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21: 32.17 Methods of Computation

§32.17 Methods of Computation

►The Painlevé equations can be integrated by Runge–Kutta methods for ordinary differential equations; see §3.7(v), Hairer et al. (2000), and Butcher (2003). …

22: 19.36 Methods of Computation

§19.36 Methods of Computation

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§19.36(i) Duplication Method

… ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … ►

§19.36(iv) Other Methods

►Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. …

23: 29.20 Methods of Computation

§29.20 Methods of Computation

… ►A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. … ►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 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. …

24: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

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Document, Code

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1st item, 4th item.

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Asymptotic methods in enumeration. SIAM Rev. 16 (4), pp. 485–515.

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External Links:

Document, MathReview (Bernard Harris), ZentralBlatt

Referenced by:

Ch.2.

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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Document, ZentralBlatt

Referenced by:

§10.73(iv).

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Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-360-9, MathReview (Hongyuan Zha), ZentralBlatt

Referenced by:

§3.11(v).

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25: 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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Asymptotic Methods in Analysis. 2nd edition, Bibliotheca Mathematica, Vol. IV, North-Holland Publishing Co., Amsterdam.

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MathReview, ZentralBlatt

Referenced by:

§2.2, Ch.2, §26.7(iv), §26.7(iv), §4.13.

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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.

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ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt

Referenced by:

§31.17(ii).

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Computing spectra of linear operators using the Floquet-Fourier-Hill method. J. Comput. Phys. 219 (1), pp. 296–321.

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ISSN 0021-9991, MathReview, ZentralBlatt

Referenced by:

§22.11.

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Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, Vol. 2, North-Holland Publishing Co., Amsterdam.

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ISBN 0-444-87634-0, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§3.7(v).

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26: 1.10 Functions of a Complex Variable

… ►Often Darboux’s method can be used to study the behavior of $p_n(x)$ as $n\to \mathrm{\infty }$. …

27: Bibliography S

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A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.

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ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt

Referenced by:

§28.32(i).

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A global Newton method for the zeros of cylinder functions. Numer. Algorithms 18 (3-4), pp. 259–276.

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

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Practical Extrapolation Methods: Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics, Vol. 10, Cambridge University Press, Cambridge.

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ISBN 0-521-66159-5, MathReview (Thomas A. Atchison), ZentralBlatt

Referenced by:

§3.9(vi).

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Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:

Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.

External Links:

ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt

Referenced by:

§3.3(vi), §3.4(i), §3.5(i).

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28: Bibliography P

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Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.

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ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§10.20(ii).

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

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ISSN 0010-4655, Document, Code

Referenced by:

6th item.

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Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.

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ISSN 0006-3835, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(vii).

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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)

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ISSN 0373-1243, MathReview, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

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A method for computing Bessel function integrals. J. Comput. Phys. 75 (2), pp. 334–344.

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ISSN 0021-9991, Document, MathReview (J. G. Herriot), ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

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29: Bibliography

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The $J$-Matrix Method. Developments and Applications. Springer-Verlag.

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External Links:

ISBN 978-1-4020-6072-4

Referenced by:

§18.39(iv).

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Pfaff’s method II: Diverse applications. J. Comput. Appl. Math. 68 (1-2), pp. 15–23.

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ISSN 0377-0427, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§17.7(iii).

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On magic squares constructed by the uniform step method. Proc. Amer. Math. Soc. 2 (4), pp. 557–565.

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ISSN 0002-9939, Fulltext, MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§27.17.

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Mathematical Methods for Physicists. 6th edition, Elsevier, Oxford.

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External Links:

ISBN 0-12-059876-0;0-12-088584-0, MathReview, ZentralBlatt

Referenced by:

§1.17(ii), §1.17(iii).

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Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York.

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Notes:

Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition

External Links:

ISBN 0-387-96890-3, MathReview

Referenced by:

§21.5(i).

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30: 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.

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

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Geometrical and Catastrophe Optics Methods in Scattering. In Physical Acoustics, A. D. Pierce and R. N. Thurston (Eds.), Vol. 21, pp. 1–234.

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Referenced by:

§36.14(ii).

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Numerical Methods for Roots of Polynomials. Part I. Studies in Computational Mathematics, Vol. 14, Elsevier, Amsterdam.

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External Links:

ISBN 978-0-444-52729-5; 0-444-52729-X, MathReview Entry, ZentralBlatt

Referenced by:

§3.8(iv).

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

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Methods and Programs for Mathematical Functions. Ellis Horwood Ltd., Chichester.

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Notes:

Includes diskette

External Links:

ISBN 0-13-578998-2, MathReview, ZentralBlatt

Referenced by:

2nd item, CEPHES (free C library).

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