Lanczos tridiagonalization of a symmetric matrix
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1: 35.1 Special Notation
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►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively and , and the special functions of matrix argument: Bessel (of the first kind) and (of the second kind) ; confluent hypergeometric (of the first kind) or and (of the second kind) ; Gaussian hypergeometric or ; generalized hypergeometric or .
►An alternative notation for the multivariate gamma function is (Herz (1955, p. 480)).
Related notations for the Bessel functions are (Faraut and Korányi (1994, pp. 320–329)), (Terras (1988, pp. 49–64)), and (Faraut and Korányi (1994, pp. 357–358)).
complex variables. | |
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complex symmetric matrix. | |
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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 with matrix argument , numerator parameters , and denominator parameters is … ►Convergence Properties
… ►A similar result for the 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 (a) be orthogonally invariant, so that is a symmetric function of , the eigenvalues of the matrix argument ; (b) be analytic in in a neighborhood of ; (c) satisfy . Subject to the conditions (a)–(c), the function is the unique solution of each partial differential equation …6: 19.16 Definitions
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§19.16(i) Symmetric Integrals
… ►A fourth integral that is symmetric in only two variables is defined by … ►which is homogeneous and of degree in the ’s, and unchanged when the same permutation is applied to both sets of subscripts . … … ►§19.16(iii) Various Cases of
…7: 3.2 Linear Algebra
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§3.2(ii) Gaussian Elimination for a Tridiagonal Matrix
… ►§3.2(vi) Lanczos Tridiagonalization of a Symmetric Matrix
… ►Define the Lanczos vectors and coefficients and by , a normalized vector (perhaps chosen randomly), , , and for by the recursive scheme …The tridiagonal matrix … ►Lanczos’ method is related to Gauss quadrature considered in §3.5(v). …8: 1.2 Elementary Algebra
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►A matrix
is: a diagonal matrix if
…a real symmetric matrix if
…a tridiagonal matrix if
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►Equation (3.2.7) displays a tridiagonal matrix in index form; (3.2.4) does the same for a lower triangular matrix.
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9: 30.16 Methods of Computation
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►For sufficiently large, construct the
tridiagonal matrix
with nonzero elements
…The eigenvalues of can be computed by methods indicated in §§3.2(vi), 3.2(vii).
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►The coefficients 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 be the
matrix given by (30.16.1) if is even, or by (30.16.6) if is odd.
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