Lanczos tridiagonalization of a symmetric matrix

(0.005 seconds)

1—10 of 895 matching pages

1: 35.1 Special Notation

$a, b$	complex variables.
…
$\mathbf{Z}$	complex symmetric matrix.
…

►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively

\Gamma_{m}\left(a\right)

and

\mathrm{B}_{m}\left(a,b\right)

, and the special functions of matrix argument: Bessel (of the first kind)

A_{\nu}\left(\mathbf{T}\right)

and (of the second kind)

B_{\nu}\left(\mathbf{T}\right)

; confluent hypergeometric (of the first kind)

{{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)

\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)

and (of the second kind)

\Psi\left(a;b;\mathbf{T}\right)

; Gaussian hypergeometric

{{}_{2}F_{1}}\left(a_{1},a_{2};b;\mathbf{T}\right)

\displaystyle{{}_{2}F_{1}}\left({a_{1},a_{2}\atop b};\mathbf{T}\right)

; generalized hypergeometric

{{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)

\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)

. ►An alternative notation for the multivariate gamma function is

\Pi_{m}(a)=\Gamma_{m}\left(a+\tfrac{1}{2}(m+1)\right)

(Herz (1955, p. 480)). Related notations for the Bessel functions are

\mathcal{J}_{\nu+\frac{1}{2}(m+1)}(\mathbf{T})=A_{\nu}\left(\mathbf{T}\right)/% A_{\nu}\left(\boldsymbol{{0}}\right)

(Faraut and Korányi (1994, pp. 320–329)),

K_{m}(0,\dots,0,\nu\mathpunct{|}\mathbf{S},\mathbf{T})=\left|\mathbf{T}\right|% ^{\nu}B_{\nu}\left(\mathbf{S}\mathbf{T}\right)

(Terras (1988, pp. 49–64)), and

\mathcal{K}_{\nu}(\mathbf{T})=\left|\mathbf{T}\right|^{\nu}B_{\nu}\left(% \mathbf{S}\mathbf{T}\right)

(Faraut and Korányi (1994, pp. 357–358)).

2: 35.5 Bessel Functions of Matrix Argument

§35.5 Bessel Functions of Matrix Argument

►

§35.5(i) Definitions

… ►

§35.5(ii) Properties

… ►

§35.5(iii) Asymptotic Approximations

►For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).

3: 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

►

§35.6(i) Definitions

… ►

Laguerre Form

… ►

§35.6(ii) Properties

… ►

§35.6(iii) Relations to Bessel Functions of Matrix Argument

…

4: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

►

§35.8(i) Definition

… ►The generalized hypergeometric function

{{}_{p}F_{q}}

with matrix argument

\mathbf{T}\in\boldsymbol{\mathcal{S}}

, numerator parameters

a_{1},\dots,a_{p}

, and denominator parameters

b_{1},\dots,b_{q}

is … ►

Convergence Properties

… ►A similar result for the

{{}_{0}F_{1}}

function of matrix argument is given in Faraut and Korányi (1994, p. 346). …

5: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

►

§35.7(i) Definition

… ►

Jacobi Form

… ►Let

f:{\boldsymbol{\Omega}}\to\mathbb{C}

(a) be orthogonally invariant, so that

f(\mathbf{T})

is a symmetric function of

t_{1},\dots,t_{m}

, the eigenvalues of the matrix argument

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

; (b) be analytic in

t_{1},\dots,t_{m}

in a neighborhood of

\mathbf{T}=\boldsymbol{{0}}

; (c) satisfy

f(\boldsymbol{{0}})=1

. Subject to the conditions (a)–(c), the function

f(\mathbf{T})={{}_{2}F_{1}}\left(a,b;c;\mathbf{T}\right)

is the unique solution of each partial differential equation …

7: 3.2 Linear Algebra

… ►

§3.2(ii) Gaussian Elimination for a Tridiagonal Matrix

… ►

§3.2(vi) Lanczos Tridiagonalization of a Symmetric Matrix

… ►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 …The tridiagonal matrix … ►Lanczos’ method is related to Gauss quadrature considered in §3.5(v). …

