Modifying the generalized singular value decomposition with application with application in direction-of-arrival finding

We consider updating and downdating problems for the generalized singular value decomposition (GSVD) of matrix pairs when new rows are added to one of the matrices or old rows are deleted. Two classes of algorithms are developed, one based on the CS decomposition formulation of the GSVD and the other based on the generalized eigenvalue decomposition formulation. In both cases we show that the updating and downdating problems can be reduced to a rank-one SVD updating problem. We also provide perturbation analysis for the cases when the added or deleted rows are subject to errors. Numerical experiments on direction-of-arrival (DOA) finding with colored noises are carried out to demonstrate the tracking ability of the algorithms. The work of the first author was supported in part by NSF grants CCR-9308399 and CCR-9619452. The work of the second author was supported in part by China State Major Key Project for Basic Researches.     

Homotopy method for generalized eigenvalue problems Ax= ΛBx

Generalized eigenvalue problems can be considered as a system of polynomials. The homotopy continuation method is used to find all the isolated zeros of the polynomial system which corresponds to the eigenpairs of the generalized eigenvalue problem. A special homotopy is constructed in such a way that there are exactly n distinct smooth curves connecting trivial solutions to desired eigenpairs. Since the curves followed by general homotopy curve following scheme are computed independently of one another, the algorithm is a likely candidate for exploiting the advantages of parallel processing to the generalized eigenvalue problems.     

Modified block preconditioner for generalized saddle point matrices with highly singular(1,1) blocks

ABSTRACT

In this paper, based on the preconditioners presented by Zhang [A new preconditioner for generalized saddle matrices with highly singular(1,1) blocks. Int J Comput Maths. 2014;91(9):2091-2101], we consider a modified block preconditioner for generalized saddle point matrices whose coefficient matrices have singular (1,1) blocks. Moreover, theoretical analysis gives the eigenvalue distribution, forms of the eigenvectors and the minimal polynomial. Finally, numerical examples show the eigenvalue distribution with the presented preconditioner and confirm our analysis.     

A fast algorithm for computing the smallest eigenvalue of a symmetric positive‐definite Toeplitz matrix

Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive‐definite (SPD) Toeplitz matrix. An algorithm for computing upper and lower bounds to the smallest eigenvalue of a SPD Toeplitz matrix has been recently derived (Linear Algebra Appl. 2007; DOI: 10.1016/j.laa.2007.05.008 ). The algorithm relies on the computation of the R factor of the QR factorization of the Toeplitz matrix and the inverse of R. The simultaneous computation of R and R^?1 is efficiently accomplished by the generalized Schur algorithm. In this paper, exploiting the properties of the latter algorithm, a numerical method to compute the smallest eigenvalue and the corresponding eigenvector of SPD Toeplitz matrices in an accurate way is proposed. Copyright © 2008 John Wiley & Sons, Ltd.     

Randomized algorithms for generalized Hermitian eigenvalue problems with application to computing Karhunen–Loève expansion

We describe randomized algorithms for computing the dominant eigenmodes of the generalized Hermitian eigenvalue problem Ax = λBx, with A Hermitian and B Hermitian and positive definite. The algorithms we describe only require forming operations Ax,Bx and B^?1x and avoid forming square roots of B (or operations of the form, B^1/2x or B^?1/2x). We provide a convergence analysis and a posteriori error bounds and derive some new results that provide insight into the accuracy of the eigenvalue calculations. The error analysis shows that the randomized algorithm is most accurate when the generalized singular values of B^?1A decay rapidly. A randomized algorithm for the generalized singular value decomposition is also provided. Finally, we demonstrate the performance of our algorithm on computing an approximation to the Karhunen–Loève expansion, which involves a computationally intensive generalized Hermitian eigenvalue problem with rapidly decaying eigenvalues. Copyright © 2015 John Wiley & Sons, Ltd.     

The principal eigenvalue of a space–time periodic parabolic operator

This paper deals with the generalized principal eigenvalue of the parabolic operator , where the coefficients are periodic in t and x. We give the definition of this eigenvalue and we prove that it can be approximated by a sequence of principal eigenvalues associated to the same operator in a bounded domain, with periodicity in time and Dirichlet boundary conditions in space. Next, we define a family of periodic principal eigenvalues associated with the operator and use it to give a characterization of the generalized principal eigenvalue. Finally, we study the dependence of all these eigenvalues with respect to the coefficients.      

Subordinacy theory for extended CMV matrices

We develop subordinacy theory for extended Cantero-Moral-Velázquez(CMV) matrices,i.e.,we provide explicit supports for the singular and absolutely continuous parts of the canonical spectral measure associated with a given extended CMV matrix in terms of the presence or absence of subordinate solutions to the generalized eigenvalue equation.Some corollaries and applications of this result are described as well.     

INDECOMPOSABLE INJECTIVE MODULES AND A THEOREM OF KAPLANSKY

Abstract

Every tripotent e of a generalized Jordan triple system of second order uniquely defines a decomposition of the space of the triple into a direct sum of eight components. This decomposition is a generalization of the Peirce decomposition for the Jordan triple system. The relations between components are studied in the case when e is a left unit.     

