On the existence and computation of -factorizations with small pivots

Author:
Tony F. Chan

Journal:
Math. Comp. **42** (1984), 535-547

MSC:
Primary 65F05; Secondary 15A23

DOI:
https://doi.org/10.1090/S0025-5718-1984-0736451-4

Corrigendum:
Math. Comp. **44** (1985), 282.

MathSciNet review:
736451

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *A* be an *n* by *n* matrix which may be singular with a one-dimensional null space, and consider the *LU*-factorization of *A*. When *A* is exactly singular, we show conditions under which a pivoting strategy will produce a zero *n*th pivot. When *A* is not singular, we show conditions under which a pivoting strategy will produce an *n*th pivot that is or , where is the smallest singular value of *A* and is the condition number of *A*. These conditions are expressed in terms of the elements of in general but reduce to conditions on the elements of the singular vectors corresponding to when *A* is nearly or exactly singular. They can be used to build a 2-pass factorization algorithm which is *guaranteed* to produce a small *n*th pivot for nearly singular matrices. As an example, we exhibit an *LU*-factorization of the *n* by *n* upper triangular matrix

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1984-0736451-4

Keywords:
LU-factorizations,
singular systems

Article copyright:
© Copyright 1984
American Mathematical Society