numerical%20approximations

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1: 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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2: 18.40 Methods of Computation

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A numerical approach to the recursion coefficients and quadrature abscissas and weights

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Stieltjes Inversion via (approximate) Analytic Continuation

… ►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. Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. … ►In Figure 18.40.2 the approximations were carried out with a precision of 50 decimal digits.

3: 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.

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

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

Referenced by:

§3.7(iv).

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4: 2.11 Remainder Terms; Stokes Phenomenon

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§2.11(i) Numerical Use of Asymptotic Expansions

… ► … ►The rest of this section is devoted to general methods for increasing this accuracy. … ►

§2.11(vi) Direct Numerical Transformations

… ►For example, extrapolated values may converge to an accurate value on one side of a Stokes line (§2.11(iv)), and converge to a quite inaccurate value on the other.

5: 3.4 Differentiation

… ► … ►With the choice $r=k$ (which is crucial when $k$ is large because of numerical cancellation) the integrand equals ${\mathrm{e}}^k$ at the dominant points $\theta =0,2\pi$, and in combination with the factor $k^{-k}$ in front of the integral sign this gives a rough approximation to $1/k!$. … ►

Laplacian

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Biharmonic Operator

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6: Bibliography C

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Numerical computation of real or complex elliptic integrals. Numer. Algorithms 10 (1-2), pp. 13–26.

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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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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).

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A method for numerical calculation of Fourier integrals. Numer. Math. 8 (5), pp. 422–436.

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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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ISSN 0029-599X, Document, MathReview (A. H. Stroud), ZentralBlatt

Referenced by:

§3.5(iv).

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Chebyshev approximations for the Fresnel integrals. Math. Comp. 22 (102), pp. 450–453.

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Notes:

with Microfiche Supplement

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FullText, MathReview, ZentralBlatt

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3rd item.

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7: Bibliography I

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

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Centre for Experimental and Constructive Mathematics, Simon Fraser University, Canada.

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Notes:

From a decimal approximation, find the most likely exact value.

External Links:

Inverse Symbolic Calculator

Referenced by:

3rd item.

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Highly Oscillatory Quadrature: The Story So Far. In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.), pp. 97–118.

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Notes:

Proceedings of ENuMath, Santiago de Compostela (2005)

External Links:

ISBN 3-540-34287-7, MathReview, ZentralBlatt

Referenced by:

§3.5(vii).

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A First Course in the Numerical Analysis of Differential Equations. Cambridge Texts in Applied Mathematics, No. 15, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-55376-8; 0-521-55655-4, MathReview (Luis Verde-Star), ZentralBlatt

Referenced by:

§3.7(i).

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More on electrostatic models for zeros of orthogonal polynomials. Numer. Funct. Anal. Optim. 21 (1-2), pp. 191–204.

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ISSN 0163-0563, FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§18.39(v).

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8: 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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Numerical calculation of a generalized complete elliptic integral. Rev. Mod. Phys. 10, pp. 264–269.

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External Links:

Document, ZentralBlatt

Referenced by:

§19.2(iii).

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

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

Referenced by:

§3.11(v).

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Numerical evaluation of continued fractions. SIAM Rev. 6 (4), pp. 383–421.

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External Links:

Document, MathReview (E. Frank), ZentralBlatt

Referenced by:

§3.10(iii), §3.10(ii).

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Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, Vol. 50, Birkhäuser Verlag, Basel.

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External Links:

ISBN 3-7643-1100-2, MathReview (A. Magnus), ZentralBlatt

Referenced by:

§3.11(iv).

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9: Bibliography V

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Rational approximations for exponential integrals $E_n(x)$ . Acad. Roy. Belg. Bull. Cl. Sci. (5) 56, pp. 1064–1072.

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MathReview (A. Magnus), ZentralBlatt

Referenced by:

3rd item.

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

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Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. J. Comput. Appl. Math. 213 (2), pp. 488–500.

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ISSN 0377-0427, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

item (f), §28.34(ii), §29.20(i).

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RFSFNS: A portable package for the numerical determination of the number and the calculation of roots of Bessel functions. Comput. Phys. Comm. 92 (2-3), pp. 252–266.

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Notes:

Double-precision Fortran, maximum accuracy 16S. For corrections see same journal 117 (1999), p. 290

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ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

3rd item.

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The topological degree theory for the localization and computation of complex zeros of Bessel functions. Numer. Funct. Anal. Optim. 18 (1-2), pp. 227–234.

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External Links:

ISSN 0163-0563, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

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10: Bibliography F

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

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Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

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Notes:

Fortran, minimum precision 6D.

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ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(ix).

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Numerical evaluation of the elliptic integral of the third kind. Math. Comp. 19 (91), pp. 494–496.

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FullText, MathReview, ZentralBlatt

Referenced by:

§19.36(iv).

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

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ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

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Numerical treatment of Coulomb wave functions. Rev. Mod. Phys. 27 (4), pp. 399–411.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§33.10(ii), §33.11, §33.12(i), §33.9(ii).

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