## Expressions And Bounds For The GMRES Residual (1999)

Venue: | BIT |

Citations: | 20 - 0 self |

### BibTeX

@ARTICLE{Ipsen99expressionsand,

author = {Ilse C. F. Ipsen},

title = {Expressions And Bounds For The GMRES Residual},

journal = {BIT},

year = {1999},

volume = {40},

pages = {524--533}

}

### Years of Citing Articles

### OpenURL

### Abstract

. Expressions and bounds are derived for the residual norm in GMRES. It is shown that the minimal residual norm is large as long as the Krylov basis is well-conditioned.For scaled Jordan blocks the minimal residual norm is expressed in terms of eigenvalues and departure from normality.For normal matrices the minimal residual norm is expressed in terms of products of relative eigenvalue di#erences. Key words. linear system, Krylov methods, GMRES, MINRES, Vandermonde matrix, eigenvalues, departure from normality AMS subject classi#cation. 15A03, 15A06, 15A09, 15A12, 15A18, 15A60, 65F10, 65F15, 65F20, 65F35. 1. Introduction.. The generalised minimal residual method #GMRES# #31, 36# #and MINRES for Hermitian matrices #30## is an iterative method for solving systems of linear equations Ax = b. The approximate solution in iteration i minimises the two-norm of the residual b , Az over the Krylov space spanfb;Ab;:::;A i,1 bg. The goal of this paper is to express this minimal residual norm...