computation by recursion

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1: 7.22 Methods of Computation

… ►The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$. …

2: 3.6 Linear Difference Equations

… ►with $a_n\ne 0$, $\forall n$, can be computed recursively for $n=2,3,\mathrm{\dots }$. … ►A “trial solution” is then computed by backward recursion, in the course of which the original components of the unwanted solution $g_n$ die away. … ►Then computation of $w_n$ by forward recursion is unstable. … ►

Example 1. Bessel Functions

… ►Thus $Y_n\left(1\right)$ is dominant and can be computed by forward recursion, whereas $J_n\left(1\right)$ is recessive and has to be computed by backward recursion. …

3: Bibliography W

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Algorithm 44: Bessel functions computed recursively. Comm. ACM 4 (4), pp. 177–178.

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Notes:

Includes Algol code, maximum accuracy 8S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

Encodings:

BibTeX

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4: 3.9 Acceleration of Convergence

… ►The ratio of the Hankel determinants in (3.9.9) can be computed recursively by Wynn’s epsilon algorithm: …

5: Bibliography G

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Recursive computation of the repeated integrals of the error function. Math. Comp. 15 (75), pp. 227–232.

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External Links:

FullText, MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§7.22(iii).

Encodings:

BibTeX

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The ABC of hyper recursions. J. Comput. Appl. Math. 190 (1-2), pp. 270–286.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§15.19(iv).

Encodings:

BibTeX

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6: 6.20 Approximations

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Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$-function (§13.2(i)) from which Chebyshev expansions near infinity for $E_1\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the $U$ functions. If the scheme can be used in backward direction.

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7: 3.11 Approximation Techniques

… ►For the recursive computation of ${[n+k/k]}_f$ by Wynn’s epsilon algorithm, see (3.9.11) and the subsequent text. …

8: 8.25 Methods of Computation

… ►Stable recursive schemes for the computation of $E_p\left(x\right)$ are described in Miller (1960) for $x>0$ and integer $p$. …

9: 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). …

10: 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. These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …