Probabilistic analysis of complex Gaussian elimination without pivoting*1

Abstract

We consider Gaussian elimination without pivoting applied to complex Gaussian matrices X ∈ C n×n . We first study some independence properties of the elements of the LU factors of X . Based on this, we then derive the probability distributions for all the L and U elements and obtain bounds for the probabilities of the occurrence of small pivots and large growth factors. Numerical experiments are presented to support the theoretical results and discussions are made to relate the results to the crucial practical problems of numerical stability of GE.