On the Convergence of Rational Ritz Values

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.

[1]  E. Saff,et al.  Logarithmic Potentials with External Fields , 1997 .

[2]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[3]  Tobin A. Driscoll,et al.  From Potential Theory to Matrix Iterations in Six Steps , 1998, SIAM Rev..

[4]  Arno B. J. Kuijlaars,et al.  Zero distributions for discrete orthogonal polynomials , 1998 .

[5]  Arno B. J. Kuijlaars,et al.  Superlinear CG convergence for special right-hand sides , 2002 .

[6]  G. Szegő,et al.  On the Eigen-Values of Certain Hermitian Forms , 1953 .

[7]  Dario Fasino.  Rational Krylov matrices and QR steps on Hermitian diagonal-plus-semiseparable matrices , 2005, Numer. Linear Algebra Appl..

[8]  Karl Meerbergen,et al.  Using Generalized Cayley Transformations within an Inexact Rational Krylov Sequence Method , 1998, SIAM J. Matrix Anal. Appl..

[9]  Edward B. Saff,et al.  Potential theoretic tools in polynomial and rational approximation , 2006 .

[10]  L. Knizhnerman,et al.  Extended Krylov Subspaces: Approximation of the Matrix Square Root and Related Functions , 1998, SIAM J. Matrix Anal. Appl..

[11]  Stefano Serra Capizzano,et al.  On the Asymptotic Spectrum of Finite Element Matrix Sequences , 2007, SIAM J. Numer. Anal..

[12]  Axel Ruhe.  Rational Krylov Algorithms for Nonsymmetric Eigenvalue Problems , 1994 .

[13]  D. Sorensen.  Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[14]  B. Beckermann.  Discrete orthogonal polynomials and superlinear convergence of Krylov subspace methods in numerical linear algebra , 2006 .

[15]  Adhemar Bultheel,et al.  RATIONAL KRYLOV SEQUENCES AND ORTHOGONAL RATIONAL FUNCTIONS , 2008 .

[16]  A. Böttcher,et al.  Introduction to Large Truncated Toeplitz Matrices , 1998 .

[17]  Arno B. J. Kuijlaars,et al.  Extremal Polynomials on Discrete Sets , 1999 .

[18]  Karl Meerbergen,et al.  Changing poles in the rational Lanczos method for the Hermitian eigenvalue problem , 2001, Numer. Linear Algebra Appl..

[19]  Axel Ruhe.  The rational Krylov algorithm for nonsymmetric eigenvalue problems. III: Complex shifts for real matrices , 1994 .

[20]  E. Rakhmanov,et al.  Families of equilibrium measures in an external field on the real axis , 1999 .

[21]  R. Vandebril,et al.  Matrix Computations and Semiseparable Matrices , 2007 .

[22]  Y. Okumura.  Note on the $N=2$ super Yang-Mills gauge theory in a noncommutative differential geometry , 1998 .

[23]  I. Gohberg,et al.  The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order , 2005 .

[24]  Philipp Birken,et al.  Numerical Linear Algebra , 2011, Encyclopedia of Parallel Computing.

[25]  L. Gemignani,et al.  Matrix Computations and Semiseparable Matrices. Volume I: Linear Systems, R. Vandebril, M. Van Barel, N. Mastronardi , 2010 .

[26]  Arno B. J. Kuijlaars,et al.  On The Sharpness of an Asymptotic Error Estimate for Conjugate Gradients , 2001 .

[27]  J. Coussement,et al.  A continuum limit of the relativistic Toda lattice: asymptotic theory of discrete Laurent orthogonal polynomials with varying recurrence coefficients , 2005 .

[28]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[29]  Bernhard Beckermann,et al.  A note on the convergence of Ritz values for sequences of matrices , 2000 .

[30]  Thomas Ransford,et al.  Potential Theory in the Complex Plane: Bibliography , 1995 .

[31]  Bernhard Beckermann,et al.  On a Conjecture of E. A. Rakhmanov , 2000 .

[32]  Marc Van Barel,et al.  Convergence of the Isometric Arnoldi Process , 2005, SIAM J. Matrix Anal. Appl..

[33]  E. Rakhmanov,et al.  Equilibrium measure and the distribution of zeros of the extremal polynomials of a discrete variable , 1996 .

[34]  Arno B. J. Kuijlaars,et al.  Which Eigenvalues Are Found by the Lanczos Method? , 2000, SIAM J. Matrix Anal. Appl..

[35]  Arno B. J. Kuijlaars,et al.  Superlinear Convergence of Conjugate Gradients , 2001, SIAM J. Numer. Anal..

[36]  Israel Gohberg,et al.  On the fast reduction of a quasiseparable matrix to Hessenberg and tridiagonal forms , 2007 .

[37]  Edward B. Saff,et al.  Constrained energy problems with applications to orthogonal polynomials of a discrete variable , 1997 .

[38]  Arno B. J. Kuijlaars,et al.  Convergence Analysis of Krylov Subspace Iterations with Methods from Potential Theory , 2006, SIAM Rev..

[39]  Henk A. van der Vorst,et al.  Approximate solutions and eigenvalue bounds from Krylov subspaces , 1995, Numer. Linear Algebra Appl..

[40]  Axel Ruhe.  Rational Krylov sequence methods for eigenvalue computation , 1984 .

[41]  K. Meerbergen,et al.  The implicit application of a rational filter in the RKS method , 1997 .