Random Matrices Generating Large Growth in LU Factorization with Pivoting

We identify a class of random, dense, $n\times n$ matrices for which LU factorization with any form pivoting produces growth factor typically size at least $n/(4 \log n)$ large $n$. The condition number the can be arbitrarily chosen, and also happens transpose. Previously, no all these properties were known. generated by MATLAB function gallery('randsvd',...), they are formed as product two random orthogonal from Haar distribution diagonal matrix having only one entry different 1, lies between 0 1 (the “one small singular value” case). Our explanation uses fact that maximum absolute value element distributed tends to relatively verify behavior numerically find partial actual is significantly larger than lower bound much observed elements uniform [0,1] or standard normal distributions. show more generally rank-1 perturbation an producing generates under reasonable assumptions. Finally, we demonstrate GMRES-based iterative refinement provide stable solutions $Ax = b$ when occurs in low precision factors, even cannot.