######
§27.19 Methods of Computation: Factorization

…

►Type I probabilistic algorithms include the

*Brent–Pollard rho algorithm* (also called

*Monte Carlo method*), the

*Pollard $p\mathrm{-}\mathrm{1}$ algorithm*, and the

*Elliptic Curve Method* (

ecm).
…As of January 2009 the largest prime factors found by these

methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard

$p-1$, and a 67-digit prime for

ecm.
…

######
§34.9 Graphical Method

►The graphical

method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression.
…For an account of this

method see

Brink and Satchler (1993, Chapter VII).
For specific examples of the graphical

method of representing sums involving the

$\mathit{3}j,\mathit{6}j$, and

$\mathit{9}j$ symbols, see

Varshalovich et al. (1988, Chapters 11, 12) and

Lehman and O’Connell (1973, §3.3).

######
§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).

######
§12.18 Methods of Computation

►Because PCFs are special cases of confluent hypergeometric functions, the

methods of computation described in §

13.29 are applicable to PCFs.
…

######
§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).
…

######
§16.25 Methods of Computation

►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations.
…

######
§18.40 Methods of Computation

…

►Usually, however, other

methods are more efficient, especially the numerical solution of difference equations (§

3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree.
…

######
§32.17 Methods of Computation

►The Painlevé equations can be integrated by Runge–Kutta

methods for ordinary differential equations; see §

3.7(v),

Hairer et al. (2000), and

Butcher (2003).
…

######
§29.20 Methods of Computation

…

►A second approach is to solve the continued-fraction equations typified by (

29.3.10) by Newton’s rule or other iterative

methods; see §

3.8.
…

►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.

►A fourth

method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree.
…