# computation by recursion

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## 1—10 of 21 matching pages

##### 1: 7.22 Methods of Computation

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►The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing
${\mathrm{i}}^{n}\mathrm{erfc}\left(x\right)$.
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##### 2: 3.6 Linear Difference Equations

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►with ${a}_{n}\ne 0$, $\forall n$, can be computed recursively for $n=2,3,\mathrm{\dots}$.
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►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.
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►Then computation of ${w}_{n}$ by forward recursion is unstable.
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###### 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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##### 4: 3.9 Acceleration of Convergence

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

##### 7: 3.11 Approximation Techniques

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►For the recursive computation of ${[n+k/k]}_{f}$ by Wynn’s epsilon algorithm, see (3.9.11) and the subsequent text.
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##### 8: 8.25 Methods of Computation

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►Stable recursive schemes for the computation of ${E}_{p}\left(x\right)$ are described in Miller (1960) for $x>0$ and integer $p$.
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##### 9: 18.40 Methods of Computation

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►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let $N$ be a positive integer and define
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##### 10: 34.13 Methods of Computation

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►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).
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