8: 1.2 Elementary Algebra

… ► … ►A matrix

\mathbf{A}

is: a diagonal matrix if …a real symmetric matrix if …a tridiagonal matrix if … ►Equation (3.2.7) displays a tridiagonal matrix in index form; (3.2.4) does the same for a lower triangular matrix. …

9: 30.16 Methods of Computation

… ►For

d

sufficiently large, construct the

d\times d

tridiagonal matrix

\mathbf{A}=[A_{j,k}]

with nonzero elements …The eigenvalues of

\mathbf{A}

can be computed by methods indicated in §§3.2(vi), 3.2(vii). … ►The coefficients

a^{m}_{n,r}(\gamma^{2})

are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5). ►A fourth method, based on the expansion (30.8.1), is as follows. Let

\mathbf{A}

be the

d\times d

matrix given by (30.16.1) if

n-m

is even, or by (30.16.6) if

n-m

is odd. …

10: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►Let

\alpha=-n

n=0,1,2,\dots

, and

q_{n,m}

m=0,1,\dots,n

, be the eigenvalues of the tridiagonal matrix ►

31.5.1

\begin{bmatrix}0&a\gamma&0&\dots&0\\ P_{1}&-Q_{1}&R_{1}&\dots&0\\ 0&P_{2}&-Q_{2}&&\vdots\\ \vdots&\vdots&&\ddots&R_{n-1}\\ 0&0&\dots&P_{n}&-Q_{n}\end{bmatrix},

ⓘ

Symbols:: $\gamma$ : real or complex parameter, $n$ : nonnegative integer, $a$ : complex parameter, $P_{j}$ : coefficient, $Q_{j}$ : coefficient and $R_{j}$ : coefficient
Permalink:: http://dlmf.nist.gov/31.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.5 and Ch.31

… ►

31.5.2

\mathit{Hp}_{n,m}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)=\mathit{H\!% \ell}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)

ⓘ

Defines:: $\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},\NVar{q_{n,m}};\NVar{-n},\NVar{% \beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun polynomials
Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.5 and Ch.31

►is a polynomial of degree

n

, and hence a solution of (31.2.1) that is analytic at all three finite singularities

0,1,a

. …

Lanczos tridiagonalization of a symmetric matrix

1—10 of 895 matching pages

1: 35.1 Special Notation

2: 35.5 Bessel Functions of Matrix Argument

§35.5 Bessel Functions of Matrix Argument

§35.5(i) Definitions

§35.5(ii) Properties

§35.5(iii) Asymptotic Approximations

3: 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6(i) Definitions

Laguerre Form

§35.6(ii) Properties

§35.6(iii) Relations to Bessel Functions of Matrix Argument

4: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8(i) Definition

Convergence Properties

5: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7(i) Definition

Jacobi Form

6: 19.16 Definitions

§19.16(i) Symmetric Integrals

§19.16(iii) Various Cases of $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

7: 3.2 Linear Algebra

§3.2(ii) Gaussian Elimination for a Tridiagonal Matrix

§3.2(vi) Lanczos Tridiagonalization of a Symmetric Matrix

8: 1.2 Elementary Algebra

9: 30.16 Methods of Computation

10: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

Lanczos tridiagonalization of a symmetric matrix

1—10 of 895 matching pages

1: 35.1 Special Notation

2: 35.5 Bessel Functions of Matrix Argument

§35.5 Bessel Functions of Matrix Argument

§35.5(i) Definitions

§35.5(ii) Properties

§35.5(iii) Asymptotic Approximations

3: 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6(i) Definitions

Laguerre Form

§35.6(ii) Properties

§35.6(iii) Relations to Bessel Functions of Matrix Argument

4: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8(i) Definition

Convergence Properties

5: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7(i) Definition

Jacobi Form

6: 19.16 Definitions

§19.16(i) Symmetric Integrals

§19.16(iii) Various Cases of R − a ⁡ ( 𝐛 ; 𝐳 )

7: 3.2 Linear Algebra

§3.2(ii) Gaussian Elimination for a Tridiagonal Matrix

§3.2(vi) Lanczos Tridiagonalization of a Symmetric Matrix

8: 1.2 Elementary Algebra

9: 30.16 Methods of Computation

10: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§19.16(iii) Various Cases of $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$