numerical use of

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31: 3.7 Ordinary Differential Equations

… ►For an introduction to numerical methods for ordinary differential equations, see Ascher and Petzold (1998), Hairer et al. (1993), and Iserles (1996). ►

§3.7(ii) Taylor-Series Method: Initial-Value Problems

… ► … ►

§3.7(v) Runge–Kutta Method

… ►An extensive literature exists on the numerical solution of ordinary differential equations by Runge–Kutta, multistep, or other methods. …

32: 35.9 Applications

… ►These references all use results related to the integral formulas (35.4.7) and (35.5.8). … ►The asymptotic approximations of §35.7(iv) are applied in numerous statistical contexts in Butler and Wood (2002). …

33: 3.5 Quadrature

§3.5 Quadrature

… ►For computing infinite oscillatory integrals, Longman’s method may be used. … ►

Example. Laplace Transform Inversion

… ►

§3.5(ix) Other Contour Integrals

… ►A special case is the rule for Hilbert transforms (§1.14(v)): …

34: 9 Airy and Related Functions

…

35: 17.18 Methods of Computation

… ►The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. …Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. …

36: 10.74 Methods of Computation

… ►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … ►Temme (1997) shows how to overcome this difficulty by use of the Maclaurin expansions for these coefficients or by use of auxiliary functions. … ►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods. … ►The integral representation used is based on (10.32.8). … ►Necessary values of the first derivatives of the functions are obtained by the use of (10.6.2), for example. …

37: 29.20 Methods of Computation

… ►The eigenvalues

a^{m}_{\nu}\left(k^{2}\right)

b^{m}_{\nu}\left(k^{2}\right)

, and the Lamé functions

\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)

\mathit{Es}^{m}_{\nu}\left(z,k^{2}\right)

, can be calculated by direct numerical methods applied to the differential equation (29.2.1); see §3.7. … ►The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). …The numerical computations described in Jansen (1977) are based in part upon this method. … ►The eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices

\mathbf{M}

given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). …

38: 9.17 Methods of Computation

… ►For large

|z|

the asymptotic expansions of §§9.7 and 9.12(viii) should be used instead. … ►A comprehensive and powerful approach is to integrate the defining differential equation (9.2.1) by direct numerical methods. … ►In these cases boundary-value methods need to be used instead; see §3.7(iii). … ►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. This method was used in the computation of the tables in §9.9(v). …

39: Morris Newman

… ►Department of Commerce Gold Medal in 1966 for his work on algorithms for solving integral linear systems exactly by using congruence techniques. … ►During his career, Newman also published numerous papers and served as an editor of several journals. …

40: 9.11 Products

… ►Numerically satisfactory triads of solutions can be constructed where needed on

\mathbb{R}

\mathbb{C}

by inspection of the asymptotic expansions supplied in §9.7. … ►

9.11.2

\mathscr{W}\left\{{\operatorname{Ai}}^{2}\left(z\right),\operatorname{Ai}\left% (z\right)\operatorname{Bi}\left(z\right),{\operatorname{Bi}}^{2}\left(z\right)% \right\}=2\pi^{-3}.

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter and $z$ : complex variable
Proof sketch:: Use (9.2.1) and (9.2.7).
Permalink:: http://dlmf.nist.gov/9.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §9.11(ii), §9.11 and Ch.9

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