| The application of totally positive matrices is very extensive,involving the field of computer-aided geometric design,statistics,and biological mathematics.We mainly study the solution to Bernstein-type linear systems and Said-ball-type linear systems and their applications.Firstly,we propose a refined progressive iterative approximati on to solve the Bernstein-type linear systems and Said-ball-type linear systems.Numerical experiments demonstrate that the prosed RPIA works much better than the PIA,in terms of not only the convergence rate but also the accuracy.Also,based on the fact that the Shur complement of a totally positive matrix is also totally positive and using the approximation of the Shur complement,we propose several multilevel preconditioners for Bernstein-type matrices and Said-ball-type matrices respe ctively.Numerical experiments show that the spectrum of the preconditioning matrix has a good cluster,and the CG method for solving preconditioning linear systems has a better convergence rate than for solving the original linear systems.This thesis is divided into four chapters as follows:In the first chapter we introduce the background and significance of the research on the totally positive linear systems,the research content as well as innovation of this thesis.The second chapter is the preliminaries which mainly introduces some basic defini tions and theorems used in sequel;In the third chapter we first introduce the PIA method and WPIA method,then we propose the iteration formats and algorithm of RPIA,and finally analyze its convergence.How to construct multi-level preconditioners and use CG method for Bernstein-type linear system and Said-ball-type linear system are arranged in the fourth chapter. |
