Average condition number for solving linear equations

N. Weiss; G. W. Wasilkowski; H. Woźniakowski; M. Shub

doi:10.1016/0024-3795(86)90266-1

Average condition number for solving linear equations

N. Weiss, G. W. Wasilkowski, H. Woźniakowski, M. Shub

Computer Science

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

6 Scopus citations

Abstract

We study an average condition number and an average loss of precision for the solution of linear equations and prove that the average case is strongly related to the worst case. This holds if the perturbations of the matrix are measured in Frobenius or spectral norm or componentwise. In particular, for the Frobenius norm we show that one gains about log₂n+0.9 bits on the average as compared to the worst case, n being the dimension of the system of linear equations.

Original language	English
Pages (from-to)	79-102
Number of pages	24
Journal	Linear Algebra and Its Applications
Volume	83
Issue number	C
DOIs	https://doi.org/10.1016/0024-3795(86)90266-1
State	Published - Nov 1986

Bibliographical note

Funding Information:
*This research was supported in part by the National Science Foundation DCR-82-14322 (G. W. W. and H. W.) and Grant MC%3241267 (M. S.).

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/0024-3795(86)90266-1

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title = "Average condition number for solving linear equations",

abstract = "We study an average condition number and an average loss of precision for the solution of linear equations and prove that the average case is strongly related to the worst case. This holds if the perturbations of the matrix are measured in Frobenius or spectral norm or componentwise. In particular, for the Frobenius norm we show that one gains about log2n+0.9 bits on the average as compared to the worst case, n being the dimension of the system of linear equations.",

author = "N. Weiss and Wasilkowski, {G. W.} and H. Wo{\'z}niakowski and M. Shub",

note = "Funding Information: *This research was supported in part by the National Science Foundation DCR-82-14322 (G. W. W. and H. W.) and Grant MC%3241267 (M. S.).",

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AU - Weiss, N.

AU - Wasilkowski, G. W.

AU - Woźniakowski, H.

AU - Shub, M.

N1 - Funding Information: *This research was supported in part by the National Science Foundation DCR-82-14322 (G. W. W. and H. W.) and Grant MC%3241267 (M. S.).

PY - 1986/11

Y1 - 1986/11

N2 - We study an average condition number and an average loss of precision for the solution of linear equations and prove that the average case is strongly related to the worst case. This holds if the perturbations of the matrix are measured in Frobenius or spectral norm or componentwise. In particular, for the Frobenius norm we show that one gains about log2n+0.9 bits on the average as compared to the worst case, n being the dimension of the system of linear equations.

AB - We study an average condition number and an average loss of precision for the solution of linear equations and prove that the average case is strongly related to the worst case. This holds if the perturbations of the matrix are measured in Frobenius or spectral norm or componentwise. In particular, for the Frobenius norm we show that one gains about log2n+0.9 bits on the average as compared to the worst case, n being the dimension of the system of linear equations.

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JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - C

ER -

Average condition number for solving linear equations

Abstract

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ASJC Scopus subject areas

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