A Geometric Analysis of Gaussian Elimination. II
In Part I of this work, we began a discussion of the numeric consequences of hyperplane orientation in Gaussian elimination. We continue this discussion by introducing the concept of back-substitution-phase error multipliers. These error multipliers help to explain many of the previously unproven or poorly understood observations concerning Gaussian elimination in a finite-precision environment. A new pivoting strategy designed to control both sweepout phase roundoff error and back-substitution-phase instability is also presented. This new strategy, called rook's pivoting, is only slightly more expensive than partial pivoting yet produces results comparable to those produced by complete pivoting.
Neal, Larry; and Poole, George. 1992. A Geometric Analysis of Gaussian Elimination. II. Linear Algebra and Its Applications. Vol.173(C). 239-264. https://doi.org/10.1016/0024-3795(92)90432-A ISSN: 0024-3795