Studying The Algorithm For Solving Ill-conditioned Linear System Of Equations

Studying the algorithm for solving ill-conditioned linear system of equations is an important task of numerical calculation studying. This paper has improved some conventional algorithms and presents the weighting iterative improved algorithm and PSD-PCG algorithm, basing on the analysis of ill-conditioned linear system of equations' characteristic and origin. The improved effect is proved in theory and by several typical numerical experiments. The first chapter of this paper gives out the styles and the evaluation criterions of algorithm for solving ill-conditioned linear system of equations, at the same time introduces several effective algorithms. The second chapter discusses the significance and method of distinguishing the system's condition, and gives out a practical scheme to estimate the condition number. In the third chapter two typical preconditioning algorithms are given, basing on analyzing the basic idea of preconditioning algorithms. The fourth chapter introduces theconjugate gradient (CG) algorithm and the PSD-PCG algorithm. The PSD-PCG algorithm is effective to sovle ill-conditioned linear system of equations whose coefficient matrix is symmetric and positive definite and sparse. The fifth chapter introduces some ill-conditioning problems and the noticeable proceedings in the practical projects. At the end of this paper, the direction and current of studying the algorithm for solving ill-conditioned system of equations are indicated.