Improved perturbation estimates for the matrix equations X±A∗X−1A=Q

On Hermitian positive definite solutions of the nonlinear matrix equation $$X-A^{*}e^{X}A=I$$ X - A ∗ e X A = I

“…Gao where F is a map on the set of the n × n positive definite matrices. Most authors have studied this equation with the particular map, such as F(X ) = ±X −1 , ±X ±n , ±X ±q , −( X − C) −1 and so on(see [1][2][3][4][5][6][7][8]). But the study is limited to the case that F(X ) is the power function.…”

Nonrecursive solution for the discrete algebraic Riccati equation and X + A*X-1A=L

“…Also, if the nonlinear matrix equation (2) has a unique positive definite solution X; then there exist minimal and maximal solutions X mi n and X max ; respectively, such that 0 < X mi n Ä X Ä X max for any positive definite solution X: We will refer to X mi n and X max ; as the extreme solutions of (2). Here, when X and Y are Hermitian matrices, X > Y means that X Y is a positive definite matrix denoted X Y > 0 and X Y means that X Y is a positive semidefinite matrix denoted X Y 0: The theoretical properties of the positive definite solution X max of (2) have been investigated by many authors (see [8,14,21] and the references therein) and the available methods for computing X max are algebraic (nonrecursive) [1][2][3] and recursive algorithms concerning per step or doubling [5,7,8,13,14].…”

“…The positive definite solutions X R ; X max of the nonlinear equations (1) and (2) are related and the solutions methods are combined [1,2,7,13,14].…”

“…ii) The assumptions of Theorem 2.2 guarantee that T has no eigenvalues on the unit disk [19,Theorem 3]. iii) If the block matrices W 11 ; W 21 are nonsingular, using (13)- (14) and following the same statements as in the proof of Theorem 2.2, the minimal Hermitian solution X of (1) is given by…”

On positive definite solution of a nonlinear matrix equation

“…Equation (1) with = 1 and Q = I have been considered in References [11][12][13][14][15][16]. Equation (1) with = n (n-natural) have been considered in [17][18][19][20]. Hasanov in [21] considered Equation (1) with 0< 1 and Q is a Hermitian positive definite matrix.…”