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37
Matrix transformations for computing rightmost eigenvalues of large sparse nonsymmetric eigenvalue problems
 IMA J. Numer. Anal
, 1996
"... This paper gives an overview of matrix transformations for finding rightmost eigenvalues of Ax = kx and Ax = kBx with A and B real nonsymmetric and B possibly singular. The aim is not to present new material, but to introduce the reader to the application of matrix transformations to the solution o ..."
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Cited by 26 (7 self)
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This paper gives an overview of matrix transformations for finding rightmost eigenvalues of Ax = kx and Ax = kBx with A and B real nonsymmetric and B possibly singular. The aim is not to present new material, but to introduce the reader to the application of matrix transformations to the solution of largescale eigenvalue problems. The paper explains and discusses the use of Chebyshev polynomials and the shiftinvert and Cayley ^ transforms as matrix transformations for problems that arise from the discretization df partial differential equations. A few other techniques are described. The reliability of iterative methods is also dealt with by introducing the concept of domain of confidence or trust region. This overview gives the reader an idea of the benefits and the drawbacks of several transformation techniques. We also briefly discuss the current software
CMV matrices: Five years after
, 2007
"... CMV matrices are the unitary analog of Jacobi matrices; we review their general theory. ..."
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Cited by 22 (3 self)
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CMV matrices are the unitary analog of Jacobi matrices; we review their general theory.
A Borgtype theorem associated with orthogonal polynomials on the unit circle
 J. London Math. Soc
, 2004
"... Abstract. We prove a general Borgtype result for reflectionless unitary CMV operators U associated with orthogonal polynomials on the unit circle. The spectrum of U is assumed to be a connected arc on the unit circle. This extends a recent result of Simon in connection with a periodic CMV operator ..."
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Cited by 15 (9 self)
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Abstract. We prove a general Borgtype result for reflectionless unitary CMV operators U associated with orthogonal polynomials on the unit circle. The spectrum of U is assumed to be a connected arc on the unit circle. This extends a recent result of Simon in connection with a periodic CMV operator with spectrum the whole unit circle. In the course of deriving the Borgtype result we also use exponential Herglotz representations of Caratheodory functions to prove an infinite sequence of trace formulas connected with the CMV operator U. 1.
The betaJacobi matrix model, the CS decomposition, and generalized singular value problems
 Foundations of Computational Mathematics
, 2007
"... Abstract. We provide a solution to the βJacobi matrix model problem posed by Dumitriu and the first author. The random matrix distribution introduced here, called a matrix model, is related to the model of Killip and Nenciu, but the development is quite different. We start by introducing a new matr ..."
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Cited by 14 (3 self)
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Abstract. We provide a solution to the βJacobi matrix model problem posed by Dumitriu and the first author. The random matrix distribution introduced here, called a matrix model, is related to the model of Killip and Nenciu, but the development is quite different. We start by introducing a new matrix decomposition and an algorithm for computing this decomposition. Then we run the algorithm on a Haardistributed random matrix to produce the βJacobi matrix model. The Jacobi ensemble on R n, parameterized by β> 0, a> −1, and b> −1, is the probability distribution whose density is proportional to Q β 2 i λ (a+1)−1
The Symplectic Eigenvalue Problem, the Butterfly Form, the SR Algorithm, and the Lanczos Method
 LINEAR ALGEBRA APPL
, 1998
"... We discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices. Any 2n x 2n symplectic matrix can be reduced to this condensed form which contains 8n x 4 nonzero entries and is determined by 4n x 1 parameters. The symplectic eigenvalue pr ..."
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Cited by 13 (3 self)
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We discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices. Any 2n x 2n symplectic matrix can be reduced to this condensed form which contains 8n x 4 nonzero entries and is determined by 4n x 1 parameters. The symplectic eigenvalue problem can be solved using the SR algorithm based on this condensed form. The SR algorithm preserves this form and can be modified to work only with the 4n x 1 parameters instead of the 4n² matrix elements. The reduction of symplectic matrices to symplectic butterfly form has a close analogy to the reduction of arbitrary matrices to Hessenberg form. A Lanczoslike algorithm for reducing a symplectic matrix to butterfly form is also presented.
