Numerical Characteristics, Algorithms And Preconditioning Techniques Of Some Kinds Of Special Matrices

Matrix computations and special matrix analysis have wide applications in computational mathematics, physics, economics and biology etc. In this thesis, numerical characteristics and preconditioning techniques of some kinds of special matrices are researched. The main results and innovations are as follows:1. Nonsingular H-matrices are studied and some new characteristics are found, which lead to a new algorithm for identifying nonsingular H-matrices. The new method is always convergent and needs fewer iterations than the earlier ones. It is also simple and suitable for competer implementation.2. A new subclass of the class of P-matrices (MB-matrices) is presented. Subsequently, some relationships between defined and some already known subclasses of P-matrices are established, which implies a new kind of nonsingular matrices. As a application, some subclasses of P-matrices are used to localize the real eigenvalues of a real matrix.3. Based on nonnegative matrices theories, some improvements of a Ky Fan theorem are presented for matrix eigenvalues and singular values inclusion problems. For example,where the nonnegative matrix B = (bij)∈Rn,n needs satisfy for any i≠j (see, Theorem 3.3.2 and 3.3.3).In addition, an almost optimal Gersgorin-type inclusion interval of singular values is also presented (see, Theorem 3.3.9):which contains all singular values of matrices equimodular with matrix . Theoretic analysis and numerical examples show the upper bound of this interval is optimal in some cases.4. For the Hadamard product A (?) A^-1 of an M-matrix A∈R^n,n and its inverse A^-1, we give new lower bounds for the minimum eigenvalue of A(?)A^-1 (q(A(?)A^-1)). These bounds are strong enough to prove the conjecture of Fiedler and Markham:5. For the iterative matrix M^-1N, we give a new result on its spectral radiusρ(M^-1N)(see, Theorem 4.2.2):where f =[f1, f2, ...,fn] is a G-function which satisfies some linear conditions, and . This bound is usually better than the earlier ones in some cases. Finally it is generalized to the block matrices case.6. We establish easily computable upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonai matrices, which improve the well-known bounds, and obtain the signs for all inverse elements of any nonsingular tridiagonai H-matrix (see, Theorem 5.2.1):" If A = Tridiag{ai, bi,ci} is a tridiagonai H-matrix, then A^-1 = (ci,j) exists andFinally, a new interesting algorithm to find the inverse of a tridiagonai matrix without imposing any restrictive conditions is presented.7. To obtain an efficient parallel algorithm to solve sparse block-tridiagonal linear systems, stair matrices and polynomial preconditioning are used to construct some parallel polynomial approximate inverse. These preconditioners are suitable when the desired goal is to maximize parallelism. Numerical results show that our preconditioners are usually better than the ILU(k) preconditioners.8. For the following Newton-type preconditioner where N₀ is an initial approximation for A^-1 and I is an identity matrix with the same dimension as matrix A, we improve its iterative scheme and obtaina new one with at least cubic convergence for computing A^l in some conditions:or equivalently...