Positive solutions to X = A−BX-1 B∗

Positive definite solutions of the nonlinear matrix equation X+AHX¯-1Zhou1, Cai2, Lam3 2013Applied Mathematics and Computation320190View full textAdd to dashboardCite

“…Especially, the problem of finding fixed points of the nonlinear matrix equation X + A * X −1 A = Q where A and Q > 0 are given and X is unknown, has been extensively studied in the last two decades. The interest of studying such problem mainly relies on its applications in many fields such as the analysis of ladder networks [2], dynamic programming [33], control theory [22], stochastic filtering [1] and statistics [32] (see [3], [18] and [19] for detailed introduction). Some generalized forms of the nonlinear matrix equation X + A * X −1 A = Q have received much attention in recent years (see, for example, [6], [17], [24], and [30]).…”

“…The first kind of results concentrate on providing analytical conditions on the existence of positive definite solutions and their corresponding properties. For example, the shorted operator theory was applied in [3] to study the existence of a positive definite solution; necessary and sufficient conditions in terms of symmetric factorizations of some rational matrix-valued function were derived in [18] for the existence of positive definite solutions; and some necessary and sufficient conditions were also derived [35] for the same problem in terms of some factorizations of the coefficient matrices. The other kind of results are mainly concerned with the numerical solutions of this class of nonlinear matrix equations.…”

On the Positive Definite Solutions of the Nonlinear Matrix Equation X-A^{*}X^{-s}A-B^{*}X^{-t}B=I

“…The nonlinear matrix equation Eq. (1.1), when s = 1 and t = 0, arises in the analysis of a stationary Gaussian reciprocal processes over a finite interval [2], which discussed in many papers [1,2,3,7,8,9,10,11,12]. The positive definite solutions of Eq.…”

“…The positive definite solutions of Eq. (1.1) have been studied in some special cases [1,2,3,7,8,9,10,11,12]. The properties, efficient iterative algorithms, and perturbation estimates of the positive definite solutions of Eq.…”

Iterative solution of two matrix equations

“…These two equations have been studied recently by several authors (see [1], [2], [3], [4], [16], [17]). For the application areas in which the equations arise, see the references given in [1] and [4].…”