backward%20recursion

SearchAdvancedHelp

(0.001 seconds)

11—20 of 140 matching pages

11: Bibliography O

… ►

An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

ⓘ

External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

FullText, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.74(iv), §3.6(v).

Encodings:

BibTeX

…

12: 20 Theta Functions

Chapter 20 Theta Functions

…

13: 18.40 Methods of Computation

… ►

A numerical approach to the recursion coefficients and quadrature abscissas and weights

… ►and these can be used for the recursion coefficients …See Gautschi (1983) for examples of numerically stable and unstable use of the above recursion relations, and how one can then usefully differentiate between numerical results of low and high precision, as produced thereby. … ►Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. … ►where the coefficients are defined recursively via $a_1=\frac{x_{1,N}}{x_{2,N}}-1$, and …

14: 18.22 Hahn Class: Recurrence Relations and Differences

… ►

18.22.20 $${\nabla }_x\left(\frac{{\left(\alpha +1\right)}_x{\left(\beta +1\right)}_{N-x}}{x!(N-x)!}{Q}_n(x;\alpha ,\beta ,N)\right)=\frac{N+1}{\beta }\frac{{\left(\alpha \right)}_x{\left(\beta \right)}_{N+1-x}}{x!(N+1-x)!}{Q}_{n+1}(x;\alpha -1,\beta -1,N+1).$$

ⓘ

Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\nabla }_x$: backward difference, $!$: factorial (as in $n!$), $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§16.4(iii)

Permalink:

http://dlmf.nist.gov/18.22.E20

Encodings:

pMML, png, TeX

See also:

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

… ►

18.22.22 $${\nabla }_x\left(\left(\genfrac{}{}{0.0pt}{}{N}{x}\right)p^x{(1-p)}^{N-x}{K}_n(x;p,N)\right)=\left(\genfrac{}{}{0.0pt}{}{N+1}{x}\right)p^x{(1-p)}^{N-x}{K}_{n+1}(x;p,N+1).$$

ⓘ

Symbols:

$K_n(x;p,N)$: Krawtchouk polynomial, ${\nabla }_x$: backward difference, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.22.E22

Encodings:

pMML, png, TeX

See also:

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

… ►

18.22.24 $${\nabla }_x\left(\frac{{\left(\beta \right)}_x{c}^x}{x!}{M}_n(x;\beta ,c)\right)=\frac{{\left(\beta -1\right)}_x{c}^x}{x!}{M}_{n+1}(x;\beta -1,c).$$

ⓘ

Symbols:

$M_n(x;\beta ,c)$: Meixner polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\nabla }_x$: backward difference, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E24

Encodings:

pMML, png, TeX

See also:

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

… ►

18.22.26 $${\nabla }_x\left(\frac{a^x}{x!}{C}_n(x;a)\right)=\frac{a^x}{x!}{C}_{n+1}(x;a).$$

ⓘ

Symbols:

$C_n(x;a)$: Charlier polynomial, ${\nabla }_x$: backward difference, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E26

Encodings:

pMML, png, TeX

See also:

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

…

15: 12.18 Methods of Computation

… ►These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …

16: 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 $n$. …

17: 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). … ►The monic and orthonormal OP’s, and their determination via recursion, are more fully discussed in §§3.5(v) and 3.5(vi), where modified recursion coefficients are listed for the classical OP’s in their monic and orthonormal forms. … ►Alternatives for numerical calculation of the recursion coefficients in terms of the moments are discussed in these references, and in §18.40(ii). … ►where the first indicates that the indices of the recursion coefficients ${\alpha }_n$, ${\beta }_n$ of (18.2.31) have been incremented by $1$, when compared to those of (18.2.11_5). … ►The OP’s $p_n^{(1)}(x)$ may also be calculated from the original recursion (18.2.11_5), but with independent initial conditions for $p_0,p_1$: …

18: 25.6 Integer Arguments

… ►

25.6.3 $$\zeta \left(-n\right)=-\frac{B_{n+1}}{n+1},$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function and $n$: nonnegative integer

Source:

Apostol (1976, (20), p. 266); with $n\ne 0$

A&S Ref:

23.2.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

… ►

§25.6(iii) Recursion Formulas

…

19: 3.10 Continued Fractions

… ►

Backward Recurrence Algorithm

… ►($\nabla$ is the backward difference operator.) …

20: 18.20 Hahn Class: Explicit Representations

… ►For comments on the use of the forward-difference operator ${\mathrm{\Delta }}_x$, the backward-difference operator ${\nabla }_x$, and the central-difference operator ${\delta }_x$, see §18.2(ii). … ►

18.20.1 $$p_n(x)=\frac{1}{{\kappa }_n{w}_x}{\nabla }_x^n\left(w_x\prod _{\mathrm{\ell }=0}^{n-1}F(x+\mathrm{\ell })\right),$$ $x\in X$.

ⓘ

Symbols:

${\nabla }_x$: backward difference, $\in$: element of, $p_n(x)$: polynomial of degree $n$ , $w_x$: weights, $X$: subset of $ℝ$ , $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $F(x)$: coefficient and ${\kappa }_n$: coefficient

A&S Ref:

22.17.2

Referenced by:

§18.20(i), Table 18.20.1

Permalink:

http://dlmf.nist.gov/18.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

…