## 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...

### Citations

1321 |
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
- Saad, Schultz
- 1986
(Show Context)
Citation Context ...igenvalues, 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 o... |

847 | Accuracy and Stability of Numerical Algorithms - Higham - 2002 |

320 |
Solution of sparse indefinite systems of linear equations
- Paige, Saunders
- 1975
(Show Context)
Citation Context ...ect classification. 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 \Gamma Az over the Krylov space spanfb; Ab; : ... |

100 |
How fast are nonsymmetric matrix iterations
- Nachtigal, Reddy, et al.
- 1992
(Show Context)
Citation Context ...this minimal residual norm in terms of eigenvalues and departure of A from normality. Although it is known that the convergence of GMRES for a non-normal matrix is not determined by eigenvalues alone =-=[1, 18, 27, 28, 12]-=-, our expressions for the residual norms of scaled Jordan blocks in x3 represent the rst quantitative dependence of the minimal residual norm on the non-normality of the matrix. Often the residual nor... |

95 | An efficient algorithm for computing a strong rank revealing QR factorization - Gu - 1996 |

91 | The rotation of eigenvectors by a perturbation - Davis, Kahan - 1970 |

79 |
der Vorst. Approximate solutions and eigenvalue bounds from Krylov subspaces. Numerical Linear Algebra with Applications
- Paige, Parlett, et al.
- 1995
(Show Context)
Citation Context ...per bounds on the residual norm in terms of polynomials are given in [4, 27, 31], and the tightness of these bounds is examined in [17, 19, 33]. Convergence analyses based on Ritz values are given in =-=[5, 29,35]-=-. The case of nearly singular matrices is analysed in [3], and comparisons with other methods are made in [2, 23]. The approach here is di erent because we ignore the way GMRES is implemented (e.g. vi... |

56 |
A theoretical comparison of the Arnoldi and GMRES algorithms
- Brown
- 1991
(Show Context)
Citation Context ...is examined in [17, 19, 33]. Convergence analyses based on Ritz values are given in [5, 29,35]. The case of nearly singular matrices is analysed in [3], and comparisons with other methods are made in =-=[2, 23]-=-. The approach here is di erent because we ignore the way GMRES is implemented (e.g. via Arnoldi's method) and we do not use polynomials to derive the bounds. Instead we exploit structure in the Krylo... |

49 | Any nonincreasing convergence curve is possible for GMRES
- Greenbaum, Pták, et al.
- 1996
(Show Context)
Citation Context ...this minimal residual norm in terms of eigenvalues and departure of A from normality. Although it is known that the convergence of GMRES for a non-normal matrix is not determined by eigenvalues alone =-=[1, 18, 27, 28, 12]-=-, our expressions for the residual norms of scaled Jordan blocks in x3 represent the rst quantitative dependence of the minimal residual norm on the non-normality of the matrix. Often the residual nor... |

47 | GMRES/CR and Arnoldi/Lanczos as matrix approximation problems
- GREENBAUM, TREFETHEN
- 1994
(Show Context)
Citation Context ... methods is bounded in terms of polynomials. With regard to GMRES, upper bounds on the residual norm in terms of polynomials are given in [4, 27, 31], and the tightness of these bounds is examined in =-=[17, 19, 33]-=-. Convergence analyses based on Ritz values are given in [5, 29,35]. The case of nearly singular matrices is analysed in [3], and comparisons with other methods are made in [2, 23]. The approach here ... |

40 | A hybrid GMRES algorithm for non-symmetric linear systems
- Nachtigal, Reichel, et al.
- 1992
(Show Context)
Citation Context ...this minimal residual norm in terms of eigenvalues and departure of A from normality. Although it is known that the convergence of GMRES for a non-normal matrix is not determined by eigenvalues alone =-=[1, 18, 27, 28, 12]-=-, our expressions for the residual norms of scaled Jordan blocks in x3 represent the rst quantitative dependence of the minimal residual norm on the non-normality of the matrix. Often the residual nor... |

39 | GMRES on (nearly) singular systems
- BROWN, WALKER
- 1997
(Show Context)
Citation Context ...n in [4, 27, 31], and the tightness of these bounds is examined in [17, 19, 33]. Convergence analyses based on Ritz values are given in [5, 29,35]. The case of nearly singular matrices is analysed in =-=[3]-=-, and comparisons with other methods are made in [2, 23]. The approach here is di erent because we ignore the way GMRES is implemented (e.g. via Arnoldi's method) and we do not use polynomials to deri... |

