Ivan G. Ivanov

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.

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.

Positive Definite Solutions of the Equation X+A*X-nA=I

The general nonlinear matrix equation X + A* X-n A = I is discussed (n is a positive integer). Some necessary and sufficient conditions for existence a solution are given. Two methods for iterative computing a positive definite solution are investigated. Numerical experiments to illustrate the performance of the methods are reported.

Iterations for solving a rational Riccati equation arising in stochastic control

We consider different iterative methods for computing a Hermitian or maximal Hermitian solution of two types rational Riccati equations arising in stochastic control. The classical Newton procedure and its modification applied to equations are very expensive. New less expensive iterations for these equations are introduced and some convergence properties of the new iterations are proved.

A numerical procedure to compute the stabilising solution of game theoretic Riccati equations of stochastic control

In this article, the problem of the numerical computation of the stabilising solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H ∞ control problem for a class of stochastic systems affected by state-dependent and control-dependent white noise and subjected to Markovian jumping. The stabilising solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilising solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The proposed algorithm extends to this general framework the method proposed in Lanzon, Feng, Anderson, and Rotkowitz (Lanzon, A., Feng, Y., Anderson, B.D.O., and Rotkowitz, M. (2008), ‘Computing the Positive Stabilizing Solution to Algebraic Riccati Equations with an Indefinite Quadratic Term Viaa Recursive Method,’ IEEE Transactions on Automatic Control, 53, pp. 2280–2291). In the proof of the convergence of the proposed algorithm different concepts associated the generalised Lyapunov operators as stability, stabilisability and detectability are widely involved. The efficiency of the proposed algorithm is demonstrated by several numerical experiments.

Computation of the stabilizing solution of game theoretic Riccati equation arising in stochastic H∞ control problems

In this paper, the problem of the numerical computation of the stabilizing solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H ∞  control problem for a class of stochastic systems affected by state dependent and control dependent white noise. The stabilizing solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilizing solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The efficiency of the proposed algorithm is demonstrated by several numerical experiments.

A method to solve the discrete-time coupled algebraic Riccati equations

We consider a set of discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control. Two iterations for computing a symmetric solution of this system are investigated. New iterations are based on the properties of a Stein equation. It is necessary to solve a Stein equation at each step of proposed algorithms. We adapt the conditions for convergence several previous iterations presented in the literature for solving rational Riccati equations arising in stochastic control.

A Method for Solving Hermitian Pentadiagonal Block Circulant Systems of Linear Equations

A new effective method and its two modifications for solving Hermitian pentadiagonal block circulant systems of linear equations are proposed. New algorithms based on the proposed method are constructed. Our algorithms are then compared with some classical techniques as far as implementation time is concerned, number of operations and storage. Numerical experiments corroborating the effectiveness of the proposed algorithms are also reported.