recursion relations

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11: 18.39 Applications in the Physical Sciences

… ►Table 18.39.1 lists typical non-classical weight functions, many related to the non-classical Freud weights of §18.32, and §32.15, all of which require numerical computation of the recursion coefficients (i. … ►with matrix eigenvalues $ϵ={ϵ}_i^N$, $i=1,2,\mathrm{\dots },N$, and the eigenvectors, $𝐜(ϵ)=(c_0(ϵ),c_1(ϵ),\mathrm{\dots },c_{N-1}(ϵ))$, are determined by the recursion relation (18.39.46) below. …

12: 8.25 Methods of Computation

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§8.25(v) Recurrence Relations

… ►An efficient procedure, based partly on the recurrence relations (8.8.5) and (8.8.6), is described in Gautschi (1979b, 1999). ►Stable recursive schemes for the computation of $E_p\left(x\right)$ are described in Miller (1960) for $x>0$ and integer $p$. …

13: 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. … ►Then computation of $w_n$ by forward recursion is unstable. … ►Then $w_n$ is generated by backward recursion from … ►

Example 1. Bessel Functions

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14: Bibliography G

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Recursive computation of the repeated integrals of the error function. Math. Comp. 15 (75), pp. 227–232.

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External Links:

FullText, MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§7.22(iii).

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Computational aspects of three-term recurrence relations. SIAM Rev. 9 (1), pp. 24–82.

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External Links:

Document, MathReview (E. Frank), ZentralBlatt

Referenced by:

§10.74(iv), §14.32, §3.10(iii), §3.6(iii).

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A note on the recursive calculation of incomplete gamma functions. ACM Trans. Math. Software 25 (1), pp. 101–107.

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External Links:

ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

§8.25(v).

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The ABC of hyper recursions. J. Comput. Appl. Math. 190 (1-2), pp. 270–286.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§15.19(iv).

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Numerically satisfactory solutions of hypergeometric recursions. Math. Comp. 76 (259), pp. 1449–1468.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview Entry, ZentralBlatt

Referenced by:

§15.19(iv).

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15: 18.2 General Orthogonal Polynomials

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§18.2(iii) Standardization and Related Constants

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§18.2(iv) Recurrence Relations

… ►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. … ►are OP’s with orthogonality relation … ►Alternatives for numerical calculation of the recursion coefficients in terms of the moments are discussed in these references, and in §18.40(ii). …

16: 29.6 Fourier Series

… ►In addition, if $H$ satisfies (29.6.2), then (29.6.3) applies. … ►This solution can be constructed from (29.6.4) by backward recursion, starting with $A_{2n+2}=0$ and an arbitrary nonzero value of $A_{2n}$, followed by normalization via (29.6.5) and (29.6.6). Consequently, ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$ reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). …

17: Bibliography W

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

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Recursion formulae for hypergeometric functions. Math. Comp. 22 (102), pp. 363–373.

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External Links:

FullText, Document, MathReview (S. D. Bajpai), ZentralBlatt

Referenced by:

§16.3(ii).

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Computation with Recurrence Relations. Pitman, Boston, MA.

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External Links:

ISBN 0-273-08508-5, MathReview (S. Conde), ZentralBlatt

Referenced by:

§13.29(iv), §16.25, §2.9(iii), §3.10(iii), §3.6(iii), §3.6(v), §3.6(vii).

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Algorithm 44: Bessel functions computed recursively. Comm. ACM 4 (4), pp. 177–178.

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Notes:

Includes Algol code, maximum accuracy 8S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

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Upon systems of recursions which obtain among the quotients of the Padé table. Numer. Math. 8 (3), pp. 264–269.

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External Links:

ISSN 0029-599X, Document, MathReview (F. L. Bauer), ZentralBlatt

Referenced by:

§3.11(iv).

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18: 7.17 Inverse Error Functions

… ►where $a_0=1$ and the other coefficients follow from the recursion …

19: 3.2 Linear Algebra

… ►Define the Lanczos vectors ${𝐯}_j$ and coefficients ${\alpha }_j$ and ${\beta }_j$ by ${𝐯}_0=\mathrm{𝟎}$, a normalized vector ${𝐯}_1$ (perhaps chosen randomly), ${\alpha }_1={𝐯}_1^{\mathrm{T}}𝐀{𝐯}_1$, ${\beta }_1=0$, and for $j=1,2,\mathrm{\dots },n-1$ by the recursive scheme …Its characteristic polynomial can be obtained from the recursion … ►Lanczos’ method is related to Gauss quadrature considered in §3.5(v). When the matrix $𝐀$ is replaced by a scalar $x$, the recurrence relation in the first line of (3.2.21) with $𝐮={\beta }_{j+1}{𝐯}_{j+1}$ is similar to the one in (3.5.30_5). Also, the recurrence relations in (3.2.23) and (3.5.30) are similar, as well as the matrix $𝐁$ in (3.2.22) and the Jacobi matrix ${𝐉}_n$ in (3.5.31). …

20: Bibliography S

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A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.

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External Links:

ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt

Referenced by:

§28.32(i).

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Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.

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External Links:

FullText, Document, MathReview (L. Fox), ZentralBlatt

Referenced by:

3rd item.

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Recursive evaluation of $3j$- and $6j$- coefficients. Comput. Phys. Comm. 11 (2), pp. 269–278.

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Notes:

Double-precision Fortran provided in 3 parts, $j_1$-recursion for $3j$-coefficients, $m_2$-recursion for $3j$-coefficients, and $j_1$-recursion for $6j$-coefficients

External Links:

Document, Code 1, Code 2, Code 3

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8th item.

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Exact recursive evaluation of $3j$- and $6j$-coefficients for quantum-mechanical coupling of angular momenta. J. Mathematical Phys. 16 (10), pp. 1961–1970.

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External Links:

Document, MathReview (J.-L. Calais)

Referenced by:

§34.13, §34.3(iii).

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Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros. Math. Comp. 70 (235), pp. 1205–1220.

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External Links:

ISSN 0025-5718, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.74(vi).

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