Fasta Lgorithm Of Pseudo-Cholesky Factorization And The Backward Error Analysis For Eigenvalue Problems

In chapter one, some properties of the unified rotation and unified Householder transformation are presented. Then, a fast version of applying a unified rotation to a 2 by n matrix is proposed. Base on this, a new algorithm is presented for computing the pseudo-cholesky factorization of a matrix.。 Compared with directly applying the unified rotation, the new algorithm has several advantages. First of all, the multiplications decrease approximately by one half; Secondly, the possible quick increase and decrease of the diagonal elements of the diagonal matrices generated in the processes are avoid, thereby better reliability and relational reliability are obtained。Chapter two deals with backward errors of a real matrix with respect to an approximate complex eigenpair. Both complex and real perturbations cases are considered. The conclusion is that, in general there are no essential difference between the two cases, but there are large differences in some situations. This extends to generalized and polynomial eigenvalue problems. As a by-product, a problem raised by Higham and Higham is partially solved。...