Iterative refinement for ill-conditioned linear systems

Shin'ichi Oishi; Takeshi Ogita; Siegfried M. Rump

doi:10.1007/BF03186544

Iterative refinement for ill-conditioned linear systems

Shin'ichi Oishi^*, Takeshi Ogita, Siegfried M. Rump

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

This paper treats a linear equation Aυ = b, where A ∈ F ^n×n and b ∈ Fⁿ. Here, F is a set of floating point numbers. Let u be the unit round-off of the working precision and κ(A) = ∥A∥∞∥A^-1∥∞ be the condition number of the problem. In this paper, ill-conditioned problems with 1 < uκ(A) < ∞ are considered and an iterative refinement algorithm for the problems is proposed. In this paper, the forward and backward stability will be shown for this iterative refinement algorithm.

Original language	English
Pages (from-to)	465-476
Number of pages	12
Journal	Japan Journal of Industrial and Applied Mathematics
Volume	26
Issue number	2-3
DOIs	https://doi.org/10.1007/BF03186544
Publication status	Published - 2009 Oct

Keywords

Ill-conditioned linear systems
Iterative refinement
Verified numerical computation

ASJC Scopus subject areas

Engineering(all)
Applied Mathematics

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10.1007/BF03186544

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title = "Iterative refinement for ill-conditioned linear systems",

abstract = "This paper treats a linear equation Aυ = b, where A ∈ F n×n and b ∈ Fn. Here, F is a set of floating point numbers. Let u be the unit round-off of the working precision and κ(A) = ∥A∥∞∥A-1∥∞ be the condition number of the problem. In this paper, ill-conditioned problems with 1 < uκ(A) < ∞ are considered and an iterative refinement algorithm for the problems is proposed. In this paper, the forward and backward stability will be shown for this iterative refinement algorithm.",

keywords = "Ill-conditioned linear systems, Iterative refinement, Verified numerical computation",

author = "Shin'ichi Oishi and Takeshi Ogita and Rump, {Siegfried M.}",

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T1 - Iterative refinement for ill-conditioned linear systems

AU - Oishi, Shin'ichi

AU - Ogita, Takeshi

AU - Rump, Siegfried M.

PY - 2009/10

Y1 - 2009/10

N2 - This paper treats a linear equation Aυ = b, where A ∈ F n×n and b ∈ Fn. Here, F is a set of floating point numbers. Let u be the unit round-off of the working precision and κ(A) = ∥A∥∞∥A-1∥∞ be the condition number of the problem. In this paper, ill-conditioned problems with 1 < uκ(A) < ∞ are considered and an iterative refinement algorithm for the problems is proposed. In this paper, the forward and backward stability will be shown for this iterative refinement algorithm.

AB - This paper treats a linear equation Aυ = b, where A ∈ F n×n and b ∈ Fn. Here, F is a set of floating point numbers. Let u be the unit round-off of the working precision and κ(A) = ∥A∥∞∥A-1∥∞ be the condition number of the problem. In this paper, ill-conditioned problems with 1 < uκ(A) < ∞ are considered and an iterative refinement algorithm for the problems is proposed. In this paper, the forward and backward stability will be shown for this iterative refinement algorithm.

KW - Ill-conditioned linear systems

KW - Iterative refinement

KW - Verified numerical computation

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JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

IS - 2-3

ER -

Iterative refinement for ill-conditioned linear systems

Abstract

Keywords

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