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21: 3.2 Linear Algebra

… ►where $u_j=c_j$, $j=1,2,\mathrm{\dots },n-1$, $d_1=b_1$, and …Forward elimination for solving $𝐀𝐱=𝐟$ then becomes $y_1=f_1$, …and back substitution is $x_n=y_n/d_n$, followed by … ►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 … ►Start with ${𝐯}_0=\mathrm{𝟎}$, vector ${𝐯}_1$ such that ${𝐯}_1^{\mathrm{T}}𝐒{𝐯}_1=1$, ${\alpha }_1={𝐯}_1^{\mathrm{T}}𝐀{𝐯}_1$, ${\beta }_1=0$. …

22: 3.11 Approximation Techniques

… ►Beginning with $u_{n+1}=0$, $u_n=c_n$, we apply … ►With $b_0=1$, the last $q$ equations give $b_1,\mathrm{\dots },b_q$ as the solution of a system of linear equations. … ►(3.11.29) is a system of $n+1$ linear equations for the coefficients $a_0,a_1,\mathrm{\dots },a_n$. … ►With this choice of $a_k$ and $f_j=f(x_j)$, the corresponding sum (3.11.32) vanishes. … ►Two are endpoints: $(x_0,y_0)$ and $(x_3,y_3)$; the other points $(x_1,y_1)$ and $(x_2,y_2)$ are control points. …

23: 21.1 Special Notation

… ► ►►►►►►►

$g,h$	positive integers.
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$a_j$	$j$th element of vector $𝐚$.
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$\mathrm{diag}𝐀$	Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$.
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${𝐉}_{2g}$	.
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$S_1{S}_2$	set of all elements of the form “$\text{element of }{S}_1\times \text{element of }{S}_2$”.
$S_1/S_2$	set of all elements of $S_1$, modulo elements of $S_2$. Thus two elements of $S_1/S_2$ are equivalent if they are both in $S_1$ and their difference is in $S_2$. (For an example see §20.12(ii).)
…

…

24: 19.29 Reduction of General Elliptic Integrals

… ►Let …where … ►Next, for $j=1,2$, define $Q_j(t)=f_j+g_{j}t+h_j{t}^2$, and assume both $Q$’s are positive for . …where …If $Q_1(t)=(a_1+b_{1}t)(a_2+b_{2}t)$, where both linear factors are positive for , and $Q_2(t)=f_2+g_{2}t+h_2{t}^2$, then (19.29.25) is modified so that …

25: Bibliography K

… ►

Evaluation of complex zeros of Bessel functions $J_{\nu }(z)$ and $I_{\nu }(z)$ and their derivatives. Zh. Vychisl. Mat. i Mat. Fiz. 24 (10), pp. 1497–1513.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 24(5), pp. 131–141

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi), 4th item.

Encodings:

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The asymptotic expansion of a hypergeometric function ${}_2{}^{}F_2^{}(1,\alpha ;{\rho }_1,{\rho }_2;z)$ . Math. Comp. 26 (120), pp. 963.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§16.11(ii), §16.11(ii).

Encodings:

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An extension of Saalschütz’s summation theorem for the series ${}_{r+3}{}^{}F_{r+2}^{}$ . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Christian Lavault), ZentralBlatt

Referenced by:

§16.4(ii), §16.4(ii).

Encodings:

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On the complex zeros of $H_{\mu }(z),J_{\mu }^{\prime }(z),J_{\mu }^{\prime \prime }(z)$ for real or complex order. J. Comput. Appl. Math. 40 (3), pp. 337–344.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

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Some special cases of the generalized hypergeometric function ${}_{q+1}{}^{}F_q^{}$ . J. Comput. Appl. Math. 78 (1), pp. 79–95.

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External Links:

ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§16.7.

Encodings:

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26: Bibliography E

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The penetration of a potential barrier by electrons. Phys. Rev. 35 (11), pp. 1303–1309.

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External Links:

Document, ZentralBlatt

Referenced by:

§15.18.

Encodings:

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Painlevé transcendent describes quantum correlation function of the $XXZ$ antiferromagnet away from the free-fermion point. J. Phys. A 29 (17), pp. 5619–5626.

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External Links:

ISSN 0305-4470, Document, MathReview (Estelle L. Basor), ZentralBlatt

Referenced by:

§32.16.

Encodings:

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Institutiones Calculi Integralis. Opera Omnia (1), Vol. 11, pp. 110–113.

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Referenced by:

§25.12(i).

Encodings:

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The Sobolev orthogonality and spectral analysis of the Laguerre polynomials $\{L_n^{-k}\}$ for positive integers $k$ . J. Comput. Appl. Math. 171 (1-2), pp. 199–234.

