forward series recurrence

SearchAdvancedHelp

(0.002 seconds)

5 matching pages

1: 3.10 Continued Fractions

… ►

Forward Series Recurrence Algorithm

… ►This forward algorithm achieves efficiency and stability in the computation of the convergents $C_n=A_n/B_n$, and is related to the forward series recurrence algorithm. …

2: 16.25 Methods of Computation

… ►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …

3: 2.9 Difference Equations

… ►Many special functions that depend on parameters satisfy a three-term linear recurrence relation … ►

2.9.2 $${\mathrm{\Delta }}^{2}w(n)+(2+f(n))\mathrm{\Delta }w(n)+(1+f(n)+g(n))w(n)=0,$$ $n=0,1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Delta }$: forward difference operator, $f(n)$: function, $g(n)$: function, $n$: nonnegative integer and $w(n)$: solution

Permalink:

http://dlmf.nist.gov/2.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §2.9(i), §2.9 and Ch.2

►in which $\mathrm{\Delta }$ is the forward difference operator (§3.6(i)). ►Often $f(n)$ and $g(n)$ can be expanded in series … ►For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a). …

4: 10.74 Methods of Computation

… ►

§10.74(i) Series Expansions

… ►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … ►In the interval , $J_{\nu }\left(x\right)$ needs to be integrated in the forward direction and $Y_{\nu }\left(x\right)$ in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)). … ►

§10.74(iv) Recurrence Relations

… ►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 $n$. …

5: 11.13 Methods of Computation

… ►

§11.13(ii) Series Expansions

… ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. 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. …