Preconditioned Householder-GMRES(m) Algorithm Analysis

GMRES(m) algorithm is one of the effective methods that are applied to solve alarge-scale sparse linear system. But this method has some defects in the computationalefficiency and the convergent speed and preconditioned technology come into being withthese shortcomings. In recent years, preconditioning technology has been paid attention tostudy by more and more researchers. It has been gaining popularity among them and thereare many theories about the applications of pretreatment technology. Because thepretreatment can change the distribution features of the matrix and simplify the problem,seeking an appropriate pretreatment method has become a hot topic.In this paper, firstly, the related basic theories of Householder transformation,GMRES(m) algorithm and the pretreatment technology are introduced. Secondly, on thebasis of the fully study about the above theoretical knowledge, the incomplete LUdecomposition, the incomplete LLTdecomposition and ADI and SSOR preconditionedmethod are applied to set up different preprocessors. Combining preprocessors with theHouseholder-GMRES(m) algorithm, the new pretreatment algorithms are proposed. Then,by the theoretical analysis and numerical experiments, it analyzes the feasibility andconvergence of the algorithms and the factors that have an effect on the algorithms. Theresult is very significant. It not only reduces the iterative times and improves theconvergent rate of the algorithms, but also reflects the numerical stability and efficiency ofthe algorithms. Finally, the paper compares a preconditioned algorithm which is proposedin this paper with Arnoldi-GMRES(m) algorithm and weighted GMRES(m) algorithm. Itshows that the high efficiency and precision of the new algorithm and demonstrates someadvantages of the new algorithm in computing speed. Obviously, the algorithms that thepaper proposed are practical and will play a key role in the calculation of actual problems.