# Convergence of Rump's method for inverting arbitrarily ill-conditioned matrices

Shinichi Oishi, Kunio Tanabe, Takeshi Ogita, Siegfried M. Rump

Research output: Contribution to journalArticle

19 Citations (Scopus)

### Abstract

In this paper, the problem of inverting regular matrices with arbitrarily large condition number is treated in double precision defined by IEEE 754 floating point standard. In about 1984, Rump derived a method for inverting arbitrarily ill-conditioned matrices. The method requires the possibility to calculate a dot product in higher precision. Rump's method is of theoretical interest. Rump made it clear that inverting an arbitrarily ill-conditioned matrix in single or double precision does not produce meaningless numbers, but contains a lot of information in it. Rump's method uses such inverses as preconditioners. Numerical experiments exhibit that Rump's method converges rapidly for various matrices with large condition numbers. Why Rump's method is so efficient for inverting arbitrarily ill-conditioned matrices is a little mysterious. Thus, to prove its convergence is an interesting problem in numerical error analysis. In this article, a convergence theorem is presented for a variant of Rump's method.

Original language English 533-544 12 Journal of Computational and Applied Mathematics 205 1 https://doi.org/10.1016/j.cam.2006.05.022 Published - 2007 Aug 1

### Fingerprint

Condition number
Error analysis
Floating point
Error Analysis
Scalar, inner or dot product
Convergence Theorem
Preconditioner
Numerical Analysis
Numerical Experiment
Converge
Calculate
Experiments
Standards

### Keywords

• Accurate dot product
• Ill-conditioned matrix
• Matrix inversion
• Precondition

### ASJC Scopus subject areas

• Applied Mathematics
• Computational Mathematics
• Numerical Analysis

### Cite this

Convergence of Rump's method for inverting arbitrarily ill-conditioned matrices. / Oishi, Shinichi; Tanabe, Kunio; Ogita, Takeshi; Rump, Siegfried M.

In: Journal of Computational and Applied Mathematics, Vol. 205, No. 1, 01.08.2007, p. 533-544.

Research output: Contribution to journalArticle

Oishi, Shinichi ; Tanabe, Kunio ; Ogita, Takeshi ; Rump, Siegfried M. / Convergence of Rump's method for inverting arbitrarily ill-conditioned matrices. In: Journal of Computational and Applied Mathematics. 2007 ; Vol. 205, No. 1. pp. 533-544.
@article{3c056b81943049a2a4c293a7efbc1aa3,
title = "Convergence of Rump's method for inverting arbitrarily ill-conditioned matrices",
abstract = "In this paper, the problem of inverting regular matrices with arbitrarily large condition number is treated in double precision defined by IEEE 754 floating point standard. In about 1984, Rump derived a method for inverting arbitrarily ill-conditioned matrices. The method requires the possibility to calculate a dot product in higher precision. Rump's method is of theoretical interest. Rump made it clear that inverting an arbitrarily ill-conditioned matrix in single or double precision does not produce meaningless numbers, but contains a lot of information in it. Rump's method uses such inverses as preconditioners. Numerical experiments exhibit that Rump's method converges rapidly for various matrices with large condition numbers. Why Rump's method is so efficient for inverting arbitrarily ill-conditioned matrices is a little mysterious. Thus, to prove its convergence is an interesting problem in numerical error analysis. In this article, a convergence theorem is presented for a variant of Rump's method.",
keywords = "Accurate dot product, Ill-conditioned matrix, Matrix inversion, Precondition",
author = "Shinichi Oishi and Kunio Tanabe and Takeshi Ogita and Rump, {Siegfried M.}",
year = "2007",
month = "8",
day = "1",
doi = "10.1016/j.cam.2006.05.022",
language = "English",
volume = "205",
pages = "533--544",
journal = "Journal of Computational and Applied Mathematics",
issn = "0377-0427",
publisher = "Elsevier",
number = "1",

}

TY - JOUR

T1 - Convergence of Rump's method for inverting arbitrarily ill-conditioned matrices

AU - Oishi, Shinichi

AU - Tanabe, Kunio

AU - Ogita, Takeshi

AU - Rump, Siegfried M.

PY - 2007/8/1

Y1 - 2007/8/1

N2 - In this paper, the problem of inverting regular matrices with arbitrarily large condition number is treated in double precision defined by IEEE 754 floating point standard. In about 1984, Rump derived a method for inverting arbitrarily ill-conditioned matrices. The method requires the possibility to calculate a dot product in higher precision. Rump's method is of theoretical interest. Rump made it clear that inverting an arbitrarily ill-conditioned matrix in single or double precision does not produce meaningless numbers, but contains a lot of information in it. Rump's method uses such inverses as preconditioners. Numerical experiments exhibit that Rump's method converges rapidly for various matrices with large condition numbers. Why Rump's method is so efficient for inverting arbitrarily ill-conditioned matrices is a little mysterious. Thus, to prove its convergence is an interesting problem in numerical error analysis. In this article, a convergence theorem is presented for a variant of Rump's method.

AB - In this paper, the problem of inverting regular matrices with arbitrarily large condition number is treated in double precision defined by IEEE 754 floating point standard. In about 1984, Rump derived a method for inverting arbitrarily ill-conditioned matrices. The method requires the possibility to calculate a dot product in higher precision. Rump's method is of theoretical interest. Rump made it clear that inverting an arbitrarily ill-conditioned matrix in single or double precision does not produce meaningless numbers, but contains a lot of information in it. Rump's method uses such inverses as preconditioners. Numerical experiments exhibit that Rump's method converges rapidly for various matrices with large condition numbers. Why Rump's method is so efficient for inverting arbitrarily ill-conditioned matrices is a little mysterious. Thus, to prove its convergence is an interesting problem in numerical error analysis. In this article, a convergence theorem is presented for a variant of Rump's method.

KW - Accurate dot product

KW - Ill-conditioned matrix

KW - Matrix inversion

KW - Precondition

UR - http://www.scopus.com/inward/record.url?scp=34247358712&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34247358712&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2006.05.022

DO - 10.1016/j.cam.2006.05.022

M3 - Article

AN - SCOPUS:34247358712

VL - 205

SP - 533

EP - 544

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1

ER -