Gaussian elimination without rounding

Gaussian elimination without rounding

Backward Error Analysis for Totally Positive Linear Systems

Gauss elimination applied to an n • n matrix ,4 in floating point arithmetic produces (if successful) a factorization /-U which differs from A by no more than ~ ILl ] U I, for some ~ of order n times the unit roundoff. If A is totally positive, then both computed factors /~ and U are nonnegative for sufficiently small unit roundoff and one obtains pleasantly small bounds for the perturbation in...

Alan Turing and the origins of modern Gaussian elimination∗

The solution of a system of linear equations is by far the most important problem in Applied Mathematics. It is important both in itself and because it is an intermediate step in many other relevant problems. Gaussian elimination is nowadays the standard method for solving this problem numerically on a computer and it was the rst numerical algorithm to be subjected to rounding error analysis. I...

Mixed accelerated techniques for solving dense linear systems

The rounding-error analysis of Gaussian elimination shows that the method is stable only when the elements of the matrix do not grow excessively in the course of the reduction. Usually such growth is prevented by interchanging rows and columns of the matrix so that the pivot element is acceptably large. In this paper firstly we introduce the Boosting LU factorization method based on a rank one ...

On a Direct Method for the Solution ofNearly Uncoupled Markov

abstract This note is concerned with the accuracy of the solution of nearly uncoupled Markov chains by a direct method based on the LU decomposition. It is shown that plain Gaussian elimination may fail in the presence of rounding errors. A modiication of Gaussian elimination with diagonal pivoting as well as corrections of small pivots by sums of oo-diagonal elements in the pivoting columns is...