### Abstract

Original language | English |
---|---|

Pages (from-to) | 111-130 |

Number of pages | 19 |

Journal | Linear Algebra and its Applications |

Volume | 174 |

DOIs | |

Publication status | Published - Sep 1992 |

### Fingerprint

### Keywords

- componentwise backward error
- numerical mathematics
- linear systems
- Hölder p-norms

### Cite this

*Linear Algebra and its Applications*,

*174*, 111-130. https://doi.org/10.1016/0024-3795(92)90046-D

}

*Linear Algebra and its Applications*, vol. 174, pp. 111-130. https://doi.org/10.1016/0024-3795(92)90046-D

**Componentwise perturbation theory for linear systems with multiple right-hand sides.** / Higham, D.J.; Higham, N.J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Componentwise perturbation theory for linear systems with multiple right-hand sides

AU - Higham, D.J.

AU - Higham, N.J.

PY - 1992/9

Y1 - 1992/9

N2 - Existing definitions of componentwise backward error and componentwise condition number for linear systems are extended to systems with multiple right-hand sides and to a general class of componentwise measure of perturbations involving Hölder p-norms. It is shown that for a system of order n with r right-hand sides, the componentwise backward error can be computed by finding the minimum p-norm solutions to n underdetermined linear systems, and an explicit expression is obtained in the case r = 1. A perturbation bound is derived, and from this the componentwise condition number is obtained to within a multiplicative constant. Applications of the results are discussed to invariant subspace computations, quasi-Newton methods based on multiple secant equations, and an inverse ODE problem.

AB - Existing definitions of componentwise backward error and componentwise condition number for linear systems are extended to systems with multiple right-hand sides and to a general class of componentwise measure of perturbations involving Hölder p-norms. It is shown that for a system of order n with r right-hand sides, the componentwise backward error can be computed by finding the minimum p-norm solutions to n underdetermined linear systems, and an explicit expression is obtained in the case r = 1. A perturbation bound is derived, and from this the componentwise condition number is obtained to within a multiplicative constant. Applications of the results are discussed to invariant subspace computations, quasi-Newton methods based on multiple secant equations, and an inverse ODE problem.

KW - componentwise backward error

KW - numerical mathematics

KW - linear systems

KW - Hölder p-norms

UR - http://dx.doi.org/10.1016/0024-3795(92)90046-D

U2 - 10.1016/0024-3795(92)90046-D

DO - 10.1016/0024-3795(92)90046-D

M3 - Article

VL - 174

SP - 111

EP - 130

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

SN - 0024-3795

ER -