Basic Fixed Point Iteration Research Articles

Dynamic Mode Decomposition: A New Approach for Computing the DMD Modes and Eigenvalues

In this paper we study a special case of linearly perturbed discrete-time algebraic Riccati equation. We give some sufficient conditions forthe existence of a positive definite solution of the considered equation.We propose a basic fixed point iteration and its inversion free variantfor finding a positive definite solution. Moreover, by specially choosingthe initial value in the basic fixed point iteration we prove that it con-verges to the largest solution. The theoretical results are illustratedby numerical examples.

POSITIVE DEFINITE SOLUTIONS OF A LINEARLY PERTURBED MATRIX EQUATION

In this paper we study a special case of linearly perturbed discrete-time algebraic Riccati equation. We give some sufficient conditions forthe existence of a positive definite solution of the considered equation.We propose a basic fixed point iteration and its inversion free variantfor finding a positive definite solution. Moreover, by specially choosingthe initial value in the basic fixed point iteration we prove that it con-verges to the largest solution. The theoretical results are illustratedby numerical examples.

Convergence analysis of some iterative methods for a nonlinear matrix equation

We study iterative methods for finding the largest Hermitian positive definite solution of the matrix equation X+∑i=1mAi∗X−1Ai=Q. Convergence rates of the basic fixed point iteration, inversion free variants of the basic fixed point iteration, and Stein iteration are considered. Some numerical examples are presented to illustrate the convergence behaviour of the various algorithms.

On convergence of three iterative methods for solving of the matrix equation $$X+A^{*}X^{-1}A+B^{*}X^{-1}B=Q$$ X + A ∗ X - 1 A + B ∗ X - 1 B = Q

In this paper, we give new convergence results for the basic fixed point iteration and its two inversion-free variants for finding the maximal positive definite solution of the matrix equation $$X+A^{*}X^{-1}A+B^{*}X^{-1}B=Q$$ , proposed by Long et al. (Bull Braz Math Soc 39:371–386, 2008) and Vaezzadeh et al. (Adv Differ Equ 2013). The new results are illustrated by numerical examples.

A Fixed Point Theorem for Monotone Maps and Its Applications to Nonlinear Matrix Equations

By using the fixed point theorem for monotone maps in a normal cone, we prove a uniqueness theorem for the positive definite solution of the matrix equationX=Q+A⁎f(X)A, wherefis a monotone map on the set of positive definite matrices. Then we apply the uniqueness theorem to a special equationX=kQ+A⁎(X^-C)qAand prove that the equation has a unique positive definite solution whenQ^≥Candk>1and0<q<1. For this equation the basic fixed point iteration is discussed. Numerical examples show that the iterative method is feasible and effective.

The iterative methods for solving nonlinear matrix equation "Equation missing"

AbstractIn this paper, we study the matrix equationX+A⋆X−1A+B⋆X−1B=Q, whereAandBare square matrices, andQis a positive definite matrix, and propose the iterative methods for finding positive definite solutions of the matrix equation. Also, general convergence results for the basic fixed point iteration for these equations are given. Some numerical examples are presented to show the usefulness of the iterations.MSC:65F10, 65F30, 65H10, 15A24.

Existence and Uniqueness of the Positive Definite Solution for the Matrix EquationX=Q+A∗(X^−C)−1A

We consider the nonlinear matrix equationX=Q+A∗(X^−C)−1A, whereQis positive definite,Cis positive semidefinite, andX^is the block diagonal matrix defined byX^=diag(X,X,…,X). We prove that the equation has a unique positive definite solution via variable replacement and fixed point theorem. The basic fixed point iteration for the equation is given.

A new class of nonsymmetric algebraic Riccati equations

Motivated by the study of linear quadratic differential games, we introduce a new class of nonsymmetric algebraic Riccati equations. It is shown that every equation in this class has a unique stabilizing solution, which is the solution required to find the open-loop Nash equilibrium for the differential game. We show that the doubling algorithm can be used to find this solution efficiently. The solution may also be found by the Schur method, and under further assumptions by Newton’s method and a basic fixed-point iteration.

Iterative methods for the extremal positive definite solution of the matrix equation [formula omitted

In this paper, the inversion free variant of the basic fixed point iteration methods for obtaining the maximal positive definite solution of the nonlinear matrix equation X + A * X - α A = Q with the case 0 < α ⩽ 1 and the minimal positive definite solution of the same matrix equation with the case α ⩾ 1 are proposed. Some necessary conditions and sufficient conditions for the existence of positive definite solutions for the matrix equation are derived. Numerical examples to illustrate the behavior of the considered algorithms are also given.

A new inversion free iteration for solving the equation X + A * X -1 A = Q

In this paper, we introduce a new inversion free variant of the basic fixed point iteration method for obtaining a maximal positive definite solution of the nonlinear matrix equation X + A*X-1A = Q...

A new inversion free iteration for solving the equation [formula omitted

In this paper, we introduce a new inversion free variant of the basic fixed point iteration method for obtaining a maximal positive definite solution of the nonlinear matrix equation X + A ★ X - 1 A = Q . It is more accurate than Zhan's algorithm (J. Sci. Comput. 17 (1996) 1167) and has less number of operations than the algorithm of Guo and Lancaster (Math. Comput. 68 (1999) 1589). We derive convergence conditions of the iteration and existence conditions of a solution to the problem. Finally, we give some numerical results to illustrate the behavior of the considered algorithm.

Iterative solution of two matrix equations

We study iterative methods for finding the maximal Hermitian positive definite solutions of the matrix equationsX+A∗X−1A=QX+A^*X^{-1}A=QandX−A∗X−1A=QX-A^*X^{-1}A=Q, whereQQis Hermitian positive definite. General convergence results are given for the basic fixed point iteration for both equations. Newton’s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation. Numerical results are reported to illustrate the convergence behaviour of various algorithms.