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1: 16.25 Methods of Computation

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

2: 5.21 Methods of Computation

… ►Or we can use forward recurrence, with an initial value obtained e. …

3: 11.13 Methods of Computation

… ►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. …

4: 3.6 Linear Difference Equations

… ►where $\mathrm{\Delta }{w}_{n-1}=w_n-w_{n-1}$, ${\mathrm{\Delta }}^2{w}_{n-1}=\mathrm{\Delta }{w}_n-\mathrm{\Delta }{w}_{n-1}$, and $n\in \mathbb{Z}$. … ►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 computation of $w_n$ by forward recursion is unstable. … ►(This part of the process is equivalent to forward elimination.) … ►Within this framework forward and backward recursion may be regarded as the special cases $\mathrm{\ell }=0$ and $\mathrm{\ell }=k$, respectively. …

5: 3.9 Acceleration of Convergence

… ►

3.9.2 $$S=\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{2}^{-k-1}{\mathrm{\Delta }}^k{a}_0,$$

ⓘ

Symbols:

$\mathrm{\Delta }$: forward difference operator and $S$: a convergent series

Permalink:

http://dlmf.nist.gov/3.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

… ►Here $\mathrm{\Delta }$ is the forward difference operator: ►

3.9.3 $${\mathrm{\Delta }}^k{a}_0={\mathrm{\Delta }}^{k-1}{a}_1-{\mathrm{\Delta }}^{k-1}{a}_0,$$ $k=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Delta }$: forward difference operator

A&S Ref:

3.6.27

Permalink:

http://dlmf.nist.gov/3.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

… ►

3.9.4 $${\mathrm{\Delta }}^k{a}_0=\sum _{m=0}^k{(-1)}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)a_{k-m}.$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $\mathrm{\Delta }$: forward difference operator

A&S Ref:

3.6.27

Permalink:

http://dlmf.nist.gov/3.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

…

6: 3.10 Continued Fractions

… ►

Forward Recurrence Algorithm

… ►In general this algorithm is more stable than the forward algorithm; see Jones and Thron (1974). ►

Forward Series Recurrence Algorithm

… ►In Gautschi (1979c) the forward series algorithm is used for the evaluation of a continued fraction of an incomplete gamma function (see §8.9). … ►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. …

7: 7.22 Methods of Computation

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

8: 18.1 Notation

… ►Forward differences: ►

$${\mathrm{\Delta }}_x\left(f(x)\right)=f(x+1)-f(x),$$

►

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

…

9: 18.22 Hahn Class: Recurrence Relations and Differences

… ►

18.22.19 $${\mathrm{\Delta }}_x{Q}_n(x;\alpha ,\beta ,N)=-\frac{n(n+\alpha +\beta +1)}{(\alpha +1)N}{Q}_{n-1}(x;\alpha +1,\beta +1,N-1),$$

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $\mathrm{\Delta }$: forward difference operator, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§16.4(iii), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E19

Encodings:

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See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

… ►

18.22.21 $${\mathrm{\Delta }}_x{K}_n(x;p,N)=-\frac{n}{pN}{K}_{n-1}(x;p,N-1),$$

ⓘ

Symbols:

$K_n(x;p,N)$: Krawtchouk polynomial, $\mathrm{\Delta }$: forward difference operator, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.22.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

… ►

18.22.23 $${\mathrm{\Delta }}_x{M}_n(x;\beta ,c)=-\frac{n(1-c)}{\beta c}{M}_{n-1}(x;\beta +1,c),$$

ⓘ

Symbols:

$M_n(x;\beta ,c)$: Meixner polynomial, $\mathrm{\Delta }$: forward difference operator, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.22.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

… ►

18.22.25 $${\mathrm{\Delta }}_x{C}_n(x;a)=-\frac{n}{a}{C}_{n-1}(x;a),$$

ⓘ

Symbols:

$C_n(x;a)$: Charlier polynomial, $\mathrm{\Delta }$: forward difference operator, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

…

10: 10.74 Methods of Computation

… ►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)). … ►Similarly, to maintain stability in the interval the integration direction has to be forwards in the case of $I_{\nu }\left(x\right)$ and backwards in the case of $K_{\nu }\left(x\right)$, with initial values obtained in an analogous manner to those for $J_{\nu }\left(x\right)$ and $Y_{\nu }\left(x\right)$. … ►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 $n$. …