Lanczos tridiagonalization of a symmetric matrix

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11: 3.5 Quadrature

… ►with a function

f

that is analytic in a strip containing

\mathbb{R}

. … ►where

a<\xi<b

. … ►The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi). ►The Gauss nodes

x_{k}

(the zeros of

p_{n}

) are the eigenvalues of the (symmetric tridiagonal) Jacobi matrix of order

n\times n

: … ►a complex Gauss quadrature formula is available. …

13: Bibliography S

… ►

B. E. Sagan (2001) The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. 2nd edition, Graduate Texts in Mathematics, Vol. 203, Springer-Verlag, New York.

ⓘ

Notes:: First edition published by Wadsworth & Brooks/Cole Advanced Books & Software, 1991.
External Links:: ISBN 0-387-95067-2, MathReview, ZentralBlatt
Referenced by:: §26.19.
Encodings:: BibTeX

… ►

B. I. Schneider, X. Guan, and K. Bartschat (2016) Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation. Adv. Quantum Chem. 72, pp. 95–127.

ⓘ

Referenced by:: §18.38(i).
Encodings:: BibTeX

… ►

R. P. Stanley (1989) Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 (1), pp. 76–115.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (Dennis White), ZentralBlatt
Referenced by:: §18.37(iii).
Encodings:: BibTeX

… ►

J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

ⓘ

Notes:: For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.
External Links:: Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

G. W. Stewart (2001) Matrix Algorithms. Vol. 2: Eigensystems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

ⓘ

External Links:: ISBN 0-89871-503-2, MathReview Entry, ZentralBlatt
Referenced by:: §3.2(vi).
Encodings:: BibTeX

…

14: 29.20 Methods of Computation

… ►A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. …(Equation (29.6.3) serves as a check.) … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). … ►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. … ►The eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices

\mathbf{M}

given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). …

15: 3.7 Ordinary Differential Equations

… ►where

\mathbf{A}(\tau,z)

is the matrix … ►Let

\mathbf{A}_{P}

be the

(2P)\times(2P+2)

band matrix … ►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. … ►Let

(a,b)

be a finite or infinite interval and

q(x)

be a real-valued continuous (or piecewise continuous) function on the closure of

(a,b)

. … ►This converts the problem into a tridiagonal matrix problem in which the elements of the matrix are polynomials in

\lambda

; compare §3.2(vi). …

16: 18.39 Applications in the Physical Sciences

… ►defines the potential for a symmetric restoring force

-kx

for displacements from

x=0

. … ►The technique to accomplish this follows the DVR idea, in which methods are based on finding tridiagonal representations of the co-ordinate,

x

. Here tridiagonal representations of simple Schrödinger operators play a similar role. …is tridiagonalized in the complete

L^{2}

non-orthogonal (with measure

\,\mathrm{d}r

r\in[0,\infty)

) basis of Laguerre functions: … ►While

s

in the basis of (18.39.44) is simply a variational parameter, care must be taken, or the relationship between the results of the matrix variational approximation and the Pollaczek polynomials is lost, although this has no effect on the use of the variational approximations Reinhardt (2021a, b). …

17: 21.5 Modular Transformations

… ►is a symplectic matrix, that is, … ►(

\mathbf{A}

invertible with integer elements.) …(

\mathbf{B}

symmetric with integer elements and even diagonal elements.) …(

\mathbf{B}

symmetric with integer elements.) …For a

g\times g

matrix

{\bf A}

we define

\operatorname{diag}\mathbf{A}

, as a column vector with the diagonal entries as elements. …

18: 35.4 Partitions and Zonal Polynomials

… ►A partition

\kappa=(k_{1},\dots,k_{m})

is a vector of nonnegative integers, listed in nonincreasing order. Also,

|\kappa|

denotes

k_{1}+\dots+k_{m}

, the weight of

\kappa

;

\ell(\kappa)

denotes the number of nonzero

k_{j}

;

a+\kappa

denotes the vector

(a+k_{1},\dots,a+k_{m})

. … ►where

{\left(a\right)_{k}}=a(a+1)\cdots(a+k-1)

. … ►Therefore

Z_{\kappa}\left(\mathbf{T}\right)

is a symmetric polynomial in the eigenvalues of

\mathbf{T}

. … ►For

\mathbf{T}\in{\boldsymbol{\Omega}}

and

\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)

, …

19: 21.1 Special Notation

… ► ►►►►►►►

$g, h$	positive integers.
…
$\boldsymbol{{\Omega}}$	$g\times g$ complex, symmetric matrix with $\Im\boldsymbol{{\Omega}}$ strictly positive definite, i.e., a Riemann matrix.
…
$a_{j}$	$j$ th element of vector $\mathbf{a}$ .
$A_{jk}$	$(j,k)$ th element of matrix $\mathbf{A}$ .
…
$\boldsymbol{{0}}_{g}$	$g\times g$ zero matrix.
$\mathbf{I}_{g}$	$g\times g$ identity matrix.
…

…

20: 35.3 Multivariate Gamma and Beta Functions

… ►

35.3.1

\Gamma_{m}\left(a\right)=\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-% \mathbf{X}\right)\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\,\mathrm{d}{% \mathbf{X}},

\Re\left(a\right)>\frac{1}{2}(m-1)

ⓘ

Symbols:: $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $a$ : complex variable, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.3.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §35.3(i), §35.3 and Ch.35

►

35.3.2

\Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{\boldsymbol{\Omega}}% \operatorname{etr}\left(-\mathbf{X}\right)\left|\mathbf{X}\right|^{s_{m}-\frac% {1}{2}(m+1)}\prod_{j=1}^{m-1}|(\mathbf{X})_{j}|^{s_{j}-s_{j+1}}\,\mathrm{d}{% \mathbf{X}},

s_{j}\in\mathbb{C}

\Re\left(s_{j}\right)>\frac{1}{2}(j-1)

j=1,\dots,m

►

35.3.3

\mathrm{B}_{m}\left(a,b\right)=\int\limits_{\boldsymbol{{0}}<\mathbf{X}<% \mathbf{I}}\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}-% \mathbf{X}\right|^{b-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},

\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)

ⓘ

Defines:: $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function
Symbols:: $\int$ : integral, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, $b$ : complex variable, $<$ : less-than for matrices, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Keywords:: definition, multivariate beta function
Permalink:: http://dlmf.nist.gov/35.3.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §35.3(i), §35.3 and Ch.35

… ►

35.3.6

\Gamma_{m}\left(a,\dots,a\right)=\Gamma_{m}\left(a\right).

ⓘ

Symbols:: $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $a$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.3.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §35.3(ii), §35.3 and Ch.35

… ►

35.3.8

\mathrm{B}_{m}\left(a,b\right)=\int_{\boldsymbol{\Omega}}\left|\mathbf{X}% \right|^{a-\frac{1}{2}(m+1)}\left|\mathbf{I}+\mathbf{X}\right|^{-(a+b)}\,% \mathrm{d}{\mathbf{X}},

\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)

ⓘ

Symbols:: $\int$ : integral, $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $b$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.3.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §35.3(ii), §35.3 and Ch.35

Lanczos tridiagonalization of a symmetric matrix

11—20 of 895 matching pages

11: 3.5 Quadrature

12: 35.2 Laplace Transform

§35.2 Laplace Transform

Definition

Inversion Formula

Convolution Theorem