Error Analysis of Direct Methods of Matrix Inversion

@article{Wilkinson1961ErrorAO,
  title={Error Analysis of Direct Methods of Matrix Inversion},
  author={James Hardy Wilkinson},
  journal={J. ACM},
  year={1961},
  volume={8},
  pages={281-330}
}
1. In order to assess the relative effectiveness of methods of inverting a matrix it is useful to have a priori bounds for the errors in the computed inverses. In this paper we determine such error bounds for a number of the most effective direct methods. To illustrate fully the techniques we have used, some of the analysis has been done for floating-point computat ion and some for fixed-point. In all cases it has been assumed tha t the computat ion has been performed using a precision of t… Expand
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References

SHOWING 1-9 OF 9 REFERENCES
Error analysis of floating-point computation
This paper consists of two main sections. In the first the bounds are derived for the rounding errors made in the fundamental floating-point arithmetic operations. In the second, these results areExpand
ROUNDING-OFF ERRORS IN MATRIX PROCESSES
A number of methods of solving sets of linear equations and inverting matrices are discussed. The theory of the rounding-off errors involved is investigated for some of the methods. In all casesExpand
Unitary Triangularization of a Nonsymmetric Matrix
TLDR
This note points out that the same result can be obtained with fewer arithmetic operations, and, in particular, for inverting a square matrix of order N, at most 2(N-1) square roots are required. Expand
Numerical inverting of matrices of high order
PREFACE 188 CHAPTER VIII. Probabilistic estimates for bounds of matrices 8.1 A result of Bargmann, Montgomery and von Neumann 188 8.2 An estimate for the length of a vector 191 8.3 The fundamentalExpand
The hnear equations prob|em
  • Technical Report no. 3. Applied Mathe- matics and Statistics Series, Nonr
  • 1959
Computational methods of hnear algebra
  • Computational methods of hnear algebra
  • 1959
The hnear equations prob|em Technical Report no. 3. Applied Mathematics and Statistics Series
  • Nonr
  • 1959
Solution of eigenvalue problems with the L-R transformation
  • In National Bureau of Standards Apphed Math Series
  • 1958