numerical inversion

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21: Bibliography I

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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

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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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Proceedings of ENuMath, Santiago de Compostela (2005)

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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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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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22: Bibliography T

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Asymptotic inversion of incomplete gamma functions. Math. Comp. 58 (198), pp. 755–764.

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ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§8.12, §8.12.

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Numerical algorithms for uniform Airy-type asymptotic expansions. Numer. Algorithms 15 (2), pp. 207–225.

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§10.19(iii), §10.20(i), §10.74(i).

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Numerical and asymptotic aspects of parabolic cylinder functions. J. Comput. Appl. Math. 121 (1-2), pp. 221–246.

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Numerical analysis in the 20th century, Vol. I, Approximation theory

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ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt

Referenced by:

§12.10(vii), §12.10(vi), §12.10(vi), §12.18.

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High Speed Numerical Integration of Fermi Dirac Integrals. Master’s Thesis, Naval Postgraduate School, Monterey, CA.

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FullText

Referenced by:

§25.21(vii).

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Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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ISBN 0-89871-361-7, MathReview (Moody T. Chu), ZentralBlatt

Referenced by:

§3.2(iii), §3.2(vii).

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23: Bibliography B

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

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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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Document, MathReview (E. Frank), ZentralBlatt

Referenced by:

§3.10(iii), §3.10(ii).

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Numerical aspects of Mathieu eigenvalues. Rend. Circ. Mat. Palermo (2) 15, pp. 51–97.

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Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

item (e).

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Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7 (1), pp. 78–90.

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Notes:

Algol 60 programs are included for real and complex elliptic integrals, and for Jacobian elliptic functions of real argument.

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ISSN 0029-599X, Document, MathReview (M. Feliciano), ZentralBlatt

Referenced by:

§19.1, §19.2(iii), §19.36(ii), §19.39(iii), §22.22(ii).

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24: Bibliography J

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On the numerical calculation of polylogarithms. Nordisk Tidskr. Informationsbehandling (BIT) 12 (4), pp. 581–585.

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Document, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

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Sur l’inversion de $y^{\alpha }{e}^y$ au moyen des nombres de Stirling associés. C. R. Acad. Sci. Paris Sér. I Math. 320 (12), pp. 1449–1452.

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ISSN 0764-4442, MathReview (F. Beukers), ZentralBlatt

Referenced by:

(4.13.10).

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Numerical calculation of Bessel, Hankel and Airy functions. Computer Physics Communications 183 (3), pp. 506–519.

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Document, FullText, ZentralBlatt

Referenced by:

§10.19(iii), §10.19(iii).

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Numerical stability in evaluating continued fractions. Math. Comp. 28 (127), pp. 795–810.

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FullText, MathReview (N. N. Abdelmalek), ZentralBlatt

Referenced by:

§3.10(iii).

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25: Bibliography K

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Numerical Methods and Software. Prentice Hall, Englewood Cliffs, N.J..

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Notes:

Includes collection of single- and double-precision Fortran subroutines.

External Links:

ISBN 0-13-627258-4, GAMS (package NMS), ZentralBlatt

Referenced by:

NMS (free collection of Fortran subroutines).

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The second Painlevé equation in electric probe theory. Some numerical solutions. Zh. Vychisl. Mat. Mat. Fiz. 38 (6), pp. 992–1000 (Russian).

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Notes:

In Russian. English translation: Comput. Math. Math. Phys. 38(1998), no. 6, pp. 950–958

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ISSN 0044-4669, MathReview, ZentralBlatt

Referenced by:

§32.17.

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The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface. Technical Physics 49 (1), pp. 1–7.

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Document

Referenced by:

§32.17.

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On Nielsen’s generalized polylogarithms and their numerical calculation. Nordisk Tidskr. Informationsbehandling (BIT) 10, pp. 38–73.

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Document, MathReview (M. Nassif), ZentralBlatt

Referenced by:

§25.12(i).

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Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge.

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ISBN 0-521-37320-4; 0-521-58646-1, MathReview (Makoto Idzumi), ZentralBlatt

Referenced by:

§26.20.

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26: Bibliography R

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Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.

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ISSN 0029-599X, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§3.11(i).

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Fourier analysis and signal processing by use of the Möbius inversion formula. IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.

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ZentralBlatt

Referenced by:

§27.17.

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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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Notes:

arXiv:1812.02196

Referenced by:

§18.40(ii).

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Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.

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MathReview, ZentralBlatt

Referenced by:

§19.36(i), §19.36(ii).

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Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.

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English translation in U.S.S.R. Computational Math. and Math. Phys. 13(6), pp. 58–67

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ISSN 0044-4669, Document, MathReview (H. Zegeling), ZentralBlatt

Referenced by:

§10.74(iii).

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27: 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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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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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).)

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ZentralBlatt

Referenced by:

§21.5(i).

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Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.

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Document

Referenced by:

§15.18.

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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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28: 3.10 Continued Fractions

… ►For example, by converting the Maclaurin expansion of $\arctan z$ (4.24.3), we obtain a continued fraction with the same region of convergence ($\left|z\right|\le 1$, $z\ne \pm \mathrm{i}$), whereas the continued fraction (4.25.4) converges for all $z\in ℂ$ except on the branch cuts from $\mathrm{i}$ to $\mathrm{i}\mathrm{\infty }$ and $-\mathrm{i}$ to $-\mathrm{i}\mathrm{\infty }$. … ►

§3.10(iii) Numerical Evaluation of Continued Fractions

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Forward Recurrence Algorithm

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Backward Recurrence Algorithm

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Forward Series Recurrence Algorithm

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29: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

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Regula Falsi

… ►Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation: … ►Because the method requires only one function evaluation per iteration, its numerical efficiency is ultimately higher than that of Newton’s method. … ►Corresponding numerical factors in this example for other zeros and other values of $j$ are obtained in Gautschi (1984, §4). …

30: 10.21 Zeros

… ►The functions ${\rho }_{\nu }(t)$ and ${\sigma }_{\nu }(t)$ are related to the inverses of the phase functions ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ defined in §10.18(i): if $\nu \ge 0$, then … ►For the first zeros rounded numerical values of the coefficients are given by … ►For numerical coefficients for $m=2,3,4,5$ see Olver (1951, Tables 3–6). … ►Next, $z(\zeta )$ is the inverse of the function $\zeta =\zeta (z)$ defined by (10.20.3). … ►The latter reference includes numerical tables of the first few coefficients in the uniform asymptotic expansions. …