backward recursion

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1: 3.6 Linear Difference Equations

… ► … ►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. … ►If, as

n\to\infty

, the wanted solution

w_{n}

grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable. … ►Then

w_{n}

is generated by backward recursion from … ►Within this framework forward and backward recursion may be regarded as the special cases

\ell=0

and

\ell=k

, respectively. …

2: 7.22 Methods of Computation

… ►See Gautschi (1977a), where forward and backward recursions are used; see also Gautschi (1961). …

3: 29.20 Methods of Computation

… ►Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. …

4: 29.6 Fourier Series

… ►This solution can be constructed from (29.6.4) by backward recursion, starting with

A_{2n+2}=0

and an arbitrary nonzero value of

A_{2n}

, followed by normalization via (29.6.5) and (29.6.6). …

5: 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 $|\operatorname{ph}z|<\pi$ the scheme can be used in backward direction.

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