Vejdi I. Hasanov

IMPROVED METHODS AND STARTING VALUES TO SOLVE THE MATRIX EQUATIONS X ± A∗X−1A = I ITERATIVELY

The two matrix iterations X k+1 = I A*X -1 k A are known to converge linearly to a positive definite solution of the matrix equations X ± A*X -1 A = I, respectively, for known choices of X 0 and under certain restrictions on A. The convergence for previously suggested starting matrices X 0 is generally very slow. This paper explores different initial choices of X 0 in both iterations that depend on the extreme singular values of A and lead to much more rapid convergence. Further, the paper offers a new algorithm for solving the minus sign equation and explores mixed algorithms that use Newtons method in part.

On matrix equations X±A*X−2A=I

The two matrix equations X + A ∗ X −2 A = I and X − A ∗ X −2 A = I are studied. We construct iterative methods for obtaining positive definite solutions of these equations. Sufficient conditions for the existence of two different solutions of the equation X + A ∗ X −2 A = I are derived. Sufficient conditions for the existence of positive definite solutions of the equation X − A ∗ X −2 A = I are given. Numerical experiments are discussed.

Solutions and perturbation estimates for the matrix equations X±A * X -n A=Q

In this paper we investigate the equations X+/-A^*X^-^nA=Q for the existence of positive definite solutions and perturbation bounds for these solutions are derived. A sufficient condition for uniqueness of a unique positive definite solution of the equation X-A^*X^-^nA=Q is given. The results are illustrated by using numerical examples.

On some families of multi-point iterative methods for solving nonlinear equations

Some semi-discrete analogous of well known one-point family of iterative methods for solving nonlinear scalar equations dependent on an arbitrary constant are proposed. The new families give multi-point iterative processes with the same or higher order of convergence. The convergence analysis and numerical examples are presented.

Improved perturbation estimates for the matrix equations X±A∗X−1A=Q☆

Abstract We give new and improved perturbation estimates for the solution of the matrix quadratic equations X±A ∗ X −1 A=Q . Some of the estimates depend and some do not depend on knowledge of the exact solution X. These bounds are compared numerically against other known bounds from the literature.