Journal ArticleDOI
Scaling for Numerical Stability in Gaussian Elimination
TLDR
Roundoff error m the solution of near algebraic systems is studied using a more reahstsc notion of what st means to perturb a problem, namely, that each datum is subject to a relatwely small change.Abstract:
Roundoff error m the solution of hnear algebraic systems is stud,ed using a more reahstsc notion of what st means to perturb a problem, namely, that each datum :s subject to a relatwely small change Th:s ,s particularly appropriate for sparse linear systems The condition number :s determined for th:s approach The effect of scahng on the stabdlty of Gaussmn ellmmat,on is stud:ed, and st is d:scovered that the proper way to scale a system depends on the right-hand s:de However, ff only the norm of the error is of concern, then there ~s a good way to scale that does not depend on the right-hand stderead more
Citations
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Book
Linear and Nonlinear Optimization
TL;DR: This chapter discusses the foundations of optimization, and some of the methods for unconstrained optimization, as well as topics from linear algebra, including the simplex method and other fundamentals.
Journal ArticleDOI
Improving Multifrontal Methods by Means of Block Low-Rank Representations
Patrick R. Amestoy,Cleve Ashcraft,Olivier Boiteau,Alfredo Buttari,Jean-Yves L'Excellent,Clement Weisbecker +5 more
TL;DR: A low-rank format called Block Low-Rank (BLR) is proposed, and it is explained how it can be used to reduce the memory footprint and the complexity of direct solvers for sparse matrices based on the multifrontal method.
Book
Solving sparse linear systems with sparse backward error
TL;DR: It is shown that one step of iterative refinement, even with single precision accumulation of residuals, guarantees such a small backward error if the final matrix is not too ill-conditioned and the solution components do not vary too much in magnitude.
Journal ArticleDOI
Iterative refinement implies numerical stability for Gaussian elimination
TL;DR: 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.
Journal ArticleDOI
Backward error and condition of structured linear systems
TL;DR: It is shown that when the structure comprises linear dependence on a set of parameters, the structured componentwise backward error is given by the solution of minimal $\infty $ -norm to an underdetermined linear system.
References
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Journal ArticleDOI
Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides
W. Oettli,W. Prager +1 more
TL;DR: In this article, conditions are established under which a given approximate solution of a system of n linear equations withn unknowns is the exact solution of the modified system whose coefficients and right-hand sides are within a given neighborhood of those of the original system.
Journal ArticleDOI
C. Forsythe and C. B. Moler, Computer Solution of Linear Algebraic Systems. (Series in Automatic Computation) XI + 148 S. Englewood Cliffs, N.J. 1967. Prentice-Hall, Inc. Preis geb. 54 s. net
Journal ArticleDOI
Numerical linear algebra
TL;DR: The primordial problems of linear algebra are the solution of a system of linear equations and the eigenvalue problem for the eigvalues λk, and corresponding eigenvectors of a given matrix A as discussed by the authors.