探讨线性方程组Ax =b的高阶2PPJ迭代收敛的充要条件.假设系数矩阵A的Jacobi矩阵特征值的平方为零或纯虚数,利用A的Jacobi矩阵特征值与高阶2PPJ迭代矩阵的关系,结合外插迭代引理,进一步给出此类方程组高阶2PPJ迭代收敛的充要条件.给出了2个数例对结论加以验证.

In order to study the convergence of GAOR iterative method on the basis of Hermitian positive and negative definite matrices,firstly the Householder-John theorem is introduced and generalized to the case of negative definite matrices.Then a sufficient and necessary condition for the convergence of GAOR iterative method is given under the negative definite condition.By using the Housholder-John theorem,the convergent conclusion of GAOR iterative method is improved.Finally,the convergence of GAOR iterative method under the Hermitian negative definite condition is analyzed through the generalized Householder-John theorem.

为给出非负不可约矩阵的谱半径上 、下界的新估计,首先构造一个新的矩阵形式及两个收敛的序列,之后利用矩阵特征值和特征向量的关系,进一步给出非负不可约矩阵谱半径的易于计算的上 、下界.最后通过数值实例验证了所得结论的有效性.

探究线性方程组Ax=b的系数矩阵A的Jacobi矩阵的特征值的平方为纯虚数或者是零时2PPJ迭代的收敛性问题,得出此类方程组2PPJ迭代的收敛性区间.并且给出一个数例对结论加以说明.

In order to study the convergence of the PSD iterative method under the irreducible L-matrix,a new preconditioned matrix P=I+Sis proposed.Under the condition of the coefficient matrix is not zero element L array,the size of the spectral radius of traditional PSD iterative and the spectral radius of the preconditioned PSD iterative method is compared by using eigenvector method.The divergence of the preconditioned PSD iterative method is obtained.Some numerical examples are given for demonstration.

The location for all the right eigenvalues of a n×n quaternionic matrix is discussed.According to the Gergorin type theorem that has been given,it gets a better conclusion that for every right eigenvalueλ,all the right eigenvalues are containned in the union of the Gergorin balls if there exists a quaternionη∈[λ]and|λ-aii|=|η-aii|.In additon,the conclusions when all the right eigenvalues or the elements of main diagonal of a quaternionic matrix are given.Finally,it gives a numerical examples to prove the conclusion to be correct in general.

In order to study the nature of the MASOR iteration method and prove that MASOR can converge on the condition ofμk>1,the method of MASOR was introduced and then the relationship between the eigenvalue and eigenvector of MASOR iteration method and Jacobbi iteration method was established.At last under the condition of a given eigenvalueμ2k=mki,by respectly discussingσ1=-σ2,σ1=σ2,the scope of the convergence of MASOR iteration method was obtained under the condition of mk<1,mk=1,mk>1.

To find a simpler and more accurate expression for the inverse of the symmetric block tridiagonal matrices,applying the recursive relations between blocks of two-consecutive-terms and algebraic operational method of matrices and the previous calculating formula of the inverse of the symmetric block tridiagonal matrices,we derive the new espression for the inverse of the symmetric block tridiagonal matrices. The new expression has more advantages in computational complexity than the one obtained previously. Finally,its validity is validated with a numerical example.

In order to further research the schur complements properties of strictly γ‐diagonally dominant matrix ,the triangle‐schur complements of strictly γ‐diagonally dominant matrix is obtained by extending the schur complements and diagonal‐schur complements .It is proved that the triangle‐schur complements is still strictly γ‐diagonally dominant matrix by properties of strictly γ‐diagonally dominant matrix .Final‐ly ,the numerical example is given to illustrates the effectiveness of the criteria .

The estimations of the lower bound of ρ(A-1 ) is discussed when matrix A is under a certain conditions and A is a strictly doubly diagonally dominant matrix .T he |akk|-Rk≤ |ai |-Ri of every k ,i subjected to n≥ k≥ i≥1 is obtained .Further the new lower bound min i≠ j| aj |+ Ri (A)| ai × aj |- Ri (A) × Rj (A) is received .It is proved that this new estimations is better than the lower existing bound ,and the numercial example illustrates the effectiveness of the criteria .

Under the premise of the linear equations Ax=b coefficient matrix A is tridiagonal L matrix,the new preconditioned matrice P=I+S1 is provided.Using USSOR iterative methods and the theory of matrix splitting,the new comparison theorems are obtained.Finally,the main results are demonstrated by numerical examples.

The symmetric MSOR iterative method for solving linear systems of AX=b is discussed where A is a 2-cyclic coefficient matrices.The optimum parameters of the symmetric MSOR method was given for solving the linear system AX=b with 2-cyclic coefficient matrices when the characteristics of the Jacobi iteration matrix is real number or purely imaginary.And the results was illustrated by example.

With the algebra,the necessary and sufficient condition for the convergence of the USAOR method as ω2=γ2=1 and ω1=γ1=1 are discussed for solving the linear system Ax=b with(q,r)=(2,1) consistently ordered matrix.Finally,the optimum parameter and the most superior spectrum radius are obtained under certain condions.

The upper bound for spectral radius of iterative matrix M-1 N is estimated when M is an α- diagnally dominant matrix. The esimations of p(A-1 ) and p(J) are obtained by calculating sufficient conditions for | λ(M-1N) |, and the numerical example illustrates the effectiveness of the criteria.

In order to improve the convergence rate of linear equations iterative method,using the appropriate pretreatment is necessary,that is the case PAx=Pb.After putting preconditioned matrix P_=I+ into USSOR iterative method and discussing the convergence when coefficient matrix is a nonsingular M matrix by using the theory of matrix splitting,a comparison theorem is obtained.And then the explanations are given as well as some specific numerical examples.

Combining geometry and algebra,the optimal parameters and the optimal spectral radius of the PSD method for rank deficient linear systems Ax=b are acquired,where A∈Cm×nr and rmin{m,n},x∈Cn,b∈Cm,and then examples are given to illustrate the results.

The linear system Ax=b where A∈Rn×nwith A is nonsingular can be solved by splitting A into three matrices P,R and S with P is nonsingular,which can be expressed by A=P-R+S.In this paper,the nonnegative double splitting method is derived by using the matrix and the algebra theories.The convergence and comparison theorems for this method are obtained.Examples are given to illustrate the process at last.

The symmetric MSOR iterative method for solving linear systems of AX=b is discussed where A is a 2-cyclic coefficient matrices.The necessary and sufficient condition for the convergence of the symmetric MSOR method was given for solving the linear system AX=b with 2-cyclic coefficient matrices when the characteristics of the Jacobi iteration matrix is real number or purely imaginary.And the advantages of the results were illustrated by example.

In order to improve the convergence rate of linear equations iterative method,using the appropriate pretreatment is necessary,that is the case PAx=Pb.The new preconditioned matrices P=I+C′α was used.When the coefficient matrix A is nonsingular M-matrix,using the USSOR iterative methods and the theory of matrix splitting,the comparison theorem is obtained.Finally,the main result is demonstrated by a numerical example.

Combining the geometry with the algebra,the convergence and divergence properties of SOR iterative methord are discussed for solving the linear system Ax =b with(1,2)consistently ordered matrix,when all the eigenvalues of the B3J are nonpositive and nonnegative respectively.Finally,the analogous results are provided for the case when all the eigenvalues of the B3J are real,then examples are given to illustrate the results,where BJ is the associated Jacobi iterative matrix,A∈Cn×n,x∈Cn,b∈Cn.