Numerically Safe Gaussian Elimination with No Pivoting

Gaussian elimination with partial pivoting is performed routinely, millions times per day around the world, but partial pivoting (that is, row interchange of an input matrix) is communication intensive and has become the bottleneck of the elimination algorithm in the present day computer environment, in both cases of matrices of large and small size. Gaussian elimination with no pivoting as well as block Gaussian elimination are highly attractive alternatives. (Hereafter we use the acronyms GENP and BGE.) It has been proved decades ago that GENP and BGE are likely to proceed safely, that is, involving no division by 0 or inversion of singular blocks, if the input is pre-processed with random or even random structured matrix multipliers. Here and hereafter “likely” means with a probability 1 or close to 1. In 2015 we proved that, with Gaussian random multipliers (hereafter we say just Gaussian), both GENP and BGE are likely to be numerically safe, that is, are likely to be safe and to encounter no numerical stability problems. In all these results the multipliers are proven to be universal, that is, GENP and BGE with such randomized preprocessing • are safe with probability 1 if an input matrix is nonsingular and • are likely to be numerically safe if an input matrix is also well-conditioned. The latter assumptions about the input matrices is necessary for GENP, but not for BGE. We prove some nontrivial positive and negative results about the universality of random structured preprocessing, including multiplicative and additive preprocessing and augmentation for GENP and BGE, but we also provide some new insights into the subject and motivate new policies of preprocessing, which are not universal but highly efficient according to the results of our extensive tests. 2000 Math. Subject Classification: 15A06, 15A52, 15A12, 65F05, 65F35