Xuefeng Duan

On solutions of the quaternion matrix equation AX=B and their applications in color image restoration

By using the complex representation of quaternion matrices, and the Moore-Penrose generalized inverse, we derive the expressions of the least squares solution with the least norm, the least squares pure imaginary solution with the least norm, and the least squares real solution with the least norm for the quaternion matrix equation AX=B, respectively. Finally, we discuss their applications in color image restoration.

On the matrix equation arising in an interpolation problem

In this article, we consider the nonlinear matrix equation arising in an interpolation problem. We use the Thompson metric to prove that the matrix equation always has a unique positive definite solution. An iterative method is constructed to compute the unique positive definite solution and its error estimation formula is given. Based on the matrix differentiation, we give a precise perturbation bound for the unique positive definite solution. The new results are illustrated by some numerical examples in the end.

Positive definite solution of a class of nonlinear matrix equation

In this paper, we consider a new nonlinear matrix equation , which is a special stochastic algebraic Riccati equation arising in stochastic control theory. Based on Brower’s fixed point theorem and Bhaskar–Lakshmikantham’s fixed point theroem, we derive some new sufficient conditions for the existence of a (unique) positive definite solution, which are easy to check. An iterative method is proposed to compute the unique positive definite solution. The error estimation of the iterative method is also given. Numerical examples show that the iterative method is feasible.

On the Low-Rank Approximation Arising in the Generalized Karhunen-Loeve Transform

We consider the low-rank approximation problem arising in the generalized Karhunen-Loeve transform. A sufficient condition for the existence of a solution is derived, and the analytical expression of the solution is given. A numerical algorithm is proposed to compute the solution. The new algorithm is illustrated by numerical experiments.

Iterative algorithm for the symmetric and nonnegative tensor completion problem

ABSTRACT In this paper, we consider the symmetric and nonnegative tensor completion problem. We first reformulate this problem as the minimization problem of the nuclear norm and then design the alternating direction method (ADM) to solve this problem. The -subproblem is treated by the singular value truncation method, and the -subproblem is solved by the nonmonotone spectral projected gradient method. The convergence of ADM method is given. Numerical examples illustrate that the new method is feasible.

Numerical methods for solving some matrix feasibility problems

In this paper, we design two numerical methods for solving some matrix feasibility problems, which arise in the quantum information science. By making use of the structured properties of linear constraints and the minimization theorem of symmetric matrix on manifold, the projection formulas of a matrix onto the feasible sets are given, and then the relaxed alternating projection algorithm and alternating projection algorithm on manifolds are designed to solve these problems. Numerical examples show that the new methods are feasible and effective.

On two classes of mixed-type Lyapunov equations ☆

Abstract The Hermitian positive definite solutions of the mixed-type Lyapunov equations X = AXB ∗ + BXA ∗ + Q and AX + XA ∗ + BXB ∗ + Q = 0 are studied in this paper. Based on the Bhaskar and Lakshmikantham’s fixed point theorem, new sufficient conditions for the existence of Hermitian positive definite solutions are derived. Iterative methods are proposed to compute the Hermitian positive definite solutions. Numerical examples are used to illustrate the convergence of the new methods.

Perturbation Analysis for the Matrix Equation

We consider the perturbation analysis of the matrix equation . Based on the matrix differentiation, we first give a precise perturbation bound for the positive definite solution. A numerical example is presented to illustrate the sharpness of the perturbation bound.