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NUMERICAL algorithmS 1996年第1-2期12卷 97-110页

作者： Ye, Q UNIV MANITOBA DEPT APPL MATH WINNIPEG MB R3T 2N2 CANADA

A generalization of the block lanczos algorithm will be given, which allows the block size to be increased during the iteration process. In particular, the algorithm can be implemented with the block size chosen adaptively according to clustering of Ritz values. In this way, multiple and clustered eigenvalues can be found and the difficulty of choosing the block size is eased. Residual bounds for clustered eigenvalues are given. Numerical examples are presented to illustrate the adaptive algorithm.

关键词： block lanczos algorithm adaptive algorithm misconvergence

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Efficient Parallel Computing of Multifrontal Linear Solver in block lanczos algorithm for Large-Scale Structural Eigenproblems

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CMES-COMPUTER MODELING IN ENGINEERING & SCIENCES 2012年第6期86卷 551-576页

作者： Byun, Wanil Kim, Seung Jo Seoul Natl Univ Sch Aerosp & Mech Engn Seoul South Korea Korea Aerosp Res Inst Taejon South Korea

A structural eigensolver for large-scale finite element analysis is developed. The algorithms and data structures implemented in this paper are well suited for a distributed memory environment. As an eigenvalue extracting algorithm, the well-known M orthogonal block lanczos iteration incorporated with a parallel multifrontal solver (PMFS) was chosen. Basically, for the better performance of this algorithm in parallel computation, lanczos vector allocation, mass matrix multiplication, and M inner product procedures were efficiently implemented. And the PMFS for a linear equation which is the most time-consuming part during lanczos iterations was improved. The idea was to optimize network topologies of parallel matrix subroutines which are working in a 2-dimensional block-cyclic processor map, as well as to reduce both communication volume and idling time of parallel matrix subroutines. To reduce the communication volume, we condensed the parallel matrix multiplication subroutine from which duplicated communications are observed in the Cholesky factorization phase. To reduce the idling time, we adopted the least common multiple (LCM) concept by inverting a frontal matrix in the triangular system.

关键词： Parallel performance finite element method natural frequency block lanczos algorithm parallel multifrontal solver

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Compression of unitary rank-structured matrices to CMV-like shape with an application to polynomial rootfinding

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2015年 278卷 326-335页

作者： Bevilacqua, Roberto Del Corso, Gianna M. Gemignani, Luca Univ Pisa Dipartimento Informat I-56127 Pisa Italy

This paper is concerned with the reduction of a unitary matrix U to CMV-like shape. A lanczos-type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of U, i.e., of the matrix U + U-H. By elaborating on the lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U is rank-structured then the same property holds for its Hermitian part and, therefore, the block tridiagonalization process can be performed using the rank-structured matrix technology with reduced complexity. Our interest in the CMV-like reduction is motivated by the unitary and almost unitary eigenvalue problem. In this respect, finally, we discuss the application of the CMV-like reduction for the design of fast companion eigensolvers based on the customary QR iteration. (C) 2014 Elsevier B.V. All rights reserved.

关键词： CMV matrices Unitary matrices Rank-structured matrices block tridiagonal reduction QR iteration block lanczos algorithm

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New block quadrature rules for the approximation of matrix functions

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LINEAR ALGEBRA AND ITS APPLICATIONS 2016年 502卷 299-326页

作者： Reichel, Lothar Rodriguez, Giuseppe Tang, Tunan Kent State Univ Dept Math Sci Kent OH 44242 USA Univ Cagliari Dipartimento Matemat & Informat Viale Merello 92 I-09123 Cagliari Italy

Golub and Meurant have shown how to use the symmetric block lanczos algorithm to compute block Gauss quadrature rules for the approximation of certain matrix functions. We describe new block quadrature rules that can be computed by the symmetric or nonsymmetric block lanczos algorithms and yield higher accuracy than standard block Gauss rules after the same number of steps of the symmetric or nonsymmetric block lanczos algorithms. The new rules are block generalizations of the generalized averaged Gauss rules introduced by Spalevic. Applications to network analysis are presented. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Matrix functions Gauss quadrature block lanczos algorithm Complex networks

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block GAUSS AND ANTI-GAUSS QUADRATURE WITH APPLICATION TO NETWORKS

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2013年第4期34卷 1655-1684页

作者： Fenu, C. Martin, D. Reichel, L. Rodriguez, G. Univ Cagliari Dipartimento Matemat & Informat I-09123 Cagliari Italy Kent State Univ Dept Math Sci Kent OH 44242 USA

