}{\mathrm{e}}^{(2\theta (x)-\pi )\tau (x)}{\left(4(1-x^2)\right)}^{l+\frac{1}{2}}{\left|\mathrm{\Gamma }\left(l+1+\mathrm{i}\tau (x)\right)\right|}^2,$$ $\theta (x)=\arccos (x)$, $\tau (x)=-\frac{2Z}{s}\sqrt{\frac{1-x}{1+x}}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\arccos z$: arccosine function, $w(x)$: weight function, $l$: nonnegative integer and $x$: real variable

Referenced by:

Figure 18.39.2, Figure 18.39.2, §18.39(iv), §18.40(ii)

Permalink:

http://dlmf.nist.gov/18.39.E50

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Annotations for §18.39(iv), §18.39(iv), §18.39 and Ch.18

… ►

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Figure 18.39.2: Coulomb–Pollaczek weight functions, $x\in [-1,1]$, (18.39.50) for $s=10$, $l=0$, and $Z=\pm 1$. … Magnify

… ►The equivalent quadrature weight, $w_i/w^{\mathrm{CP}}(x_i)$, also forms the foundation of a novel inversion of the Stieltjes–Perron moment inversion discussed in §18.40(ii). … ►As this follows from the three term recursion of (18.39.46) it is referred to as the J-Matrix approach, see (3.5.31), to single and multi-channel scattering numerics. …

20: Bibliography C

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The inverse of the error function. Pacific J. Math. 13 (2), pp. 459–470.

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External Links:

MathReview (D. P. Banerjee), ZentralBlatt

Referenced by:

§7.17(ii).

Encodings:

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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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Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

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External Links:

ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

§19.25(v), §22.15(ii).

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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).

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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).

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