Comparison of subdominant eigenvalues of some linear search schemes

The subdominant eigenvalue of the transition probability matrix of a Markov chain is a determining factor in the speed of transition of the chain to a stationary state. However, these eigenvalues can be difficult to estimate in a theoretical sense. In this paper we revisit the problem of dynamically organizing a linear list. Items in the list are selected with certain unknown probabilities and then returned to the list according to one of two schemes: the move-to-front scheme or the transposition scheme. The eigenvalues of the transition probability matrix Q of the former scheme are well-known but those of the latter T are not. Nevertheless the transposition scheme gives rise to a reversible Markov chain. This enables us to employ a generalized Rayleigh-Ritz theorem to show that the subdominant eigenvalue of T is at least as large as the subdominant eigenvalue of Q.     

Locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity

The Wilkinson distance of a matrix A is the two-norm of the smallest perturbation E so that A + E has a multiple eigenvalue. Malyshev derived a singular value optimization characterization for the Wilkinson distance. In this work we generalize the definition of the Wilkinson distance as the two-norm of the smallest perturbation so that the perturbed matrix has an eigenvalue of prespecified algebraic multiplicity. We provide a singular value characterization for this generalized Wilkinson distance. Then we outline a numerical technique to solve the derived singular value optimization problems. In particular the numerical technique is applicable to Malyshev’s formula to compute the Wilkinson distance as well as to retrieve a nearest matrix with a multiple eigenvalue.     

Eigenvalues and Pole Functions of Hamiltonian Systems with Eigenvalue Depending Boundary Conditions

This paper consists of two chapters. The first chapter concerns matrix functions belonging to the generalized Nevanlinna class N_k^{m × m}. We present results about the operator representation of such functions. These representations are then used to obtain information about the (generalized) poles of generalized Nevanlinna functions. The second chapter may be viewed as a continuation of our paper [DLS3] and treats Hamiltonian systems of differential equations with boundary conditions depending on the eigenvalue parameter. In particular we study the eigenvalues, both isolated and embedded eigenvalues.     

A Structured Kronecker form for the Palindromic Eigenvalue Problem

We consider a structured generalized eigenvalue problem of the form Ax = λ (–A^T )x, called palindromic eigenvalue problem. We will explain this name and give applications. To characterize the spectral properties of this problem we introduce a structured version of the Kronecker canonical form. As an application we present a characterization of the set of pencils B + λC that are equivalent to a palindromic pencil A + λA^T . (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equilibrium Problems with Applications to Eigenvalue Problems

In this paper, we consider equilibrium problems and introduce the concept of (S)₊ condition for bifunctions. Existence results for equilibrium problems with the (S)₊ condition are derived. As special cases, we obtain several existence results for the generalized nonlinear variational inequality studied by Ding and Tarafdar (Ref. 1) and the generalized variational inequality studied by Cubiotti and Yao (Ref. 2). Finally, applications to a class of eigenvalue problems are given.     

On the number of eigenvalues of a matrix operator

We consider a matrix operator H in the Fock space. We prove the finiteness of the number of negative eigenvalues of H if the corresponding generalized Friedrichs model has the zero eigenvalue (0 = min σ _ess(H)). We also prove that H has infinitely many negative eigenvalues accumulating near zero (the Efimov effect) if the generalized Friedrichs model has zero energy resonance. We obtain asymptotics for the number of negative eigenvalues of H below z as z → −0.     

Graphs with Least Eigenvalue −2: The Star Complement Technique

Let G be a connected graph with least eigenvalue –2, of multiplicity k. A star complement for –2 in G is an induced subgraph H = G – X such that |X| = k and –2 is not an eigenvalue of H. In the case that G is a generalized line graph, a characterization of such subgraphs is used to decribe the eigenspace of –2. In some instances, G itself can be characterized by a star complement. If G is not a generalized line graph, G is an exceptional graph, and in this case it is shown how a star complement can be used to construct G without recourse to root systems.     

A Characterization of Positive Matrices

It is shown that an n-by-n matrix has a strictly dominant positive eigenvalue with positive left and right eigenvectors and this property is inherited by principle submatrices if and only if it is entry-wise positive. This limits the extent to which attractive Perron-Frobenius properities may be generalized outside the positive matrices.Mathematics Subject Classification (2000): 15A48     

Pseudoperturbation method for computation of E. Schmidt eigenvalues

In two problems with approximately given n -multiple generalized E. Schmidt eigenvalue with relevant Jordan chains pseudoperturbation method is applied for their sharpening. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on the normwise perturbation theory for the regular generalized eigenproblem

In this paper, we present a normwise perturbation theory for the regular generalized eigenproblem Ax = λBx, when λ is a semi-simple and finite eigenvalue, which departs from the classical analysis with the chordal norm [9]. A backward error and a condition number are derived for a choice of flexible measure to represent independent perturbations in the matrices A and B. The concept of optimal backward error associated with an eigenvalue only is also discussed. © 1998 John Wiley & Sons, Ltd.     

A parallel eigenvalue problem solver for computational nanoelectronics

A new parallel eigenvalue solver for finding the interior eigenvalues of a standard Hermitian eigenvalue problem arising in atomistic simulations in nanoelectronics is presented. It is based on the Tracemin algorithm which finds the p smallest eigenpairs of a generalized Hermitian eigenvalue problem. The original problem is modified using spectrum folding or a quadratic mapping so that the interior eigenvalues are mapped onto the smallest or the largest, respectively. In the latter case the solution of systems in every iteration of Tracemin is avoided and Chebyshev polynomials are used to speedup convergence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)