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11: Bibliography G

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Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.

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Referenced by:

§18.39(iii).

Encodings:

BibTeX

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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

Encodings:

BibTeX

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Computing special functions by using quadrature rules. Numer. Algorithms 33 (1-4), pp. 265–275.

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Notes:

International Conference on Numerical Algorithms, Vol. I (Marrakesh, 2001)

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§3.5(ix).

Encodings:

BibTeX

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.

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Notes:

Loose microfiche suppl. A1–A10

External Links:

FullText, MathReview (F. Stenger), ZentralBlatt

Referenced by:

§18.39(iii), §3.5(vi).

Encodings:

BibTeX

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12: 1.11 Zeros of Polynomials

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Descartes’ Rule of Signs

… ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. …

13: 3.4 Differentiation

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$$B_{-2}^5=\frac{1}{120}(6-10t-15t^2+20t^3-5t^4),$$

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$$B_{-3}^6=-\frac{1}{720}(12-8t-45t^2+20t^3+15t^4-6t^5),$$

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$$B_{-2}^6=\frac{1}{60}(9-9t-30t^2+20t^3+5t^4-3t^5),$$

… ►The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2). … ►As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands. …

14: 3.5 Quadrature

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§3.5(i) Trapezoidal Rules

… ►The composite trapezoidal rule is … ►

§3.5(ii) Simpson’s Rule

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§3.5(iv) Interpolatory Quadrature Rules

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15: 8 Incomplete Gamma and Related
Functions

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16: 28 Mathieu Functions and Hill’s Equation

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17: 8.25 Methods of Computation

… ►See Allasia and Besenghi (1987b) for the numerical computation of $\mathrm{\Gamma }(a,z)$ from (8.6.4) by means of the trapezoidal rule. … ►A numerical inversion procedure is also given for calculating the value of $x$ (with 10S accuracy), when $a$ and $P(a,x)$ are specified, based on Newton’s rule (§3.8(ii)). …

18: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

19: 23 Weierstrass Elliptic and Modular
Functions

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20: 9.17 Methods of Computation

… ►The trapezoidal rule (§3.5(i)) is then applied. … ►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. …