backward recurrence

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1: 5.21 Methods of Computation

… ►An effective way of computing $\mathrm{\Gamma }\left(z\right)$ in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). …For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). …

2: 11.13 Methods of Computation

… ►In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …

3: 10.74 Methods of Computation

… ►Then $J_n\left(x\right)$ and $Y_n\left(x\right)$ can be generated by either forward or backward recurrence on $n$ when , but if $n>x$ then to maintain stability $J_n\left(x\right)$ has to be generated by backward recurrence on $n$, and $Y_n\left(x\right)$ has to be generated by forward recurrence on $n$. …

4: 3.10 Continued Fractions

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Backward Recurrence Algorithm

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5: Bibliography O

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Note on backward recurrence algorithms. Math. Comp. 26 (120), pp. 941–947.

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External Links:

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Referenced by:

§10.74(iv), §3.6(v).

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6: 16.25 Methods of Computation

… ►In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …

7: 15.19 Methods of Computation

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§15.19(iv) Recurrence Relations

… ►For example, in the half-plane $\mathrm{\Re }z\le \frac{1}{2}$ we can use (15.12.2) or (15.12.3) to compute $F(a,b;c+N+1;z)$ and $F(a,b;c+N;z)$, where $N$ is a large positive integer, and then apply (15.5.18) in the backward direction. …

8: 3.6 Linear Difference Equations

… ► … ►Because the recessive solution of a homogeneous equation is the fastest growing solution in the backward direction, it occurred to J. …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 $w_n$ is generated by backward recursion from … ►Thus in the inhomogeneous case it may sometimes be necessary to recur backwards to achieve stability. …

9: 18.2 General Orthogonal Polynomials

… ►If the orthogonality discrete set $X$ is $\{0,1,\mathrm{\dots },N\}$ or $\{0,1,2,\mathrm{\dots }\}$, then the role of the differentiation operator $d/dx$ in the case of classical OP’s (§18.3) is played by ${\mathrm{\Delta }}_x$, the forward-difference operator, or by ${\nabla }_x$, the backward-difference operator; compare §18.1(i). … ►

§18.2(iv) Recurrence Relations

… ►the monic recurrence relations (18.2.8) and (18.2.10) take the form … ►Then, with the coefficients (18.2.11_4) associated with the monic OP’s $p_n$, the orthonormal recurrence relation for $q_n$ takes the form … ►The recurrence relations (18.2.10) can be equivalently written as …