Approximations of matrix-valued functions of the form W-T f(A) W, where A is an element of R-mxm is symmetric, W is an element of R-mxk, with m large and k << m, has orthonormal columns, and f is a function, can be computed by applying a few steps of the symmetric block lanczos method to A with initial block-vector W is an element of R-mxk. Golub and Meurant have shown that the approximants obtained in this manner may be considered block Gauss quadrature rules associated with a matrix-valued measure. This paper generalizes anti-Gauss quadrature rules, introduced by Laurie for real-valued measures, to matrix-valued measures, and shows that under suitable conditions pairs of block Gauss and block anti-Gauss rules provide upper and lower bounds for the entries of the desired matrix-valued function. Extensions to matrix-valued functions of the form W-T f(A) V, where A is an element of R-mxm may be nonsymmetric, and the matrices V, W is an element of R-mxk satisfy (VW)-W-T = I (k) are also discussed. Approximations of the latter functions are computed by applying a few steps of the nonsymmetric block lanczos method to A with initial block-vectors V and W. We describe applications to the evaluation of functions of a symmetric or nonsymmetric adjacency matrix for a network. Numerical examples illustrate that a combination of block Gauss and anti-Gauss quadrature rules typically provides upper and lower bounds for such problems. We introduce some new quantities that describe properties of nodes in directed or undirected networks, and demonstrate how these and other quantities can be computed inexpensively with the quadrature rules of the present paper.

关键词： Gauss quadrature anti-Gauss quadrature block lanczos algorithm nonsymmetric block lanczos algorithm adjacency matrix network analysis graph

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block tridiagonal reduction of perturbed normal and rank structured matrices

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LINEAR ALGEBRA AND ITS APPLICATIONS 2013年第11期439卷 3505-3517页

作者： Bevilacqua, Roberto Del Corso, Gianna M. Gemignani, Luca Univ Pisa Dipartimento Informat I-56127 Pisa Italy

It is well known that if a matrix A is an element of C-nxn solves the matrix equation f(A, A(H)) = 0, where f(x, y) is a linear bivariate polynomial, then A is normal;A and A(H) can be simultaneously reduced in a finite number of operations to tridiagonal form by a unitary congruence and, moreover, the spectrum of A is located on a straight line in the complex plane. In this paper we present some generalizations of these properties for almost normal matrices which satisfy certain quadratic matrix equations arising in the study of structured eigenvalue problems for perturbed Hermitian and unitary matrices. (C) 2013 Elsevier Inc. All rights reserved.

关键词： block tridiagonal reduction Rank structured matrix block lanczos algorithm

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On the complexity of computing determinants

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COMPUTATIONAL COMPLEXITY 2005年第3-4期13卷 91-130页

作者： Kaltofen, E Villard, G N Carolina State Univ Dept Math Raleigh NC 27695 USA Ecole Normale Super Lyon Lab LIP F-69364 Lyon 07 France

We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n x n matrix A with integer entries in ( n(3.2) log \\A\\s)(1+o( 1)) and ( n(2.697263) log \\A\\) (1+o( 1)) bit operations;here \\A\\ denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors C-1( log n) (C2)( loglog \\A\\)(C3) for positive real constants C1, C2, C3. The bit complexity ( n(3.2) log \\A\\)(1+o( 1)) results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n(3.2+o( 1)) and O(n(2.697263)) ring additions, subtractions and multiplications.

关键词： integer matrix matrix determinant characteristic polynomial Smith normal form bit complexity division-free complexity randomized algorithm multivariable control theory realization matrix sequence block Wiedemann algorithm block lanczos algorithm

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A block QMR method for computing multiple simultaneous solutions to complex symmetric systems

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 1996年第1期17卷 263-274页

作者： Boyse, WE Seidl, AA Lockheed Palo Alto Res. Laboratories Palo Alto CA 94304-1191 3251 Hanover Street United States

The solution of complex symmetric indefinite systems of equations where multiple solutions are required is considered. The quasi-minimum residual (QMR) method, ideally suited for these matrices, is generalized using the block lanczos algorithm to solve multiple solutions simultaneously. This modification alone is shown, through numerical examples involving large sparse matrices from finite element discretization of Maxwell's equations, to accelerate the convergence by a factor almost as great as the number of simultaneous solutions. A natural convergence criterion for this method is presented that is shown to be as effective as, and easier to compute than, the usual equation residual. Finally, a numerical comparison of the classical incomplete Cholesky and a variant of the ILU(T) preconditioners is given showing superior performance by the latter.

关键词： QMR complex symmetric matrices block lanczos algorithm preconditioner Maxwell's equations

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Efficient DOA estimation using signal eigenvector method without eigendecomposition

Efficient DOA estimation using signal eigenvector method wit...

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IEEE International Conference on Universal Personal Communications (ICUPC 98)

作者： Kim, KS Elect & Telecommun Res Inst Yusong Gu Taejon 305350 South Korea

ISBN: (纸本)0780351061

In this paper, we propose the new method of SEM using block lanczos algorithm for improving the resolution in bearing angle estimation of narrowband coherent signals at a uniform linear array. The signal eigenvector method use the signal subspace that is composed of the eigenvectors associated with the largest eigenvalues of the array spatial correlation matrix. The proposed method estimates the signal subspace using the block lanczos algorithm instead of Eigendecomposition. The results of computer simulation are shown the superior performance of this proposed method to the conventional SEM under narrowband coherent signal environment.

关键词： direction-of-arrival estimation array signal processing signal resolution eigenvalues and eigenfunctions efficient DOA estimation signal eigenvector method eigendecomposition SEM block lanczos algorithm resolution bearing angle estimation narrowband coherent signals uniform linear array signal subspace array spatial correlation matrix performance narrowband coherent signal

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