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11: 17.18 Methods of Computation

§17.18 Methods of Computation

… ►Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. ►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9. Shanks (1955) applies such methods in several $q$-series problems; see Andrews et al. (1986).

12: Bibliography E

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Links to software for elliptic curve methods of factorization and primality testing.

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Home Page

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4th item.

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Ten Physical Applications of Spectral Zeta Functions. Lecture Notes in Physics. New Series m: Monographs, Vol. 35, Springer-Verlag, Berlin.

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ISBN 3-540-60230-5, MathReview (Alfred Actor), ZentralBlatt

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§25.17.

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A comparison of some methods for the evaluation of highly oscillatory integrals. J. Comput. Appl. Math. 112 (1-2), pp. 55–69.

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ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

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§3.5(vii).

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The Sobolev orthogonality and spectral analysis of the Laguerre polynomials $\{L_n^{-k}\}$ for positive integers $k$ . J. Comput. Appl. Math. 171 (1-2), pp. 199–234.

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ISSN 0377-0427, Document, Link, MathReview (Khélifa Trimèche)

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§18.36(v).

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13: Bibliography G

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On high precision methods for computing integrals involving Bessel functions. Math. Comp. 33 (147), pp. 1049–1057.

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ISSN 0025-5718, FullText, MathReview (K. Jetter), ZentralBlatt

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§10.74(vii).

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WKB and turning point theory for second-order difference equations. In Spectral Methods for Operators of Mathematical Physics, Oper. Theory Adv. Appl., Vol. 154, pp. 101–138.

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FullText, MathReview (Ioannis K. Purnaras), ZentralBlatt

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§18.15(v), §18.15(v).

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A method for evaluating certain Bessel integrals. Z. Angew. Math. Phys. 30 (4), pp. 722–723.

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ISSN 0044-2275, Document, MathReview

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§10.71(ii).

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Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA.

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CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26

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MathReview (O. Widlund), ZentralBlatt

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§18.38(i).

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Spectral Methods and Their Applications. World Scientific Publishing Co. Inc., River Edge, NJ-Singapore.

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ISBN 981-02-3333-7, MathReview (K. Moszyński), ZentralBlatt

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§18.38(i).

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14: 31.8 Solutions via Quadratures

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$ϵ=-m_3+\frac{1}{2}$ , $m_0,m_1,m_2,m_3=0,1,2,\mathrm{\dots }$,

►the Hermite–Darboux method (see Whittaker and Watson (1927, pp. 570–572)) can be applied to construct solutions of (31.2.1) expressed in quadratures, as follows. … ►The variables $\lambda$ and $\nu$ are two coordinates of the associated hyperelliptic (spectral) curve $\mathrm{\Gamma }:{\nu }^2={\prod }_{j=1}^{2g+1}(\lambda -{\lambda }_j)$. …

15: 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

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§27.18.

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An efficient derivative-free method for solving nonlinear equations. ACM Trans. Math. Software 11 (3), pp. 250–262.

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Corrigendum, ibid. 15(3) (1989), 287

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ISSN 0098-3500, Document, MathReview (T. Feagin), ZentralBlatt

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§3.8(iii).

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Introduction to spectral theory: selfadjoint ordinary differential operators. Translations of Mathematical Monographs, Vol. 39, American Mathematical Society, Providence, R.I..

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Translated from the Russian by Amiel Feinstein

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MathReview Entry

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§1.18(x).

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The spectral analysis of three families of exceptional Laguerre polynomials. J. Approx. Theory 202, pp. 5–41.

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ISSN 0021-9045, Document, Link, MathReview (Edmundo J. Huertas)

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§18.36(vi).

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Linear prediction of bandlimited processes with flat spectral densities. IEEE Trans. Signal Process. 49 (7), pp. 1564–1569.

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ISSN 1053-587X, Document, MathReview, ZentralBlatt

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§30.15(v).

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16: Bibliography S

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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

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§10.74(vi).

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Spectral Methods in Chemistry and Physics. Applications to Kinetic Theory and Quantum Mechanics. Scientific Computation, Springer-Verlag, Dordrecht.

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ISBN 978-94-017-9453-4; 978-94-017-9454-1, Document, Link, MathReview (Gabriel Stoltz)

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§18.38(i), §18.39(iii), §18.39(iii), §18.39(iii).

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Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.

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ISBN 0-8218-3675-7, MathReview (P. L. Duren), ZentralBlatt

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§18.33(i), §18.33(vi).

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Szegő’s Theorem and Its Descendants. Spectral Theory for $L^2$ Perturbations of Orthogonal Polynomials. M. B. Porter Lectures, Princeton University Press, Princeton, NJ.

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ISBN 978-0-691-14704-8, MathReview (Harry Dym)

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§18.2(xi).

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Multiparameter spectral theory in Hilbert space. Research Notes in Mathematics, Vol. 22, Pitman (Advanced Publishing Program), Boston, Mass.-London.

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ISBN 0-8224-8414-5, MathReview (F. H. Szafraniec), ZentralBlatt

Referenced by:

Profile Brian D. Sleeman.

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17: 12.18 Methods of Computation

§12.18 Methods of Computation

►Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs. …

18: 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

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§10.20(ii).

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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

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§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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International conference on special functions: theory and computation (Turin, 1984)

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ISSN 0373-1243, MathReview, ZentralBlatt

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§10.74(vii), §10.74(vii).

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Fast analytic formulas for the modified Bessel functions of imaginary order for spectral line broadening calculations. J. Quantit. Spec. and Rad. Trans. 62 (4), pp. 389–395.

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Document

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§10.76(ii).

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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

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§10.74(vii), §10.74(vii).

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19: Bibliography K

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

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

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

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NMS (free collection of Fortran subroutines).

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Periodic Orbits, Spectral Statistics, and the Riemann Zeros. In Supersymmetry and Trace Formulae: Chaos and Disorder, J. P. Keating, D. E. Khmelnitskii, and I. V. Lerner (Eds.), pp. 1–15.

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§25.17.

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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Document, MathReview (Matti Jutila), ZentralBlatt

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§25.18(i).

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The Rayleigh function: Theory and computational methods. Zh. Vychisl. Mat. Mat. Fiz. 39 (12), pp. 1962–2006.

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Translation in Comput. Math. Math. Phys. 39 (1999), No. 12, pp. 1883–1925.

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ISSN 0044-4669, MathReview (Walter Gautschi), ZentralBlatt

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§10.21(xiii).

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An indirect method for evaluating certain infinite integrals. Z. Angew. Math. Phys. 29 (3), pp. 380–386.

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ISSN 0044-2275, Document, MathReview, ZentralBlatt

Referenced by:

§10.71(ii).

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20: 34.13 Methods of Computation

§34.13 Methods of Computation

►Methods of computation for $\mathit{3}j$ and $\mathit{6}j$ symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). ►For $\mathit{9}j$ symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989). A review of methods of computation is given in Srinivasa Rao and Rajeswari (1993, Chapter VII, pp. 235–265). …