預條件J-共軛梯度法求解漢米爾頓線性系統和最小平方法問題

Abstract

主旨在找一穩定不變子空間使得漢米爾頓線性系統之解落在其中,此篇論 文的目地在找一類似共軛梯度法的方法來解漢米爾頓線性系統,我們將這 種新方法命名為J-共軛梯度法 (JCG). 漢米爾頓矩陣在 J-Lanczos 方法 使用一系列的 symplectic 向量轉換成一 J-tridiagonal矩陣, 然後可從 這 J-tridiagonal 矩陣導出四項遞迴式, 然後將分析 JCG方法其收斂情 形及証明它將在有限步收斂, 還要討論 JCG應用在解預條件最小平方法問 題, 最後使用 JCG所得之數值結果和一些有名的方法來做一些比較. The object of solving Hamiltonian linear systems is to find the stable invarient subspace corresponding to the solution lying in. The purpose of this thesis is to propose a conjugate gradient- type algorithm for computing the solution of Hamiltonian linear systems. We name this new approach J- conjugate Gradient (JCG) method. With this algorithm, the Hamiltonian coefficient matrix is first reduced implicity to a J-tridiagonal matrix based on J- Lanczos algorithm using a sequence of symplectic similarity transformations. Then a four- term recurrence formulation is derived by updating the LU-like factorization of the J- tridiagonal matrix. Convergence behavior is analyzed and finite termination theorem is proved. Application of JCG to least squares problems and associated preconditioner is discussed. We report the numerical performance of JCG method and compare it with some well-known iterative methods.