# Fast verification of solutions of matrix equations

Shinichi Oishi, Siegfried M. Rump

Research output: Contribution to journalArticle

35 Citations (Scopus)

### 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 2/3n3 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.

Original language English 755-773 19 Numerische Mathematik 90 4 https://doi.org/10.1007/s002110100310 Published - 2002 Feb

### Fingerprint

LU decomposition
Matrix Equation
Decomposition
Nonsingularity
Error Bounds
Fast Algorithm
Computational Cost
Approximate Solution
Costs

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics
• Computational Mathematics

### Cite this

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

In: Numerische Mathematik, Vol. 90, No. 4, 02.2002, p. 755-773.

Research output: Contribution to journalArticle

Oishi, Shinichi ; Rump, Siegfried M. / Fast verification of solutions of matrix equations. In: Numerische Mathematik. 2002 ; Vol. 90, No. 4. pp. 755-773.
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