Research On GAOR Iteration Method And Sine-Schur Complement Of Hermite Positive And Negative Definite Matrices

In the applied sciences and engineering technologies,many researches are finally reduced to solving one or more linear algebraic equations.Generally speaking,different iterative methods are used to solve different linear equations.In recent years,some scholars have pointed out that the GAOR iterative method can be used to solve the linear equations whose coefficient matrix is positive definite Hermitian matrix.The convergent problem of the GAOR iterative method can be attributed to the relationship between the spectral radius of iterative matrix and 1.If the spectral radius is less than 1,the iteration will be convergent,conversely will be not.In chapter 1,the iterative matrix of GAOR iterative method is replaced by two similar matrices.It is more convenient to study the convergence of GAOR iterative method.In chapter 2,first the Householder-John theorem is introduced.Then it is used to give the convergent condition of GAOR iterative method under the condition of positive definite Hermitian matrices.As long as the condition of iterative parameters is satisfied,the GAOR iterative method will be convergent.Finally,some concrete examples are used to verify the validity of the conclusion in this chapter.Compared with all previous conclusions,the conclusion in this chapter is more perfect.The conclusion in chapter 2 is only for the positive definite Hermitian matrices,is not suitable for negative definite Hermitian matrices.In order to study the con-vergence of GAOR iterative method under the negative definite Hermitian matrices,in chapter 3,first the Householder-John theorem is generalized to the condition of negative definite Hermitian matrices.When MH +N is a negative definite Hermi-tian matrix,the negative definiteness of matrix A can be used as a necessary and sufficient condition for p(M-1N)<1.Secondly,the convergent condition of GAOR iterative method is given on the basis of negative definite Hermitian matrices.Fi-nally,some concrete examples are used to verify the validity of the conclusions in this chapter.In theory of matrices,there are many matrices with special structures or prop-erties,such as three diagonal matrices,strictly diagonally dominant matrices and positive or negative definite Hermitian matrices mentioned in the previous chapters.For these matrices with special properties,it is always tend to know whether some special structures or properties are inherited by their sub-matrices or related matri-ces.It is known that the Schur complements of positive definite Hermitian matrices are still positive definite Hermitian matrices.In chapter 4,the sinusoidal parameter is increased to introduce the definition of sinusoidal-Schur complements.Then it is used to prove that the sinusoidal-Schur complements of positive definite Hermitian matrices are still positive definite Hermitian matrices.After adding constraints re-lated to sub-matrices,the positive definiteness of matrices can be analysed by the positive definiteness of their sinusoidal-Schur complements.Similar to the positive definite Hermitian matrix,negative definite Hermitian matrix also plays an important role in theory of matrices.In chapter 5 of this paper,first the negative definiteness of sinusoidal-Schur complements is researched.Then adding the related constraints,the negative definiteness of matrices can be received from the negative definiteness of sinusoidal-Schur complements.Finally,some ex-amples are used to illustrate the rationality and effectiveness of the conclusions.