The field of values bound on ideal GMRES

A widely known result of Elman, and its improvements due to Starke, Eiermann and Ernst, gives a bound on the worst-case GMRES residual norm using quantities related to the field of values of the given matrix and its inverse. We prove that these bounds also hold for the ideal GMRES approximation, and we derive and discuss some improvements of the bounds.

[1]  S. Eisenstat,et al.  Variational Iterative Methods for Nonsymmetric Systems of Linear Equations , 1983 .

[2]  H. Elman.  Iterative methods for large, sparse, nonsymmetric systems of linear equations , 1982 .

[3]  Thomas A. Manteuffel,et al.  Minimal Residual Method Stronger than Polynomial Preconditioning , 1996, SIAM J. Matrix Anal. Appl..

[4]  Lloyd N. Trefethen,et al.  GMRES/CR and Arnoldi/Lanczos as Matrix Approximation Problems , 2018, SIAM J. Sci. Comput..

[5]  Anne Greenbaum,et al.  Max-Min Properties of Matrix Factor Norms , 1994, SIAM J. Sci. Comput..

[6]  A. Greenbaum.  Generalizations of the field of values useful in the study of polynomial functions of a matrix , 2002 .

[7]  Z. Strakos,et al.  Krylov Subspace Methods: Principles and Analysis , 2012 .

[8]  Norms and the spectral radius of matrices , 1962 .

[9]  Eugene E. Tyrtyshnikov,et al.  Some Remarks on the Elman Estimate for GMRES , 2005, SIAM J. Matrix Anal. Appl..

[10]  Wayne Joubert,et al.  A Robust GMRES-Based Adaptive Polynomial Preconditioning Algorithm for Nonsymmetric Linear Systems , 1994, SIAM J. Sci. Comput..

[11]  A minimax inequality for operators and a related numerical range , 1971 .

[12]  Zdenek Strakos,et al.  Krylov Subspace Methods , 2012 .

[13]  M. Eiermann,et al.  Geometric aspects of the theory of Krylov subspace methods , 2001, Acta Numerica.

[14]  Jörg Liesen,et al.  Convergence analysis of Krylov subspace methods , 2004 .

[15]  Olof B. Widlund,et al.  Domain Decomposition Algorithms for Indefinite Elliptic Problems , 2017, SIAM J. Sci. Comput..

[16]  Dianne P. O'Leary,et al.  Complete stagnation of gmres , 2003 .

[17]  J. Curtiss.  Faber Polynomials and the Faber Series , 1971 .

[18]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[19]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[20]  Michel Crouzeix,et al.  Numerical range and functional calculus in Hilbert space , 2007 .

[21]  Kim-Chuan Toh,et al.  GMRES vs. Ideal GMRES , 1997, SIAM J. Matrix Anal. Appl..

[22]  G. Starke.  Field-of-values analysis of preconditioned iterative methods for nonsymmetric elliptic problems , 1997 .