numerical evaluation

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21: Bibliography H

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A fast FFT-based discrete Legendre transform. IMA J. Numer. Anal. 36 (4), pp. 1670–1684.

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External Links:

ISSN 0272-4979, Document, Link, MathReview (Denis N. Sidorov)

Referenced by:

§18.16(iii).

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A Course of Pure Mathematics. 10th edition, Cambridge University Press.

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Notes:

Numerous reprintings exist, including the Centenary Edition with Foreword by T. W. Körner (Cambridge Univerity Press, 2008).

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.4(ii), §1.4(iii), §1.4(iv), §1.4(v), §1.4(vi), §1.4(vii), §4.4(iii), §4.6(i), §4.6(ii).

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Analytic evaluation of two-center STO electron repulsion integrals via ellipsoidal expansion. Internat. J. Quantum Chem. 88 (6), pp. 701–734.

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Document

Referenced by:

§8.24(iii).

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Numerical Tools for the Study of Finite Gap Solutions of Integrable Systems. Ph.D. Thesis, Technischen Universität Berlin.

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Notes:

(All relevant material, plus corrections, are included in Deconinck et al. (2004).)

External Links:

ZentralBlatt

Referenced by:

§21.5(i).

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Introduction to Numerical Analysis. 2nd edition, McGraw-Hill Book Co., New York.

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Notes:

Reprinted by Dover Publications Inc., New York, 1987.

External Links:

ISBN 0-486-65363-3, MathReview, ZentralBlatt

Referenced by:

§3.3(iii), §3.3(iv), §3.3(v), §3.4(i), §3.8(v), Ch.3.

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22: 18.40 Methods of Computation

… ►The question is then: how is this possible given only $F_N(z)$, rather than $F(z)$ itself? $F_N(z)$ often converges to smooth results for $z$ off the real axis for $\mathrm{\Im }z$ at a distance greater than the pole spacing of the $x_n$, this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to $F_N(z)$ and evaluating these on the real axis in regions of higher pole density that those of the approximating function. …

23: 2.11 Remainder Terms; Stokes Phenomenon

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§2.11(i) Numerical Use of Asymptotic Expansions

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§2.11(vi) Direct Numerical Transformations

… ►The numerically smallest terms are the 5th and 6th. … ►Optimum truncation occurs just prior to the numerically smallest term, that is, at $s_4$. … ►However, direct numerical transformations need to be used with care. …

24: 28.36 Software

… ►In this section we provide links to the research literature describing the implementation of algorithms in software for the evaluation of functions described in this chapter. … ►See also Clemm (1969), Delft Numerical Analysis Group (1973), Rengarajan and Lewis (1980), and Schäfke and Schmidt (1966). … ►See also Clemm (1969), Delft Numerical Analysis Group (1973), Rengarajan and Lewis (1980), Van Buren and Boisvert (2007), and Ziener et al. (2012).

25: 3.5 Quadrature

… ►About $2^9=512$ function evaluations are needed. (With the 20-point Gauss–Laguerre formula (§3.5(v)) the same precision can be achieved with 15 function evaluations.) … ►In adaptive algorithms the evaluation of the nodes and weights may cause difficulties, unless exact values are known. … ►Convergence acceleration schemes, for example Levin’s transformation (§3.9(v)), can be used when evaluating the series. … ►A second example is provided in Gil et al. (2001), where the method of contour integration is used to evaluate Scorer functions of complex argument (§9.12). …

26: 19.36 Methods of Computation

… ►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. … ► $F(\varphi ,k)$ can be evaluated by using (19.25.5). $E(\varphi ,k)$ can be evaluated by using (19.25.7), and $R_D$ by using (19.21.10), but cancellations may become significant. … ►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. …

27: Bibliography Z

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Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the Faddeyeva Function. ACM Trans. Math. Softw. 44 (2), pp. 22:1–22:9.

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ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

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4th item, §7.25(iii).

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Numerical analysis of Struve functions with applications to other special functions. Ann. Numer. Math. 2 (1-4), pp. 199–208.

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ISSN 1021-2655, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§11.7(ii).

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Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2 (1-4), pp. 457–472.

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ISSN 1021-2655, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§13.9(i), §15.13.

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Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

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Notes:

Includes hypergeometric and $q$-hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).

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Fast evaluation of elementary mathematical functions with correctly rounded last bit. ACM Trans. Math. Software 17 (3), pp. 410–423.

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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

§4.45(i).

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28: 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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On the computation of Mathieu functions. J. Engrg. Math. 7, pp. 39–61.

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Notes:

Variable-precision Algol program, maximum precision 15D

External Links:

Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§28.36(ii), §28.36(iii).

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Accurate and efficient evaluation of Schur and Jack functions. Math. Comp. 75 (253), pp. 223–239.

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Notes:

For Fortran software for these functions, see Plamen Koev’s home page.

External Links:

Home Page, ISSN 0025-5718, Document, MathReview Entry, ZentralBlatt

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1st item.

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Texas Instruments, Inc..

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Notes:

Symbolic and extended-precision numerical computing system.

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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29: 10.74 Methods of Computation

… ►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … ►It should be noted, however, that there is a difficulty in evaluating the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, from the explicit expressions (10.20.10)–(10.20.13) when $z$ is close to $1$ owing to severe cancellation. … ►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods. … ►For applications of generalized Gauss–Laguerre quadrature (§3.5(v)) to the evaluation of the modified Bessel functions $K_{\nu }\left(z\right)$ for and see Gautschi (2002a). … ►For evaluation of $K_{\nu }\left(z\right)$ from (10.32.14) with $\nu =n$ and $z$ complex, see Mechel (1966). …

30: 3.3 Interpolation

… ►By using this approximation to $x$ as a new point, $x_3=x$, and evaluating $\left[f_0,f_1,f_2,f_3\right]x=\mathrm{1.12388\hspace{0.33em}6190}$, we find that $x=-\mathrm{2.33810\hspace{0.33em}7409}$, with 9 correct digits. … ►Then by using $x_3$ in Newton’s interpolation formula, evaluating $\left[x_0,x_1,x_2,x_3\right]f=-\mathrm{0.26608\hspace{0.33em}28233}$ and recomputing $f^{\prime }(x)$, another application of Newton’s rule with starting value $x_3$ gives the approximation $x=\mathrm{2.33810\hspace{0.33em}7373}$, with 8 correct digits. … ►For theory and applications see Stenger (1993, Chapter 3).