### Abstract

In this paper, we are concerned with a matrix equation Ax = b where A is an n × n real matrix and x and b are n-vectors. Assume that an approximate solution x is given together with an approximate LU decomposition. We will present fast algorithms for proving nonsingularity of A and for calculating rigorous error bounds for ∥A^{-1} b - x̃∥_{∞}. The emphasis is on rigour of the bounds. The purpose of this paper is to propose different algorithms, the fastest with 2/3n^{3} flops computational cost for the verification step, the same as for the LU decomposition. The presented algorithms exclusively use library routines for LU decomposition and for all other matrix and vector operations.

Original language | English |
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Pages (from-to) | 755-773 |

Number of pages | 19 |

Journal | Numerische Mathematik |

Volume | 90 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2002 Feb |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Computational Mathematics

### Cite this

*Numerische Mathematik*,

*90*(4), 755-773. https://doi.org/10.1007/s002110100310

**Fast verification of solutions of matrix equations.** / Oishi, Shinichi; Rump, Siegfried M.

Research output: Contribution to journal › Article

*Numerische Mathematik*, vol. 90, no. 4, pp. 755-773. https://doi.org/10.1007/s002110100310

}

TY - JOUR

T1 - Fast verification of solutions of matrix equations

AU - Oishi, Shinichi

AU - Rump, Siegfried M.

PY - 2002/2

Y1 - 2002/2

N2 - In this paper, we are concerned with a matrix equation Ax = b where A is an n × n real matrix and x and b are n-vectors. Assume that an approximate solution x is given together with an approximate LU decomposition. We will present fast algorithms for proving nonsingularity of A and for calculating rigorous error bounds for ∥A-1 b - x̃∥∞. The emphasis is on rigour of the bounds. The purpose of this paper is to propose different algorithms, the fastest with 2/3n3 flops computational cost for the verification step, the same as for the LU decomposition. The presented algorithms exclusively use library routines for LU decomposition and for all other matrix and vector operations.

AB - In this paper, we are concerned with a matrix equation Ax = b where A is an n × n real matrix and x and b are n-vectors. Assume that an approximate solution x is given together with an approximate LU decomposition. We will present fast algorithms for proving nonsingularity of A and for calculating rigorous error bounds for ∥A-1 b - x̃∥∞. The emphasis is on rigour of the bounds. The purpose of this paper is to propose different algorithms, the fastest with 2/3n3 flops computational cost for the verification step, the same as for the LU decomposition. The presented algorithms exclusively use library routines for LU decomposition and for all other matrix and vector operations.

UR - http://www.scopus.com/inward/record.url?scp=0036003238&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036003238&partnerID=8YFLogxK

U2 - 10.1007/s002110100310

DO - 10.1007/s002110100310

M3 - Article

AN - SCOPUS:0036003238

VL - 90

SP - 755

EP - 773

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 4

ER -