## Differences in the effects of rounding errors in Krylov solvers for symmetric indefinite linear systems (1999)

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Citations: | 15 - 0 self |

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@MISC{Sleijpen99differencesin,

author = {Gerard L.G. Sleijpen and Henk A. Van Der Vorst and Jan Modersitzki},

title = {Differences in the effects of rounding errors in Krylov solvers for symmetric indefinite linear systems},

year = {1999}

}

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### Abstract

The 3-term Lanczos process leads, for a symmetric matrix, to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of symmetric indefinite linear systems, by solving the reduced system in one way or another. This leads to well-known methods: MINRES, GMRES, and SYMMLQ. We will discuss in what way and to what extent these approaches differ in their sensitivity to rounding errors. In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the different methods, and we will not consider the errors in the Lanczos process itself. We will show that the method of solution may lead, under certain circumstances, to large additional errors, that are not corrected by continuing the iteration process. Our findings are supported and illustrated by numerical examples. 1 Introduction We will consider iterative methods for the construction of approximate solutions, starting with...

### Citations

1323 |
GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems
- Saad, Schultz
- 1986
(Show Context)
Citation Context ...m the Krylov subspace have to be saved (in fact, MINRES works with transformed basis vectors; this will be explained in x2.3). For the implementation of MINRES that we have used, see Fig. 2. 2. GMRES =-=[13]-=-: This method also minimizes, for y k 2 R k , the residual kb \Gamma Ax k k 2 . GMRES was designed for unsymmetric matrices, for which the orthogonalisation of the Krylov basis is done with Arnoldi's ... |

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

764 | The Symmetric Eigenvalue Problem - Parlett - 1980 |

710 | Method Of Conjugate Gradient for Solving Linear Equations - Hestenes, E - 1952 |

491 | der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition - Berry, Demmel, et al. - 1994 |

324 | Iterative Methods for Solving Linear Systems - Greenbaum - 1997 |

320 |
Solution of sparse indefinite systems of linear equations
- Paige, Saunders
- 1975
(Show Context)
Citation Context ... precision, but it will give us some insight in answering practical questions such as: -- When and why is MINRES less accurate than SYMMLQ? This question was already posed in the original publication =-=[11]-=-, but the answer in [11, p.625] is largely speculative. -- Is MINRES suspect for ill-conditioned systems, because of the minimal residual approach (see [11, p.619])? Hints are given for the explanatio... |

187 | Matrix computations, third edition - Golub, Loan - 1996 |

82 | Condition numbers and equilibration of matrices - Sluis - 1969 |

80 | der Vorst. Approximate solutions and eigenvalue bounds from Krylov subspaces. Numerical Linear Algebra with Applications - Paige, Parlett, et al. - 1995 |

56 |
A theoretical comparison of the Arnoldi and GMRES algorithms
- Brown
- 1991
(Show Context)
Citation Context ...calars s and c represent the Givens transformation used in the kth step of MINRES. This relation is a special case of the slightly more general relation between GMRES and FOM residuals, formulated in =-=[1, 16]-=-. For symmetric A, GMRES is equivalent with MINRES, and FOM is equivalent with CG. Since r CG k = kr CG k k 2 v k+1 ? r MR k\Gamma1 2 K k (A; r 0 ), it follows that r MR k = s 2 r MR k\Gamma1 + flv k+... |

49 | Polynomial based iteration methods for symmetric linear systems - FISCHER - 1996 |

42 |
Error analysis of the Lanczos algorithm for tridiagonalizing a symmetric matrix
- Paige
- 1976
(Show Context)
Citation Context ...zos process can be recast in matrix formulation as AV k = V k+1T k ; (1) in which V j is defined as the n by j matrix with columns v 1 , : : :, v j , and T k is a k + 1 by k tridiagonal matrix. Paige =-=[9]-=- has shown that in finite precision arithmetic, the Lanczos process can be implemented so that the computed V k+1 and T k satisfy AV k = V k+1 T k + F k ; (2) with, under mild conditions for k, kF k k... |