38 | Manifestations of the Schur complement - Cottle - 1974 |

35 | From potential theory to matrix iterations in six steps
- Driscoll, Toh, et al.
- 1998
(Show Context)
Citation Context |

34 | Bounds for iterates, inverses, spectral variation and field of values of non–normal matrices - Henrici - 1962 |

29 |
Max-min properties of matrix factor norms
- GREENBAUM, GURVITS
- 1994
(Show Context)
Citation Context ... methods is bounded in terms of polynomials. With regard to GMRES, upper bounds on the residual norm in terms of polynomials are given in [4, 27, 31], and the tightness of these bounds is examined in =-=[17, 19, 33]-=-. Convergence analyses based on Ritz values are given in [5, 29,35]. The case of nearly singular matrices is analysed in [3], and comparisons with other methods are made in [2, 23]. The approach here ... |

26 | Comparison of splittings used with the conjugate gradient algorithm - GREENBAUM - 1979 |

23 | GMRES vs. ideal GMRES
- TOH
- 1997
(Show Context)
Citation Context ... methods is bounded in terms of polynomials. With regard to GMRES, upper bounds on the residual norm in terms of polynomials are given in [4, 27, 31], and the tightness of these bounds is examined in =-=[17, 19, 33]-=-. Convergence analyses based on Ritz values are given in [5, 29,35]. The case of nearly singular matrices is analysed in [3], and comparisons with other methods are made in [2, 23]. The approach here ... |

22 | How bad are Hankel matrices - TYRTYSHNIKOV - 1994 |

20 | Lower bounds for the condition number of Vandermonde matrices - Gautschi, Ingese - 1988 |

19 |
A simpler GMRES
- Walker, Zhou
- 1994
(Show Context)
Citation Context ...igenvalues, 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 o... |

18 |
The superlinear convergence behaviour of
- Vorst, Vuik
- 1993
(Show Context)
Citation Context ...per bounds on the residual norm in terms of polynomials are given in [4, 27, 31], and the tightness of these bounds is examined in [17, 19, 33]. Convergence analyses based on Ritz values are given in =-=[5, 29,35]-=-. The case of nearly singular matrices is analysed in [3], and comparisons with other methods are made in [2, 23]. The approach here is di erent because we ignore the way GMRES is implemented (e.g. vi... |

14 | The angle between complementary subspaces - Ipsen, Meyer - 1995 |

13 |
Krylov sequences of maximal length and convergence
- ARIOLI, PTÁK, et al.
- 1998
(Show Context)
Citation Context |

13 | Error analysis of Krylov methods in a nutshell
- HOCHBRUCK, LUBICH
- 1998
(Show Context)
Citation Context ...is examined in [17, 19, 33]. Convergence analyses based on Ritz values are given in [5, 29,35]. The case of nearly singular matrices is analysed in [3], and comparisons with other methods are made in =-=[2, 23]-=-. The approach here is di erent because we ignore the way GMRES is implemented (e.g. via Arnoldi's method) and we do not use polynomials to derive the bounds. Instead we exploit structure in the Krylo... |

11 | M.: Some new bounds on perturbation of subspaces - Kahan - 1969 |

10 |
Solution of sparse inde nite systems of linear equations
- Paige, Saunders
- 1975
(Show Context)
Citation Context ...ject 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... |

8 | Vandermonde matrices on the circle: spectral properties and conditioning - Córdova, Gautschi, et al. - 1990 |

8 | A singular value inequality for block matrices - Govaerts, Pryce - 1989 |

4 | Perturbation theory for the solution of systems of linear equations - Chandrasekaran, Ipsen - 1991 |

4 | Collinearity and least squares regression, Statist - Stewart - 1987 |

4 | A different approach to bounding the minimal residual norm in Krylov methods - Ipsen - 1998 |

2 |
A note on the convergence behaviour of GMRES
- Cao
- 1997
(Show Context)
Citation Context ...per bounds on the residual norm in terms of polynomials are given in [4, 27, 31], and the tightness of these bounds is examined in [17, 19, 33]. Convergence analyses based on Ritz values are given in =-=[5, 29,35]-=-. The case of nearly singular matrices is analysed in [3], and comparisons with other methods are made in [2, 23]. The approach here is di erent because we ignore the way GMRES is implemented (e.g. vi... |

1 | Bounds for iterates, inverses, spectral variation and elds of values of non-normal matrices - Henrici - 1962 |

1 | A di erent approach to bounding the minimal residual norm in Krylov methods - Ipsen - 1998 |