Iterative Computation For Some Special Solutions Of Matrix Equations

Linear matrix equations with some constrained conditions and corresponding least-squares problems are hot topics in computational mathematics, and are widely applied in many fields such as biology, electricity, optics, automatic control theory,linear optimal control, etc.First of all, by using the real representation of quaternion matrix and its properties, we give a block Jacobi algorithm for linear equation Ax =b and some convergent conditions of the method; and then basing on the definition of the quaternion matrix norm, we consider the quaternionic least-squares(QLS) problem for the pure imaginary quaternion matrix solution. By means of transforming the vector iteration of the classical LSQR algorithm into the matrix form, we present an iterative method for the minimum norm solution of the above problem. Finally,numerical examples are reported which shows the favorable numerical properties of the methods.In addition, we aim at finding the symmetric arrowhead matrix solution of the matrix equation By constructing a linear operator and a projection, we present a conjugate gradient(CG) algorithm and an alternating projection(APM) algorithm to solve the above problem in the premise of consistency. In the end, numerical examples are given to illustrate the effectiveness and efficiency of four methods in various circumstances by comparisons.Furthermore, we discuss the symmetric arrowhead matrix solution of the least-squares problem In the first place, according to the matrix form of the classical LSQR algorithm, we give two iterative methods for the like-minimum norm solution and minimum norm solution of the above problem. And then basing on the linear operator and conjugate gradient least-squares(CGLS) algorithm, we present an iterative method with its theoretical properties for the above problem. At last, the favorable numerical properties of the two methods are verified by numerical examples.