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Error Analysis of Direct Methods of Matrix Inversion

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J. H. WilkinsonAuthors Info & Claims

Journal of the ACM (JACM), Volume 8, Issue 3

Pages 281 - 330

https://doi.org/10.1145/321075.321076

Published: 01 July 1961 Publication History

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Cited By

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Sahraneshinsamani NCatalán SHerrero J(2025)Mixed-precision pre-pivoting strategy for the LU factorizationThe Journal of Supercomputing10.1007/s11227-024-06523-w81:1Online publication date: 1-Jan-2025
https://dl.acm.org/doi/10.1007/s11227-024-06523-w
Peca-Medlin J(2024)Distribution of the number of pivots needed using Gaussian elimination with partial pivoting on random matricesThe Annals of Applied Probability10.1214/23-AAP202334:2Online publication date: 1-Apr-2024
https://doi.org/10.1214/23-AAP2023
Peca-Medlin J(2024)Growth Factors of Orthogonal Matrices and Local Behavior of Gaussian Elimination with Partial and Complete PivotingSIAM Journal on Matrix Analysis and Applications10.1137/23M159773345:3(1599-1620)Online publication date: 22-Aug-2024
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Index Terms

Error Analysis of Direct Methods of Matrix Inversion
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Linear algebra algorithms
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on matrices

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We study the inversion analog of the well-known Gauss algorithm for multiplying complex matrices. A simple version is ${(A + i B)}^{- 1} = {(A + B A^{- 1} B)}^{- 1} - i A^{- 1} B {(A + B A^{- 1} B)}^{- 1}$ when A is invertible, which may be traced back to Frobenius but has received scant ... $^{}$
Generalized matrix inversion is not harder than matrix multiplication

Starting from the Strassen method for rapid matrix multiplication and inversion as well as from the recursive Cholesky factorization algorithm, we introduced a completely block recursive algorithm for generalized Cholesky factorization of a given ...
Error control in the numerical inversion of the eigenvector matrix

The numerical inversion of the eigenvector matrix is an essential mathematical operation in various branches of mathematics, science and engineering. There is no single method which yields an exact inverse of any given matrix. There are several ...

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Published In

Journal of the ACM Volume 8, Issue 3

July 1961

186 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/321075

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 July 1961

Published in JACM Volume 8, Issue 3

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Sahraneshinsamani NCatalán SHerrero J(2025)Mixed-precision pre-pivoting strategy for the LU factorizationThe Journal of Supercomputing10.1007/s11227-024-06523-w81:1Online publication date: 1-Jan-2025
https://dl.acm.org/doi/10.1007/s11227-024-06523-w
Peca-Medlin J(2024)Distribution of the number of pivots needed using Gaussian elimination with partial pivoting on random matricesThe Annals of Applied Probability10.1214/23-AAP202334:2Online publication date: 1-Apr-2024
https://doi.org/10.1214/23-AAP2023
Peca-Medlin J(2024)Growth Factors of Orthogonal Matrices and Local Behavior of Gaussian Elimination with Partial and Complete PivotingSIAM Journal on Matrix Analysis and Applications10.1137/23M159773345:3(1599-1620)Online publication date: 22-Aug-2024
https://doi.org/10.1137/23M1597733
Edelman AUrschel J(2024)Some New Results on the Maximum Growth Factor in Gaussian EliminationSIAM Journal on Matrix Analysis and Applications10.1137/23M157190345:2(967-991)Online publication date: 26-Apr-2024
https://doi.org/10.1137/23M1571903
Giles MSheridan-Methven O(2024)Rounding Error Using Low Precision Approximate Random VariablesSIAM Journal on Scientific Computing10.1137/23M155281446:4(B502-B526)Online publication date: 16-Jul-2024
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Noorizadegan ASchaback R(2024)Introducing the evaluation condition number: A novel assessment of conditioning in radial basis function methodsEngineering Analysis with Boundary Elements10.1016/j.enganabound.2024.105827166(105827)Online publication date: Sep-2024
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Coria IUrkullu GUriarte Hde Bustos I(2024)An efficient algorithm for the minimal least squares solution of linear systems with indefinite symmetric matricesJournal of Computational and Applied Mathematics10.1016/j.cam.2024.115783444(115783)Online publication date: Jul-2024
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Lardas EMitrouli M(2024)On the growth factor of Hadamard matrices of order 20Applied Numerical Mathematics10.1016/j.apnum.2024.01.019Online publication date: Feb-2024
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Sekhri EKapoor RTamre M(2024)Review of State-of-the-Art Microwave Filter Tuning Techniques and Implementation of a Novel Tuning Algorithm Using Expert-Based Hybrid LearningWireless Personal Communications10.1007/s11277-024-10894-x134:2(625-681)Online publication date: 21-Feb-2024
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Complex matrix inversion via real matrix inversions

Generalized matrix inversion is not harder than matrix multiplication

Error control in the numerical inversion of the eigenvector matrix

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