Eigenvalue statistics for CMV matrices: From Poisson to clock via CβE
"... Abstract. We study CMV matrices (a discrete onedimensional Diractype operator) with random decaying coefficients. Under mild assumptions we identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numer ..."
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Cited by 12 (1 self)
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Abstract. We study CMV matrices (a discrete onedimensional Diractype operator) with random decaying coefficients. Under mild assumptions we identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numerals on a clock); in the case of slow decrease, the eigenvalues are distributed according to a Poisson process. For a certain critical rate of decay we obtain the circular beta ensembles of random matrix theory. The temperature β −1 appears as the square of the coupling constant. 1.
Riemann–Hilbert methods in the theory of orthogonal polynomials
 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS, AMER. MATH. SOC
, 2006
"... In this paper we describe various applications of the RiemannHilbert method to the theory of orthogonal polynomials on the line and on the circle. ..."
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Cited by 11 (0 self)
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In this paper we describe various applications of the RiemannHilbert method to the theory of orthogonal polynomials on the line and on the circle.
The QR Algorithm Revisited
 SIAM REVIEW
, 2008
"... The QR algorithm is still one of the most important methods for computing eigenvalues and eigenvectors of matrices. Most discussions of the QR algorithm begin with a very basic version and move by steps toward the versions of the algorithm that are actually used. This paper outlines a pedagogical p ..."
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Cited by 10 (0 self)
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The QR algorithm is still one of the most important methods for computing eigenvalues and eigenvectors of matrices. Most discussions of the QR algorithm begin with a very basic version and move by steps toward the versions of the algorithm that are actually used. This paper outlines a pedagogical path that leads directly to the implicit multishift QR algorithms that are used in practice, bypassing the basic QR algorithm completely.
Convergence analysis of Krylov subspace iterations with methods from potential theory
 SIAM Review
"... Abstract. Krylov subspace iterations are among the bestknown and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices. These methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on t ..."
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Cited by 9 (2 self)
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Abstract. Krylov subspace iterations are among the bestknown and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices. These methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix. This leads to an extremal problem in polynomial approximation theory: how small can a monic polynomial of a given degree be on the spectrum? This survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices. It is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure. Then depending on the number of iterations, the Lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region, which is characterized by means of a constrained equilibrium problem from potential theory. The same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems. Key words. Krylov subspace iterations, Ritz values, eigenvalue distribution, equilibrium measure, contrained equilibrium, potential theory AMS subject classifications. 15A18, 31A05, 31A15, 65F15 1. Introduction. Krylov
Breakdowns In The Implementation Of The Lanczos Method For Solving Linear Systems
, 1993
"... The L'anczos method for solving systems of linear equations is based on formal orthogonal polynomials. Its implementation is realized via some recurrence relationships between polynomials of a family of orthogonal polynomials or between those of two adjacent families of orthogonal polynomials. A div ..."
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Cited by 8 (7 self)
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The L'anczos method for solving systems of linear equations is based on formal orthogonal polynomials. Its implementation is realized via some recurrence relationships between polynomials of a family of orthogonal polynomials or between those of two adjacent families of orthogonal polynomials. A division by zero can occur in such recurrence relations, thus causing a breakdown in the algorithm which has to be stopped. In this paper, two types of breakdowns are discussed. The true breakdowns which are due to the nonexistence of some polynomials and the ghost breakdowns which are due to the recurrence relationship used. Among all the recurrence relationships which can be used and all the algorithms for implementing the L'anczos method which came out from them, the only reliable algorithm is L'anczos/Orthodir which can only suffer from true breakdowns. It is showed how to avoid true breakdowns in this algorithm. Other algorithms are also discussed and the case of nearbreakdown is treated. The same treatment applies to other methods related to L'anczos'.