The Two-step Progressive Iteration Approximation And Convergence Analysis

Totally positive matrices have applications in computer-aided geometric design,biological mathematics,engineering,etc.This paper mainly studies the solutions and applications of Bernstein-Vandermonde and Said-Ball-Vandermonde matrices.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 approximation to solve the Bernstein-type linear systems and Said-ball-type linear systems.Numerical experiments demonstrate that the prosed WPIA works much better than the PI A.In this paper,we present a new method for updating the control polygons of a fitting curve or surface.The method uses the last two previous steps of control polygons simultaneously together with the fitting error vector to get a new control polygon,we refer it to the two-step progressive iteration approximation(TSPIA).In addition,we prove that for any standard all positive base,the two-step progressive iterative approximation method(TSPIA)has better convergence than the weighted progressive iterative approximation method(WPIA).Several examples are provided to illustrate the efficiency and effectiveness of the proposed method.This thesis is divided into three chapters as follows:The first chapter introduces the research background and significance of all positive linear equations,and the innovation of this article;The second chapter is the preliminaries which mainly introduces some basic definitions and theorems used in sequel;In the third chapter we first introduce the WPIA method,then we propose the iteration formats and algorithm of TSPIA,and finally analyze its convergence,Numerical experiments show the effectiveness of the method.In the fourth chapter we summarized this article and raised possible problems.