Positive definiteness of real quadratic forms resulting from the variable-step L1-type approximations of convolution operators

The positive definiteness of real quadratic forms with convolution structures plays an important rolein stability analysis for time-stepping schemes for nonlocal operators. In this work, we present a novel analysistool to handle discrete convolution kernels resulting from variable-step approximations for convolution operators.More precisely, for a class of discrete convolution kernels relevant to variable-step L1-type time discretizations, weshow that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Ourproof is based on an elementary constructing strategy by using the properties of discrete orthogonal convolutionkernels and discrete complementary convolution kernels. To our knowledge, this is the first general result onsimple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using theunified theory, we obtain the stability for some simple nonuniform time-stepping schemes straightforwardly.

Hermite Positive Definite Solution of the Quaternion Matrix Equation Xm + B*XB = C

This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation Xm+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .

ON RUBIN'S HARMONIC ANALYSIS AND ITS RELATED POSITIVE DEFINITE FUNCTIONS

In this paper,a new formulation of the Rubin’s q-translation is given,which leads to a reliable q-harmonic analysis.Next,related q-positive definite functions are introduced and studied,and a Bochner’s theorem is proved.

Two Structure-Preserving-Doubling Like Algorithms to Solve the Positive Definite Solution of the Equation X-A^(H)X^(-1)A=Q

In this paper,we study the nonlinear matrix equation X-A^(H)X^(-1)A=Q,where A,Q∈C^(n×n),Q is a Hermitian positive definite matrix and X∈C^(n×n)is an unknown matrix.We prove that the equation always has a unique Hermitian positive definite solution.We present two structure-preserving-doubling like algorithms to find the Hermitian positive definite solution of the equation,and the convergence theories are established.Finally,we show the effectiveness of the algorithms by numerical experiments.

奇异鞍点问题中广义位移分裂迭代方法的半收敛性分析

Recently,some authors(Shen and Shi,2016)studied the generalized shiftsplitting(GSS)iteration method for singular saddle point problem with nonsymmetric positive definite(1,1)-block and symmetric positive semidefinite(2,2)-block.In this paper,we further apply the GSS iteration method to solve singular saddle point problem with nonsymmetric positive semidefinite(1,1)-block and symmetric positive semidefinite(2,2)-block,prove the semi-convergence of the GSS iteration method and analyze the spectral properties of the corresponding preconditioned matrix.Numerical experiment is given to indicate that the GSS iteration method with appropriate iteration parameters is effective and competitive for practical use.

Construction and Analysis of Structured Preconditioners for Block Two-by-Two Matrices

For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to high-quality preconditioning matrices for some typical matrices from the real-world applications.

A Two-Step Shape-Preserving Advection Scheme

This paper proposes a new two-step non-oscillatory shape-preserving positive definite finite difference advection transport scheme, which merges the advantages of small dispersion error in the simple first-order upstream scheme and small dissipation error in the simple second-order Lax-Wendroff scheme and is completely different from most of present positive definite advection schemes which are based on revising the upstream scheme results. The proposed scheme is much less time consuming than present shape-preserving or non-oscillatory advection transport schemes and produces results which are comparable to the results obtained from the present more complicated schemes. Elementary tests are also presented to examine the behavior of the scheme.

A POD REDUCED-ORDER SPDMFE EXTRAPOLATING ALGORITHM FOR HYPERBOLIC EQUATIONS

In this article, a proper orthogonal decomposition （POD） method is used to study a classical splitting positive definite mixed finite element （SPDMFE） formulation for second- order hyperbolic equations. A POD reduced-order SPDMFE extrapolating algorithm with lower dimensions and sufficiently high accuracy is established for second-order hyperbolic equations. The error estimates between the classical SPDMFE solutions and the reduced-order SPDMFE solutions obtained from the POD reduced-order SPDMFE extrapolating algorithm are provided. The implementation for solving the POD reduced-order SPDMFE extrapolating algorithm is given. Some numerical experiments are presented illustrating that the results of numerical computation are consistent with theoretical conclusions, thus validating that the POD reduced-order SPDMFE extrapolating algorithm is feasible and efficient for solving second-order hyperbolic equations.

ON THE APPROXIMATE COMPUTATION OF EXTREME EIGENVALUES AND THE CONDITION NUMBER OF NONSINGULAR MATRICES

From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determined by the parameters of the conjugate gradient. The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B. The approximation of extreme eigenvalues of A can be obtained as a 'by-product' in the computation of the conjugate gradient if a computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient. If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on AT A. Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.

