backward%20recursion

SearchAdvancedHelp

(0.002 seconds)

1—10 of 140 matching pages

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 $n\to \mathrm{\infty }$, 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 $\mathrm{\ell }=0$ and $\mathrm{\ell }=k$, respectively. …

2: 6.20 Approximations

… ►

Cody and Thacher (1968) provides minimax rational approximations for $E_1\left(x\right)$, with accuracies up to 20S.

►

Cody and Thacher (1969) provides minimax rational approximations for $\mathrm{Ei}\left(x\right)$, with accuracies up to 20S.

►

MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$, with accuracies up to 20S.

… ►

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.

…

3: 7.22 Methods of Computation

… ►The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$. See Gautschi (1977a), where forward and backward recursions are used; see also Gautschi (1961). …

4: 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. …

5: 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). …

6: 16.25 Methods of Computation

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

7: Browsers

… ►Although we have attempted to follow standards and maintain backwards compatibility with older browsers, you will generally get the best results by upgrading to the latest version of your preferred browser.

8: 11.13 Methods of Computation

… ►The solution ${𝐊}_{\nu }\left(x\right)$ needs to be integrated backwards for small $x$, and either forwards or backwards for large $x$ depending whether or not $\nu$ exceeds $\frac{1}{2}$. For ${𝐌}_{\nu }\left(x\right)$ both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). … ►In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …

9: 18.1 Notation

… ►Backward differences: ►

$${\nabla }_x\left(f(x)\right)=f(x)-f(x-1),$$

►

$${\nabla }_x^{n+1}\left(f(x)\right)={\nabla }_x\left({\nabla }_x^n(f(x))\right).$$

…

10: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function $p\left(n\right)$ for . … ►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau \left(n\right)$, and the values can be checked by the congruence (27.14.20). …