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External Links:

ISSN 0377-0427, Document, Link, MathReview (Khélifa Trimèche)

Referenced by:

§18.36(v).

Encodings:

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Note on the $X_1$-Laguerre orthogonal polynomials.

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Notes:

arXiv:0811.3559v2

Referenced by:

§18.36(vi).

Encodings:

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27: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

► ►►►

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
…
$11\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$-\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$-(2-\sqrt{3})$	$\sqrt{2}(\sqrt{3}+1)$	$-\sqrt{2}(\sqrt{3}-1)$	$-(2+\sqrt{3})$
…

►

►

4.17.1 $$\underset{z\to 0}{lim}\frac{\sin z}{z}=1,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.74

Permalink:

http://dlmf.nist.gov/4.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

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4.17.2 $$\underset{z\to 0}{lim}\frac{\tan z}{z}=1.$$

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Symbols:

$\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.75

Permalink:

http://dlmf.nist.gov/4.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

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4.17.3 $$\underset{z\to 0}{lim}\frac{1-\cos z}{z^2}=\frac{1}{2}.$$

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Symbols:

$\cos z$: cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

28: Bibliography H

… ►

La $q$-conjecture de Macdonald-Morris pour $G_2$ . C. R. Acad. Sci. Paris Sér. I Math. 303 (6), pp. 211–213 (French).

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External Links:

ISSN 0249-6291, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.14.

Encodings:

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25D Table of the First One Hundred Values of $j_{0,s}$,$J_1(j_{0,s})$, $j_{1,s}$,$J_0(j_{1,s})=J_0(j_{0,s+1}^{\prime })$,$j_{1,s}^{\prime }$,$J_1(j_{1,s}^{\prime })$ . Technical report Department of Physics, Worcester Polytechnic Institute, Worcester, MA.

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Notes:

Ms. of six pages deposited in the UMT file

Referenced by:

8th item.

Encodings:

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Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.

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External Links:

Document

Referenced by:

§15.18.

Encodings:

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… ►

Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(xiv).

Encodings:

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… ►

Algorithm 571: Statistics for von Mises’ and Fisher’s distributions of directions: $I_1(x)/I_0(x)$, $I_{1.5}(x)/I_{0.5}(x)$ and their inverses [S14]. ACM Trans. Math. Software 7 (2), pp. 233–238.

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Notes:

Single-precision Fortran, minimum accuracy 8S.

External Links:

ISSN 0098-3500, Document, GAMS (module 571; package TOMS)

Referenced by:

5th item.

Encodings:

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29: 24.19 Methods of Computation

… ►Equations (24.5.3) and (24.5.4) enable $B_n$ and $E_n$ to be computed by recurrence. …For example, the tangent numbers $T_n$ can be generated by simple recurrence relations obtained from (24.15.3), then (24.15.4) is applied. … ►For other information see Chellali (1988) and Zhang and Jin (1996, pp. 1–11). For algorithms for computing $B_n$, $E_n$, $B_n\left(x\right)$, and $E_n\left(x\right)$ see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180). ►

§24.19(ii) Values of $B_n$ Modulo $p$

…

30: Bibliography O

… ►

Studies on the Painlevé equations. III. Second and fourth Painlevé equations, $P_{\mathrm{II}}$ and $P_{\mathrm{IV}}$ . Math. Ann. 275 (2), pp. 221–255.

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External Links:

ISSN 0025-5831, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(ii), §32.10(iv), §32.6(iii), §32.6(v), §32.7(viii).

Encodings:

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Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{\mathrm{VI}}$ . Ann. Mat. Pura Appl. (4) 146, pp. 337–381.

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External Links:

ISSN 0003-4622, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(vi), §32.6(v), §32.7(vii), §32.7(viii).

Encodings:

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Studies on the Painlevé equations. II. Fifth Painlevé equation $P_{\mathrm{V}}$ . Japan. J. Math. (N.S.) 13 (1), pp. 47–76.

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External Links:

ISSN 0289-2316, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(v), §32.6(v), §32.7(viii).

Encodings:

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Studies on the Painlevé equations. IV. Third Painlevé equation $P_{\mathrm{III}}$ . Funkcial. Ekvac. 30 (2-3), pp. 305–332.

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External Links:

ISSN 0532-8721, FullText, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(iii), §32.2(iii), §32.6(iv), §32.7(viii).

Encodings:

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Numerical solution of Riemann-Hilbert problems: Painlevé II. Found. Comput. Math. 11 (2), pp. 153–179.

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External Links:

ISSN 1615-3375, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§32.17, §32.17.

Encodings:

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