computation by quadrature

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21: 29.20 Methods of Computation

§29.20 Methods of Computation

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§29.20(i) Lamé Functions

… ►The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5). … ►

§29.20(ii) Lamé Polynomials

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§29.20(iii) Zeros

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22: Bibliography B

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Automatic computation of zeros of Bessel functions and other special functions. SIAM J. Sci. Comput. 21 (4), pp. 1458–1464.

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ISSN 1064-8275, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.74(vi), §3.8(iii).

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Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.

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ISSN 0893-9659, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

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New Aspects in Numerical Quadrature. In Proc. Sympos. Appl. Math., Vol. XV, pp. 199–218.

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MathReview (S. Haber), ZentralBlatt

Referenced by:

§3.5(iii).

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Computation of the polygamma functions. Comm. Statist. B—Simulation Comput. 13 (3), pp. 409–415.

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ISSN 0361-0918, Document, MathReview (S. Conde)

Referenced by:

§5.24(iii).

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Interpolation und genäherte Quadratur. In Mathematische Hilfsmittel des Ingenieurs. Teil III, R. Sauer and I. Szabó (Eds.), Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vol. 141, pp. 232–319.

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External Links:

MathReview (E. Kreyszig)

Referenced by:

§3.3(vi).

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23: Bibliography H

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On an “equivalent quadrature” calculation of matrix elements of ${(z-p^2/2m)}^{-1}$ using an $L^2$ expansion technique. J. Comput. Phys. 13, pp. 536–550.

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External Links:

Document, Link

Referenced by:

§18.39(iv).

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24: Bibliography L

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Computing $\pi (x)$: The Meissel-Lehmer method. Math. Comp. 44 (170), pp. 537–560.

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ISSN 0025-5718, FullText, MathReview (Kenneth A. Jukes), ZentralBlatt

Referenced by:

§27.18.

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Exact computation of the $3$-$j$ and $6$-$j$ symbols. Comput. Phys. Comm. 61 (3), pp. 350–360.

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Notes:

A Fortran program is included.

External Links:

Document, ZentralBlatt

Referenced by:

§34.15.

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Exact computation of the $9$-$j$ symbols. Comput. Phys. Comm. 70 (3), pp. 544–556.

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Notes:

Double-precision Fortran

External Links:

ISSN 0010-4655, Document, Code, MathReview

Referenced by:

7th item.

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Algorithms for the Computation of Mathematical Functions. Academic Press, New York.

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External Links:

MathReview (Gh. Adam)

Referenced by:

§16.26.

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Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature. Math. Comp. 25 (113), pp. 87–104.

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FullText, MathReview (J. Bryant), ZentralBlatt

Referenced by:

§2.3(iv).

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25: Bibliography W

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Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.

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ISSN 0006-3835, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.5(iv).

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Computation with Recurrence Relations. Pitman, Boston, MA.

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External Links:

ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt

Referenced by:

§13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).

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Some explicit Padé approximants for the function ${\mathrm{\Phi }}^{\prime }/\mathrm{\Phi }$ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.

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ISSN 0036-1410, Document, MathReview (Wyman Fair), ZentralBlatt

Referenced by:

§13.31(ii).

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Algorithm 44: Bessel functions computed recursively. Comm. ACM 4 (4), pp. 177–178.

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Notes:

Includes Algol code, maximum accuracy 8S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

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Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.

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ISSN 0029-599X, Document, MathReview (A.J. Rodrigues), ZentralBlatt

Referenced by:

§10.74(vii), §3.5(vii).

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26: Bibliography C

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A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

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ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt

Referenced by:

§10.74(vii).

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On computing elliptic integrals and functions. J. Math. and Phys. 44, pp. 36–51.

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External Links:

MathReview (R. Nicolovius), ZentralBlatt

Referenced by:

§19.36(ii), §19.36(ii).

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An algorithm for computing logarithms and arctangents. Math. Comp. 26 (118), pp. 543–549.

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External Links:

FullText, MathReview (F. Gotze), ZentralBlatt

Referenced by:

§4.45(i).

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Computing elliptic integrals by duplication. Numer. Math. 33 (1), pp. 1–16.

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ISSN 0029-599X, Document, MathReview, ZentralBlatt

Referenced by:

§19.19.

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Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.

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Notes:

Fortran code.

External Links:

ISSN 0098-3500, Document

Referenced by:

§10.77(ix).

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27: 11.13 Methods of Computation

§11.13 Methods of Computation

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§11.13(i) Introduction

►Subsequent subsections treat the computation of Struve functions. …For a review of methods for the computation of ${𝐇}_{\nu }\left(z\right)$ see Zanovello (1975). … ►

§11.13(iii) Quadrature

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28: 3.2 Linear Algebra

… ►When the factorization (3.2.5) is available, the accuracy of the computed solution $𝐱$ can be improved with little extra computation. … ►Let ${𝐱}^{\ast }$ denote a computed solution of the system (3.2.1), with $𝐫=𝐛-𝐀{𝐱}^{\ast }$ again denoting the residual. … ►Lanczos’ method is related to Gauss quadrature considered in §3.5(v). … ►

§3.2(vii) Computation of Eigenvalues

►Many methods are available for computing eigenvalues; see Golub and Van Loan (1996, Chapters 7, 8), Trefethen and Bau (1997, Chapter 5), and Wilkinson (1988, Chapters 8, 9).

29: 10.74 Methods of Computation

§10.74 Methods of Computation

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§10.74(vi) Zeros and Associated Values

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Hankel Transform

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30: 33.23 Methods of Computation

§33.23 Methods of Computation

… ►The methods used for computing the Coulomb functions described below are similar to those in §13.29. … ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. … ►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. … ►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). …