Randomized Circulant and Gaussian Pre-processing

Circulant matrices have been extensively applied in Symbolic and Numerical Computations, but we study their new application, namely, to randomized pre-processing that supports Gaussian elimination with no pivoting, hereafter referred to as GENP. We prove that, with a probability close to 1, GENP proceeds with no divisions by 0 if the input matrix is pre-processed with a random circulant multiplier. This yields 4-fold acceleration in the cases of both general and structured input matrices versus pre-processing with the pair of random triangular Toeplitz multipliers, which has been the user's favorite since 1991. In that part of our paper, we assume computations with infinite precision, but in other parts with double precision, in the presence of rounding errors. In this case, GENP fails without pre-processing unless all square leading blocks of the input matrix are well-conditioned, but empirically GENP produces accurate output consistently if a well-conditioned input matrix is pre-processed with random circulant multipliers. We also support formally the latter empirical observation if we allow standard Gaussian random input and hence the average non-singular and well-conditioned input as well, but we prove that GENP fails numerically with a probability close to 1 in the case of some specific input matrix pre-processed with such multipliers. We also prove that even for the worst case well-conditioned input, GENP runs into numerical problems only with a probability close to 0, if a nonsingular and well-conditioned input matrix is multiplied by a standard Gaussian random matrix. All our results for GENP can be readily extended to the highly important block Gaussian elimination.