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31: 3.9 Acceleration of Convergence

… ►A transformation of a convergent sequence

\{s_{n}\}

with limit

\sigma

into a sequence

\{t_{n}\}

is called limit-preserving if

\{t_{n}\}

converges to the same limit

\sigma

. … ►This transformation is accelerating if

\{s_{n}\}

is a linearly convergent sequence, i. … ►Then the transformation of the sequence

\{s_{n}\}

into a sequence

\{t_{n,2k}\}

is given by … ►Then

t_{n,2k}=\varepsilon_{2k}^{(n)}

. … ►We give a special form of Levin’s transformation in which the sequence

s=\{s_{n}\}

of partial sums

s_{n}=\sum_{j=0}^{n}a_{j}

is transformed into: …

32: Bibliography C

… ►

L. Carlitz (1960) Note on Nörlund’s polynomial

B_{n}^{(z)}

. Proc. Amer. Math. Soc. 11 (3), pp. 452–455.

ⓘ

External Links:: FullText, MathReview (M. P. Drazin), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

… ►

P. A. Clarkson (2003b) The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

… ►

J. A. Cochran (1963) Further formulas for calculating approximate values of the zeros of certain combinations of Bessel functions. IEEE Trans. Microwave Theory Tech. 11 (6), pp. 546–547.

ⓘ

External Links:: ISSN 0018-9480, FullText
Referenced by:: §10.21(x).
Encodings:: BibTeX

… ►

M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

… ►

F. Cooper, A. Khare, and A. Saxena (2006) Exact elliptic compactons in generalized Korteweg-de Vries equations. Complexity 11 (6), pp. 30–34.

ⓘ

External Links:: ISSN 1076-2787, Document, MathReview
Referenced by:: §19.6(i).
Encodings:: BibTeX

…

33: 1.12 Continued Fractions

… ►

A_{n}

and

B_{n}

are called the

n

th (canonical) numerator and denominator respectively. … ►

b_{0}+\displaystyle{\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}}

is equivalent to

b^{\prime}_{0}+\displaystyle{\cfrac{a^{\prime}_{1}}{b^{\prime}_{1}+\cfrac{a^{% \prime}_{2}}{b^{\prime}_{2}+\cdots}}}

if there is a sequence

\{d_{n}\}^{\infty}_{n=0}

d_{0}=1

d_{n}\neq 0

, such that … ►Define … ►The continued fraction

\displaystyle{\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}}

converges when … ►Then the convergents

C_{n}

satisfy …

34: 21.1 Special Notation

… ► ►►►►►►►

$g, h$	positive integers.
…
$a_{j}$	$j$ th element of vector $\mathbf{a}$ .
…
$\operatorname{diag}\mathbf{A}$	Transpose of $[A_{11},A_{22},\dots,A_{gg}]$ .
…
$\mathbf{J}_{2g}$	$\begin{bmatrix}\boldsymbol{{0}}_{g}&\mathbf{I}_{g}\\ -\mathbf{I}_{g}&\boldsymbol{{0}}_{g}\end{bmatrix}$ .
…
$S_{1}S_{2}$	set of all elements of the form “ $\mbox{element of $S_{1}$}\times\mbox{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).)
…

…

35: Bibliography B

… ►

R. Barakat (1961) Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.

ⓘ

External Links:: FullText, MathReview (J. C. P. Miller), ZentralBlatt
Referenced by:: §8.7.
Encodings:: BibTeX

… ►

B. C. Berndt, S. Bhargava, and F. G. Garvan (1995) Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

ⓘ

External Links:: ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt
Referenced by:: §15.8(v), §20.11(iii).
Encodings:: BibTeX

… ►

F. Bethuel (1998) Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

ⓘ

External Links:: ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §17.17.
Encodings:: BibTeX

… ►

A. Bhattacharyya and L. Shafai (1988) Theoretical and experimental investigation of the elliptical annual ring antenna. IEEE Trans. Antennas and Propagation 36 (11), pp. 1526–1530.

