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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On the existence and computation of $LU$-factorizations with small pivots
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by Tony F. Chan PDF
Math. Comp. 42 (1984), 535-547 Request permission

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 nth pivot. When A is not singular, we show conditions under which a pivoting strategy will produce an nth pivot that is $O({\sigma _n})$ or $O({\kappa ^{ - 1}}(A))$, where ${\sigma _n}$ is the smallest singular value of A and $\kappa (A)$ is the condition number of A. These conditions are expressed in terms of the elements of ${A^{ - 1}}$ in general but reduce to conditions on the elements of the singular vectors corresponding to ${\sigma _n}$ 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 nth pivot for nearly singular matrices. As an example, we exhibit an LU-factorization of the n by n upper triangular matrix \[ T = \left [ {\begin {array}{*{20}{c}} 1 & { - 1} & {} & { - 1} & {} & \cdot & \cdot & { - 1} \\ {} & 1 & {} & \cdot & {} & {} & {} & \cdot \\ {} & {} & {} & 1 & \cdot & {} & {} & {} \\ {} & {} & {} & {} & \cdot & \cdot & {} & \cdot \\ {} & {} & 0 & {} & {} & \cdot & \cdot & \cdot \\ {} & {} & {} & {} & {} & {} & \cdot & { - 1} \\ {} & {} & {} & {} & {} & {} & {} & 1 \\ \end {array} } \right ]\] that has an nth pivot equal to $2^{-(n-2)}$.
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Math. Comp. 42 (1984), 535-547
  • MSC: Primary 65F05; Secondary 15A23
  • DOI: https://doi.org/10.1090/S0025-5718-1984-0736451-4
  • MathSciNet review: 736451