Residual bounds of the nonlinear matrix equation X + A*F(X)A = Q (original) (raw)

Solutions and perturbation estimates for the matrix equations X+/-A*X-nA=Q

Applied Mathematics and Computation, 2004

In this paper we investigate the equations X AE A Ã X Àn A ¼ 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 À A Ã X Àn A ¼ Q is given. The results are illustrated by using numerical examples.

Perturbation bounds for the matrix equation X s ±A H X t A=Q

Comptes rendus de l'Académie bulgare des sciences: sciences mathématiques et naturelles

In this paper we consider the sensitivity of the nonlinear complex matrix equation X s ± A H X t A = Q, where the exponents s and t are real numbers. Local and nonlocal perturbation bounds for the perturbation in the solution X are obtained.

Lyapunov Majorants for Perturbation Analysis of Matrix Equations

We describe some efficient perturbation techniques for algebraic matrix equations. Among them are improved first order perturbation bounds, the method of equivalent operators and the technique of Lyapunov majorants combined with application of fixed point principles.

A general framework for the perturbation theory of matrix equations

2021

A general framework is presented for the local and non-local perturbation analysis of general real and complex matrix equations in the form F(P,X)=0F(P,X) = 0F(P,X)=0, where FFF is a continuous, matrix valued function, PPP is a collection of matrix parameters and XXX is the unknown matrix. The local perturbation analysis produces condition numbers and improved first order homogeneous perturbation bounds for the norm ∣deX∣\|\de X\|∣deX∣ or the absolute value ∣deX∣|\de X|∣deX∣ of deX\de XdeX. The non-local perturbation analysis is based on the method of Lyapunov majorants and fixed point principles. % for the operator pi(p,cdot)\pi(p,\cdot)pi(p,cdot). It gives rigorous non-local perturbation bounds as well as conditions for solvability of the perturbed equation. The general framework can be applied to various matrix perturbation problems in science and engineering. We illustrate the procedure with several simple examples. Furhermore, as a model problem for the new framework we derive a new perturbation theory for continuous-time algebr...