Solution of Algebraic Lyapunov Equation on Positive-Definite Hermitian Matrices by Using Extended Hamiltonian Algorithm

This communique is opted to study the approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices.We choose the geodesic distance between􀀀AHX􀀀XA and P as the cost function,and put forward the Extended Hamiltonian algorithm(EHA)and Natural gradient algorithm(NGA)for the solution.Finally,several numerical experiments give you an idea about the effectiveness of the proposed algorithms.We also show the comparison between these two algorithms EHA and NGA.Obtained results are provided and analyzed graphically.We also conclude that the extended Hamiltonian algorithm has better convergence speed than the natural gradient algorithm,whereas the trajectory of the solution matrix is optimal in case of Natural gradient algorithm(NGA)as compared to Extended Hamiltonian Algorithm(EHA).The aim of this paper is to show that the Extended Hamiltonian algorithm(EHA)has superior convergence properties as compared to Natural gradient algorithm(NGA).Upto the best of author’s knowledge,no approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices is found so far in the literature.

Positive-Definite Sparse Precision Matrix Estimation

The positive-definiteness and sparsity are the most important property of high-dimensional precision matrices. To better achieve those property, this paper uses a sparse lasso penalized D-trace loss under the positive-definiteness constraint to estimate high-dimensional precision matrices. This paper derives an efficient accelerated gradient method to solve the challenging optimization problem and establish its converges rate as . The numerical simulations illustrated our method have competitive advantage than other methods.

A new approach to obtaining positive-definite Wigner operator for two entangled particles with different masses

Based on our previously proposed Wigner operator in entangled form, we introduce the generalized Wigner operator for two entangled particles with different masses, which is expected to be positive-definite. This approach is able to convert the generalized Wigner operator into a pure state so that the positivity can be ensured. The technique of integration within an ordered product of operators is used in the discussion.

A Class of Preconditioners Based on Positive-Definite Operator Splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations

After discretization by the finite volume method,the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like structure.The theoretical analysis gives sufficient conditions to guarantee the positive-definite property of the discretized matrix.Moreover,we develop a class of positive-definite operator splitting iteration methods for the numerical solution of fractional diffusion equations,which is unconditionally convergent for any positive constant.Meanwhile,the iteration methods introduce a new preconditioner for Krylov subspace methods.Numerical experiments verify the convergence of the positive-definite operator splitting iteration methods and show the efficiency of the proposed preconditioner,compared with the existing approaches.

LIE-TROTTER FORMULA FOR THE HADAMARD PRODUCT

Suppose that A and B are two positive-definite matrices,then,the limit of(A^p/2B^pA^p/2)1/p as p tends to 0 can be obtained by the well known Lie-Trotter formula.In this article,we generalize the usual product of matrices to the Hadamard product denoted as*which is commutative,and obtain the explicit formula of the limit(A^p*B^p)^1/p as p tends to 0.Furthermore,the existence of the limit of(A^p*B^p)^1/p as p tends to+∞is proved.

Stability of Operator-Valued Truncated Moment Problems

In this note a multidimensional Hausdorff truncated operator-valued moment problem, from the point of view of “stability concept” of the number of atoms of the obtained atomic, operator-valued representing measure for the terms of a finite, positively define kernel of operators, is studied. The notion of “stability of the dimension” in truncated, scalar moment problems was introduced in [1]. In this note, the concept of “stability” of the algebraic dimension of the obtained Hilbert space from the space of the polynomials of finite, total degree with respect to the null subspace of a unital square positive functional, in [1], is adapted to the concept of stability of the algebraic dimension of the Hilbert space obtained as the separated space of some space of vectorial functions with respect to the null subspace of a hermitian square positive functional attached to a positive definite kernel of operators. In connection with the stability of the dimension of such obtained Hilbert space, a Hausdorff truncated operator-valued moment problem and the stability of the number of atoms of the representing measure for the terms of the given operator kernel, in this note, is studied.

Fourier Transforms of Bounded Bilinear Forms on C＊（S1）×C＊（S2） of Foundation ＊-semigroups S1 and S2

The author gives a characterization of the Fourier transforms of bounded bilinear forms on C＊（S1）×C＊（S2） of two foundation semigroups S1 and S2 in terms of Jordan ＊-representations, hemimogeneous random fields, and as well as weakly harmonizable random fields of S1 and S2 into Hilbert spaces.