The Analysis Of The Precondition PSD Iteration Method Under The Irreducible L Array

To solve a series of practical problems,we often construct a linear system of equations such as Ax = b,by solving Ax = b,the problem is solved,these problems are often encountered in many fields such as mathematics,physics,and engineer-ing applications.In order to quickly and efficiently solve the above equations,the most commonly used method is the iterative method,through continuous iteration and judgment,the optimal solution under the constraint condition is finally ob-tained.The advantages and disadvantages of an iterative method often depend on its convergence and convergence speed,the convergence and convergence speeds are closely related to the coefficient matrix of linear equations.Therefore,precondition the coefficient matrix A of the linear equation set Ax = b can effectively improve the convergence and convergence speed of an iterative method.This paper first gives two new types of precondition matrices,Secondly,under the condition that the coefficient matrix of the linear equations is an irreducible L matrix,the convergence of the preconditioning PSD iteration method is discussed.at last,the coefficient matrix in the linear equations is an irreducible L matrix,preconditioner matrix satisfies appropriate conditions,and the parameters of the PSD iterative matrix and the two types of PSD preconditioned iteration matrices satisfy 0≤ω≤τ≤1,τ≠0,when the PSD iteration method and the two preconditioning iteration methods converge,in particular τ=ω= 1 the spectral radius of the PSD iteration matrices and the two types of preconditioned PSD iterative matrices are minimal.This article is divided into four chapters,the specific work is as follows:In the first chapter,some important concepts such as irreducible matrix,L matrix,nonsingular M matrix,spectral radius,and normal splitting are introduced.Then two new kinds of preconditioned matrix is given.In the second chapter and the third chapter,under the condition that coefficient matrix A in linear equations Ax = b is an irreducible L-matrix,using the method of eigenvectors,the convergence and divergence of two kinds of preconditioned PSD iteration method are discussed,the same convergence and divergence results are obtained.When the spectral radius of the traditional PSD iteration matrix is less than 1,the spectral radius of the preconditioned PSD iteration matrix is smaller than the spectral radius of the traditional PSD iterative matrix;when the spectral radius of the traditional PSD iteration matrix is equal to 1,the spectral radius of the preconditioned PSD iteration matrix is equal to the spectral radius of the conventional PSD iteration matrix;when the spectral radius of the traditional PSD iteration matrix is greater than 1,the spectral radius of the preconditioned PSD iteration matrix is larger than the spectral radius of the traditional PSD iterative matrix.Therefore,two kinds of preconditioned matrices effectively improve the convergence rate of the iterative method.Some example are given for verification.In the fourth chapter,under the condition that the coefficient matrix in the lin-ear equations is an irreducible L matrix,preconditioned matrix satisfies appropriate conditions,the parameters of the iteration matrix and the two types of precondi-tioned iteration matrices satisfy 0≤ω≤τ≤1,τ≠0,when the PSD itera-tion method and the two preconditioning iteration methods converge,in particularτ= ω=1,at the time,the spectral radius of the PSD iteration matrices and the two kinds of preconditioned iteration matrices are minimal.