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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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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
Superlinear CG Convergence for Special RightHand Sides
 Electronic Transactions on Numerical Analysis
"... Recently, we gave a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the Conjugate Gradient method. Roughly speaking, one may observe superlinear convergence while solving a sequence of (symmetric positive definite) linear ..."
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Cited by 14 (1 self)
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Recently, we gave a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the Conjugate Gradient method. Roughly speaking, one may observe superlinear convergence while solving a sequence of (symmetric positive definite) linear systems if the asymptotic eigenvalue distribution of the sequence of the corresponding matrices of coefficients is far from an equilibrium distribution. However, it is well known that the convergence of the Conjugate Gradient or other Krylov subspace methods does not only depend on the spectrum but also on the righthand side of the underlying system and the starting vector. In this paper we present a family of examples based on the discretization via finite differences of the one dimensional Poisson problem where the asymptotic distribution equals an equilibrium distribution but one may as well observe superlinear convergence according to the particular choice of the righthand sides. Our findings are related to some recent results concerning asymptotics of discrete orthogonal polynomials. An important tool in our investigations is a constrained energy problem in logarithmic potential theory, where an additional external field is used being related to our particular righthand sides. Key words. Superlinear convergence, Conjugate gradients, Krylov subspace methods, Logarithmic potential theory. AMS subject classifications. 65F10, 65E05, 31A99, 41A10. 1.
Convergence of the isometric Arnoldi process
, 2003
"... It is well known that the performance of eigenvalue algorithms such as the Lanczos and the Arnoldi method depends on the distribution of eigenvalues. Under fairly general assumptions we characterize the region of good convergence for the Isometric Arnoldi Process. We also determine bounds for the ra ..."
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Cited by 8 (2 self)
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It is well known that the performance of eigenvalue algorithms such as the Lanczos and the Arnoldi method depends on the distribution of eigenvalues. Under fairly general assumptions we characterize the region of good convergence for the Isometric Arnoldi Process. We also determine bounds for the rate of convergence and we prove sharpness of these bounds. The distribution of isometric Ritz values is obtained as the minimizer of an extremal problem. We use techniques from logarithmic potential theory in proving these results.
On the convergence of rational Ritz values
 SIAM J. Matrix Anal. Appl
, 2010
"... Abstract. Ruhe’s rational Krylov method is a popular tool for approximating eigenvalues of a given matrix, though its convergence behavior is far from being fully understood. Under fairly general assumptions we characterize in an asymptotic sense which eigenvalues of a Hermitian matrix are approxima ..."
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Abstract. Ruhe’s rational Krylov method is a popular tool for approximating eigenvalues of a given matrix, though its convergence behavior is far from being fully understood. Under fairly general assumptions we characterize in an asymptotic sense which eigenvalues of a Hermitian matrix are approximated by rational Ritz values and how fast this approximation takes place. Our main tool is a constrained extremal problem from logarithmic potential theory, where an additional external field is required for taking into account the poles of the underlying rational Krylov space. Several examples illustrate our analytic results. Key words. Rational Krylov, Ritz values, orthogonal rational functions, logarithmic potential theory. AMS subject classifications. 15A18, 31A05, 31A15, 65F15 1. Introduction. In order to approximate parts of the spectrum Λ(A) of a Hermitian matrix A ∈ CN×N, a widely used approach is to project A onto an ndimensional subspace of CN, with n being small compared to N. Given a matrix Vn ∈ CN×n with orthonormal columns, the eigenvalues of the projected counterpart V ∗nAVn ∈ Cn×n are called Ritz values of order n. These Ritz values are often good
On the sharpness of an asymptotic error estimate for Conjugate Gradients
 BIT
, 2000
"... Recently, the authors obtained an upper bound on the error for the conjugate gradient method, which is valid in an asymptotic setting as the size of the linear systems tends to infinity. The estimate depends on the asymptotic distribution of eigenvalues, and the ratio between the size and the number ..."
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Cited by 6 (4 self)
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Recently, the authors obtained an upper bound on the error for the conjugate gradient method, which is valid in an asymptotic setting as the size of the linear systems tends to infinity. The estimate depends on the asymptotic distribution of eigenvalues, and the ratio between the size and the number of iterations. Such error bounds are related to the existence of polynomials with value 1 at 0 whose supnorm on the spectrum is as small as possible. A possible strategy for constructing such a polynomial p is to select a set S, to specify that every eigenvalue outside S is a zero of p, and then to minimize the supnorm of p on S. Here we show that this strategy can never give a better asymptotic upper bound than the one we obtained before. We also discuss the case where equality is met.
A numerical solution of the constrained Energy Problem
, 2004
"... An algorithm is proposed to solve the constrained energy problem from potential theory. Numerical examples are presented, showing the accuracy of the algorithm. The algorithm is also compared with another numerical method for the same problem. ..."
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Cited by 2 (0 self)
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An algorithm is proposed to solve the constrained energy problem from potential theory. Numerical examples are presented, showing the accuracy of the algorithm. The algorithm is also compared with another numerical method for the same problem.
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"... Noname manuscript No. (will be inserted by the editor) Superlinear convergence of the rational Arnoldi method for the approximation of matrix functions ..."
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Noname manuscript No. (will be inserted by the editor) Superlinear convergence of the rational Arnoldi method for the approximation of matrix functions