On the Nonlinear Matrix Equation X＋A~＊f_1（X）A＋B~＊f_2（X）B=Q

In this paper, nonlinear matrix equations of the form X ＋ A＊f1 （X）A ＋ B＊f2 （X）B = Q are discussed. Some necessary and sufficient conditions for the existence of solutions for this equation are derived. It is shown that under some conditions this equation has a unique solution, and an iterative method is proposed to obtain this unique solution. Finally, a numerical example is given to identify the efficiency of the results obtained.

A MATRIX EQUATION FROM AN INVERSE PROBLEM OF VIBRATION THEORY

The symmetric,positive semidefinite,and positive definite real solutions of the matrix equation XA=YAD from an inverse problem of vibration theory are considered.When D=T the necessary and sufficient conditions for the existence of such solutions and their general forms are derived.

A SHIFT-SPLITTING PRECONDITIONER FOR NON-HERMITIAN POSITIVE DEFINITE MATRICES

A shift splitting concept is introduced and, correspondingly, a shift-splitting iteration scheme and a shift-splitting preconditioner are presented, for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned non-Hermitian positive definite matrix. The convergence property of the shift-splitting iteration method and the eigenvalue distribution of the shift-splitting preconditioned matrix are discussed in depth, and the best possible choice of the shift is investigated in detail. Numerical computations show that the shift-splitting preconditioner can induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the large sparse non-Hermitian positive definite systems of linear equations.

ON HERMITIAN POSITIVE DEFINITE SOLUTION OF NONLINEAR MATRIX EQUATION X＋A^＊X^-2A=Q

Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q, where Q is a square Hermitian positive definite matrix and A＊ is the conjugate transpose of the matrix A. We also demonstrate some essential properties and analyze the sensitivity of this solution. In addition, we derive computable error bounds about the approximations to the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q. At last, we further generalize these results to the nonlinear matrix equation X＋A^＊X^-nA=Q, where n≥2 is a given positive integer.

Superconvergence Analysis of Splitting Positive Definite Nonconforming Mixed Finite Element Method for Pseudo-hyperbolic Equations

In this paper, a new splitting positive definite nonconforming mixed finite element method is proposed for pseudo-hyperbolic equations, in which a quasi-Wilson quadrilateral element is used for the flux p, and the bilinear element is used for u. Superconvergence results in ｜｜·｜｜div,h norm for p and optimal error estimates in L2 norm for u are derived for both semi-discrete and fully discrete schemes under almost uniform meshes.

The α-Geometric Structures on Manifold of Positive Definite Hermite Matrices

Geometric structures of a manifold of positive definite Hermite matrices are considered from the viewpoint of information geometry.A Riemannian metric is defined and dual α-connections are introduced.Then the fact that the manifold is ±l-flat is shown.Moreover,the divergence of two points on the manifold is given through dual potential functions.Furthermore,the optimal approximation of a point onto the submanifold is gotten.Finally,some simulations are given to illustrate our results.

ON THE CONVERGENCE OF THE RELAXATION METHODS FOR POSITIVE DEFINITE LINEAR SYSTEMS

We establish the convergence theories of the symmetric relaxation methods for the system of linear equations with symmetric positive definite coefficient matrix, and more generally, those of the unsymmetric relaxation methods for the system of linear equations with positive definite matrix.

ON HERMITIAN POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATION X-A^＊X^-2 A=I

The Hermitian positive definite solutions of the matrix equation X-A^＊X^-2 A=I are studied. A theorem for existence of solutions is given for every complex matrix A. A solution in case A is normal is given. The basic fixed point iterations for the equation are discussed in detail. Some convergence conditions of the basic fixed point iterations to approximate the solutions to the equation are given.

LIMIT CYCLES FOR A CLASS OF DIFFERENTIAL SYSTEM WITH POSITIVE DEFINITE POLYNOMIAL

In this paper we consider a class of differential systems with positive definite polynomial having exactly one and two limit cycles. Such a system is more extensive than paper [1,2].

Laplace-Beltrami Differentiability of Positive Definite Kernels on the Sphere

This contribution gives results on the action of the Laplace-Beltrami derivative on suffi- ciently smooth kernels on the sphere, those defined by absolutely and uniformly expansions generated by a family of at least continuous functions. Among other things, the results show that convenient Laplace-Beltrami derivatives of positive definite kernels on the sphere are positive definite too. We also include similar results on the action of the Laplace-Beltrami derivative on condensed spherical harmonic expansions.

On Conditionally Positive Definite Dot Product Kernels

Let m and n be fixed, positive integers and P a space composed of real polynomials in m variables. The authors study functions f ： R →R which map Gram matrices, based upon n points of R^m, into matrices, which are nonnegative definite with respect to P Among other things, the authors discuss continuity, differentiability, convexity, and convexity in the sense of Jensen, of such functions