| The contents of this thesis are divided into two parts:the first part is concerned with algorithm researches on two classes of quaternion problems, which are included in Chap-ter 2 and Chapter 3; the second one is devoted to perturbation analyses for generalized inverses, see Chapter 4 for detail.1. Quadratic quaternion polynomial equations We deduce necessary and sufficient conditions that general bilateral quadratic quaternion polynomial equation (QQPE) has solutions, and provide two ideas for solving the general bilateral QQPE. The first idea is transforming it into a paramet-ric linear system with quadratic constraint; the second one is transforming it into corresponding equivalent real quadratic forms, by defining four real representation matrices of a quaternion.2. Quaternion least squares problems By real representation of quaternion matrices, we study quaternion least squares (QLS) problems. We provide iteration algorithms for finding general solutions and Hermitian tridiagonal solutions with the least norm of the QLS problem, and propose a preconditioning strategy for the algorithm LSQR-Q in [99] and our algorithms. Numerical experiments are provided to verify the effectiveness of the precondition-ing strategy.3. Perturbation analyses for generalized inverses We study the perturbation problems of{1}-inverse and associated oblique projec-tions. Let A∈Cm×n, A= A+E, where the perturbation matrix E is sufficiently small. For any given A-∈A{1}, in the sense of Frobenius norm and matrix spectral norm, we find out the corresponding A-∈A{1} that are closest to the given A-and its oblique projections, respectively. In what follows, we deduce the perturbation bounds for the{1}-inverse and its oblique projections under the constraint of rank invariant perturbation; Similar problems on{1,2}-,{1,3}-,{1,4}-,{1,2,3}-,{1,2,4}-, {1,3,4}-inverses are also discussed. In summary, we found a new method for study-ing the perturbation problems of generalized inverses.
|
PDF Full Text Request