Iterative refinement implies numerical stability for Gaussian elimination
TLDR
It is shown that even a single iteration of iterative refinement in single precision is enough to make Gaussian elimination stable in a very strong sense and row pivoting is inferior to column pivoting in situations where the norm of the residual is important.Abstract:
Because of scaling problems, Gaussian elimination with pivoting is not always as accurate as one might reasonably expect. It is shown that even a single iteration of iterative refinement in single precision is enough to make Gaussian elimination stable in a very strong sense. Also, it is shown that without iterative refinement row pivoting is inferior to column pivoting in situations where the norm of the residual is important.read more
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Linear Programming 1: Introduction
TL;DR: Encompassing all the major topics students will encounter in courses on the subject, the authors teach both the underlying mathematical foundations and how these ideas are implemented in practice, making this an ideal textbook for all those coming to the subject for the first time.
Journal ArticleDOI
Computer Solution of Linear Algebraic Systems. By G. Forsythe and C. B. Moler. Pp. xi, 148. 1967. (Prentice-Hall.)
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Accurate Sum and Dot Product
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SuperLU Users'' Guide
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Verification methods for dense and sparse systems of equations
TL;DR: In this paper, the authors describe verification methods for dense and large sparse systems of linear and nonlinear equations, and present a fast interval library having been developed at the author's institute.
References
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Introduction to matrix computations
TL;DR: Rounding-Error Analysis of Solution of Triangular Systems and of Gaussian Elimination.