Research On Rigorous Perturbation Bounds For Some Matrix Factorizations

This thesis is concerned with the perturbation analysis for the generalizedCholesky factorization, Cholesky-like factorization and SR factorization problem. Thesematrix factorizations are among the most fundamental and important tools in somefields. The goal of such an analysis is to determine rigorous bounds for the changes inthe factors of a matrix when the matrix is respect to additive perturbation ormultiplicative perturbation.(1) The rigorous perturbation bounds for the generalized Cholesky factorizationand Cholesky-like factorization. We derive the rigorous perturbation bounds for thegeneralized Cholesky factorization and Cholesky-like factorization by the combinationof the matrix equation and the fined matrix equation method;(2) The multiplicative rigorous perturbation bounds for the generalized Choleskyfactorization and Cholesky-like factorization. We present the rigorous perturbationmultiplicative bounds for the two factorizations. The corresponding first ordermultiplicative perturbation bounds, as special cases, are also presented. Some of theseresults improve the corresponding ones given in this literature;(3) Rigorous perturbation bounds for the SR factorization. We obtain somerigorous perturbation bounds for the SR factorization with normwise or componentwiseperturbations in the given matrix, where the componentwise perturbation has the formof backward error resulting from the SR factorization algorithms. As special cases,some first order perturbation bounds are presented. Furthermore, the rigorouscolumnwise perturbation bounds for the factor R are derived;(4) Multiplicative perturbation bounds for the SR factorization. We present the firstorder perturbation bounds for the SR factorization with the respect to left multiplicativeperturbation, and the first order and rigorous perturbation bounds for this factorizationwith respect to right multiplicative perturbation. Moreover, taking the properties ofthe SR factors into consideration, some refined perturbation bounds are also provided.