ⓘ

External Links:: Document
Referenced by:: 5th item.
Encodings:: BibTeX

… ►

R. L. Bishop (1981) Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).

ⓘ

External Links:: ISSN 0035-9149, Document, MathReview
Referenced by:: Sidebar 9.SB1.
Encodings:: BibTeX

…

36: 3.7 Ordinary Differential Equations

… ►The path is partitioned at

P+1

points labeled successively

z_{0},z_{1},\dots,z_{P}

, with

z_{0}=a

z_{P}=b

. … ►Write

\tau_{j}=z_{j+1}-z_{j}

j=0,1,\dots,P

, expand

w(z)

and

w^{\prime}(z)

in Taylor series (§1.10(i)) centered at

z=z_{j}

, and apply (3.7.2). … ►If, for example,

\beta_{0}=\beta_{1}=0

, then on moving the contributions of

w(z_{0})

and

w(z_{P})

to the right-hand side of (3.7.13) the resulting system of equations is not tridiagonal, but can readily be made tridiagonal by annihilating the elements of

\mathbf{A}_{P}

that lie below the main diagonal and its two adjacent diagonals. … ►The values

\lambda_{k}

are the eigenvalues and the corresponding solutions

w_{k}

of the differential equation are the eigenfunctions. … ►where

h=z_{n+1}-z_{n}

and …

37: Software Index

… ► ►►►►►►►

	Open Source											With Book			Commercial
…
8.28(iv) $\mathrm{B}_{x}\left(a,b\right)$ , $I_{x}\left(a,b\right)$ , $x,a,b\in\mathbb{R}$	✓							✓
…
11 Struve and Related Functions
…
24.21(ii) $B_{n}$ , $B_{n}\left(x\right)$ , $E_{n}$ , $E_{n}\left(x\right)$	✓	✓					✓		✓	a	✓	✓	✓	✓		✓	✓	✓	Derive, MuPAD
…
25.21(v) $\operatorname{Li}_{2}\left(z\right)$ , $\operatorname{Li}_{s}\left(z\right)$	✓		✓	✓	✓	✓		✓		✓	✓	✓	✓			✓	✓	✓
…
30 Spheroidal Wave Functions
…

…

38: 3.6 Linear Difference Equations

… ►Given numerical values of

w_{0}

and

w_{1}

, the solution

w_{n}

of the equation …These errors have the effect of perturbing the solution by unwanted small multiples of

w_{n}

and of an independent solution

g_{n}

, say. … ►The unwanted multiples of

g_{n}

now decay in comparison with

w_{n}

, hence are of little consequence. … ►The latter method is usually superior when the true value of

w_{0}

is zero or pathologically small. … ►beginning with

e_{0}=w_{0}

. …

39: 18.8 Differential Equations

… ►

Table 18.8.1: Classical OP’s: differential equations

A(x)f^{\prime\prime}(x)+B(x)f^{\prime}(x)+C(x)f(x)+\lambda_{n}f(x)=0

► ►►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda_{n}$
…
11	${\mathrm{e}}^{-n^{-1}x}x^{\ell+1}L^{(2\ell+1)}_{n-\ell-1}\left(2n^{-1}x\right)$	$1$	$0$	$\frac{2}{x}-\frac{\ell(\ell+1)}{x^{2}}$	$-\frac{1}{n^{2}}$
12	$H_{n}\left(x\right)$	$1$	$-2x$	$0$	$2n$
…
14	$\mathit{He}_{n}\left(x\right)$	$1$	$-x$	$0$	$n$

► ►Item 11 of Table 18.8.1 yields (18.39.36) for

Z=1

40: 3.2 Linear Algebra

… ►where

u_{j}=c_{j}

j=1,2,\dots,n-1

d_{1}=b_{1}

, and …Forward elimination for solving

\mathbf{A}\mathbf{x}=\mathbf{f}

then becomes

y_{1}=f_{1}

, …and back substitution is

x_{n}=y_{n}/d_{n}

, followed by … ►Define the Lanczos vectors

\mathbf{v}_{j}

and coefficients

\alpha_{j}

and

\beta_{j}

\mathbf{v}_{0}=\boldsymbol{{0}}

, a normalized vector

\mathbf{v}_{1}

(perhaps chosen randomly),

\alpha_{1}=\mathbf{v}_{1}^{\rm T}\mathbf{A}\mathbf{v}_{1}

\beta_{1}=0

, and for

j=1,2,\ldots,n-1

by the recursive scheme … ►Start with

\mathbf{v}_{0}=\boldsymbol{{0}}

, vector

\mathbf{v}_{1}

such that

\mathbf{v}_{1}^{\rm T}\mathbf{S}\mathbf{v}_{1}=1

\alpha_{1}=\mathbf{v}_{1}^{\rm T}\mathbf{A}\mathbf{v}_{1}

\beta_{1}=0

. …