Mixed, Componentwise Condition Numbers Of Generalized Quadratic Matrix Equations

This paper studies mixed and componentwise condition numbers of algebraic ma-trix equations, the generalized quadratic equation (GQME) is also discussed with con-dition numbers of*-Sylvester matrix equations.The main work is to research the mixed and componentwise condition numbers of GQME, and the relevant conclusions of two special equations are presented. Condition number is a measure of the sensitivity of the solution to the problem with respect to perturbation on its data. Research on condition number is an important topic issue of perturbation analysis. In most literatures, norm condition number is often used to measure the sensitivity of the perturbation, but it has several deficiencies. In order to obtain sharper bounds, the mixed condition number and componentwise condition number are took into consideration, as well as the effective condition number of*-Sylvester matrix equations. The numerical examples illustrate that these three condition numbers are all smaller than normwise condition number, which can also give sharper bounds than the normwise one. This paper is organized as follows:Introduces relevant concepts and background of normwise condition number, mixed condition number, componentwise condition number, and the effective condition number. It also presents the background and the derivation of the corresponding process.The mixed condition number, componentwise condition number and the effective condition number of*-Sylvester equations are studied, and the explicit expression and upper bounds are discussed, respectively. Numerical examples are illustrated that all of these three condition numbers can get sharper perturbation bounds than normwise condition number.The paper introduces the background and mixed condition number and com-ponentwise condition number of the generalized quadratic matrix equation-s(GQME). The numerical example illustrates that it can get sharper bounds. The conclusion of two special quadratic equations:symmetric algebraic Riccati equa-tions and nonsymmetric algebraic Riccati equations are presented in the last part of paper.