Iterative Methods And Applications For Some Nonlinear Problems

In this dissertation, we study some problems on the solving of nonlinear equations F(x)=0; large non-Hermitian linear systems with multiple right-hand sides AX=B; quadratic inverse eigenvalue problem for damped gyroscopic systems Q(λ)=λ2M+λ(D+ G)+K;linear response eigen-problems. Our main results are as follows.1. By using modified Newton method as the outer solver, preconditioned modified Her-mitian and skew-Hermitian splitting (PMHSS) method as the inner solver, we estab-lish a modified Newton-PMHSS method for solving nonlinear equations with large sparse complex symmetric Jacobian matrix. We analyze the local convergence prop-erties under the Holder continuous condition, which is weaker than the Lipschtiz continuous condition used in many references.2. For the Harmonic mean Newton method, we establish the Newton-Kantorovich-type convergence theorem by using majorizing functions. Compared to use recurrence relations to establish the convergence, we can provide a larger convergence radius and bigger R-order.3. Theoretical results indicate simpler block GMRES is less stable due to the ill-conditioning of the basis used. Then we propose an adaptive Simpler block GMRES algorithm for large linear systems with multiple right-hand sides. After that, we con-sider how to restart and precondition the adaptive simpler block GMRES algorithm, and propose a flexible and adaptive simpler block GMRES algorithm with deflated restarting.4. Given k≤n pairs of complex numbers and vectors, we consider the quadratic inverse eigenvalue problem. First we construct n×nreal matrices M, D, G, K, where M>0, K and D are symmetric, G is skew-symmetric, so that the damped gyroscopic system Q(λ)=λ2M+λ(D+G)+K has the given k pairs as eigenpairs. Then, for undamped gyroscopic system and the same system with K< 0, we construct a general solution and a particular solution, respectively.5. We present weighted Golub-Kahan-Lanczos bidiagonal algorithm to solve eigen-problems of KM with K and M are both symmetric and positive definite. We also give the convergence analysis of the extreme eigenvalues. As an application, we use our new algorithms to solve the linear response eigen-problems, together with the contact between new algorithms with classic CG method.