# Lanczos tridiagonalization of a symmetric matrix

(0.003 seconds)

## 1—10 of 895 matching pages

##### 1: 35.1 Special Notation
 $a,b$ complex variables. … 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) ${{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)$ or $\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)$ and (of the second kind) $\Psi\left(a;b;\mathbf{T}\right)$; Gaussian hypergeometric ${{}_{2}F_{1}}\left(a_{1},a_{2};b;\mathbf{T}\right)$ or $\displaystyle{{}_{2}F_{1}}\left({a_{1},a_{2}\atop b};\mathbf{T}\right)$; generalized hypergeometric ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ or $\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)$. 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(iii) Asymptotic Approximations
For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).
##### 4: 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
###### 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 …
##### 6: 19.16 Definitions
###### §19.16(i) Symmetric Integrals
A fourth integral that is symmetric in only two variables is defined by … which is homogeneous and of degree $-a$ in the $z$’s, and unchanged when the same permutation is applied to both sets of subscripts $1,\dots,n$. … …
##### 7: 3.2 Linear Algebra
###### §3.2(vi) LanczosTridiagonalization of aSymmetricMatrix
Define the Lanczos vectors $\mathbf{v}_{j}$ and coefficients $\alpha_{j}$ and $\beta_{j}$ by $\mathbf{v}_{0}=\boldsymbol{{0}}$, 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 matrixLanczos’ 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 $\begin{bmatrix}0&a\gamma&0&\dots&0\\ P_{1}&-Q_{1}&R_{1}&\dots&0\\ 0&P_{2}&-Q_{2}&&\vdots\\ \vdots&\vdots&&\ddots&R_{n-1}\\ 0&0&\dots&P_{n}&-Q_{n}\end{bmatrix},$
is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. …