Preconditioned Weight-GMRES(m) Algorithm Analysis

The solving large sparse linear equations is one of the important techniques innumerical simulation of many problems in the field of natural science and engineering.GMRES algorithm is one of the most effective methods for solving large sparsenonsymmetric linear equations. However, in the operation of the overall GMRES, thecomputation and storage capacity increases with the rise of the number of iterations, it isnot acceptable. In order to overcome this shortcoming, many scholars are focus onimprove and improve this algorithm. In recent years, preconditioning technology has beenapplied widely. This preconditioning technology can change the distributioncharacteristics of coefficient matrix, which could simplify the problem. Therefore, Seekingthe suitable preconditioning methods has become the focus of many scholars.In this paper, in chapter1firstly introduces the background, development andapplication, research significance of generalized minimal residual algorithm.Chapter2introduces the relevant basic theory of GMRES algorithm、restart GMRES algorithm andpreprocessing technology. In chapter3introduces the pretreatment of SSOR method onthe basic theory of knowledge, and establish preconditioning matrix, combining with theweighted GMRES (m) algorithm, and proposed a new SSOR-weighted GMRES (m)algorithm, demonstrates the feasibility and the convergence of the new algorithm by thetheoretical analysis and numerical examples. Besides, analyzes the factors affecting thecalculation efficiency and precision of the algorithm. In chapter4products thepolynomials preconditioning weighted GMRES(m) algorithm, show the feasibility andsuperiority of the new algorithm by the theoretical analysis and numerical examples.Reflect the numerical stability and efficiency of the new algorithms.The new algorithm is visible in solving the practical problem, and will play animportant role in the computation of practical life.