41 |
The superlinear convergence behaviour of GMRES
- Vorst, Vuik
- 1993
(Show Context)
Citation Context ...calars s and c represent the Givens transformation used in the kth step of MINRES. This relation is a special case of the slightly more general relation between GMRES and FOM residuals, formulated in =-=[1, 16]-=-. For symmetric A, GMRES is equivalent with MINRES, and FOM is equivalent with CG. Since r CG k = kr CG k k 2 v k+1 ? r MR k\Gamma1 2 K k (A; r 0 ), it follows that r MR k = s 2 r MR k\Gamma1 + flv k+... |

38 | A survey of preconditioned iterative methods - Bruaset - 1995 |

34 |
Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences
- Greenbaum
- 1989
(Show Context)
Citation Context ...e would like to replace R k in the error bounds by something that can directly be related to A. Therefore, we note that R T k R k = T T k T k ; ignoring errors in the order of u. It has been shown in =-=[5, 7]-=- that the matrix T k that has been obtained in finite precision arithmetic, may be interpreted as the exact Lanczos matrix obtained from a matrix e A in which eigenvalues of A are replaced by multiple... |

28 | Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem - Paige - 1980 |

26 |
Predicting the behavior of finite precision Lanczos and conjugate gradient computations
- Greenbaum, Strakoˇs
- 1992
(Show Context)
Citation Context ...o know how the generating formulas behave in finite precision arithmetic. The errors in the underlying Lanczos process have been analysed by Paige [9, 10]. It has been proven by Greenbaum and Strakos =-=[7]-=- that rounding errors in the Lanczos process may have a delaying effect on the convergence of iterative solvers, but do not prevent eventual convergence in general. Usually, a rigorous error analysis ... |

22 | Relaxationsmethoden bester strategie zur lösung linearer gleichungssysteme - Stiefel - 1955 |

21 | der Vorst. Reliable updated residuals in hybrid Bi-CG methods - Sleijpen, van |

18 |
The superlinear convergence behaviour of
- Vorst, Vuik
- 1993
(Show Context)
Citation Context ...calars s and c represent the Givens transformation used in the kth step of MINRES. This relation is a special case of the slightly more general relation between GMRES and FOM residuals, formulated in =-=[1, 16]-=-. For symmetric A, GMRES is equivalent with MINRES, and FOM is equivalent with CG. Since r CG k = kr CG k k 2 v k+1 ? r MR k\Gamma1 2 K k (A; r 0 ), it follows that r MR k = s 2 r MR k\Gamma1 + flv k+... |

16 |
Efficient High Accuracy Solutions with GMRES(m
- Turner, Walker
- 1992
(Show Context)
Citation Context ...+z j . The procedure could be repeated and eventually this leads to approximations for x so that the relative error in the residual is in the order of machine precision (for more details on this, see =-=[14]-=-). However, if we would use MINRES then, after restart, we have to ROUNDING ERRORS IN KRYLOV SOLVERS 25 carry out at least a number of iterations for the reduction by a factor equal to the condition n... |

5 | A Survey of Preconditioned Iterative - Bruaset - 1995 |

3 |
Rozlo zn' ik, and Z. Strako s, Numerical stability of GMRES
- a, Greenbaum, et al.
- 1995
(Show Context)
Citation Context ...RES has been restricted to certain parts of the algorithm. For an analysis of all errors in the original GMRES, including those in the Arnoldi process and the Givens rotations, for unsymmetric A, see =-=[3]-=-. 2.3. Error analysis for MINRES. For MINRES we have to study the errors in the evaluation in finite precision of \Gamma V k R \Gamma1 k \Delta z k . We will first analyze the floating point errors in... |

3 | Accuracy and e#ectiveness of the Lanczos algorithm for the symmetric eigenproblem - Paige - 1980 |

2 | Condition numbers and equilibration of matrices - Sluis - 1969 |

1 | Rozlo zn k, and Z. Strako s, Numerical stability of GMRES - a, Greenbaum, et al. - 1995 |