Improved methods and starting values to solve the matrix equations $X\pm A^*X^{-1}A=I$ iteratively

Abstract

Abstract. The two matrix iterations X k+1 = I ∓ A * X −1 k A are known to converge linearly to a positive definite solution of the matrix equations X ± A * X −1 A = I, respectively, for known choices of X 0 and under certain restrictions on A. The convergence for previously suggested starting matrices X 0 is generally very slow. This paper explores different initial choices of X 0 in both iterations that depend on the extreme singular values of A and lead to much more rapid convergence. Further, the paper offers a new algorithm for solving the minus sign equation and explores mixed algorithms that use Newton's method in part.

Solving systems of nonlinear matrix equations involving Lipshitzian mappings

“…In the last few years, there has been a constantly increasing interest in developing the theory and numerical approaches for HPD (Hermitian positive definite) solutions to different classes of nonlinear matrix equations (see [8][9][10][11][12][13][14][15][16][17][18][19][20][21]). In this study, we consider the following problem: Find (X 1 , X 2 , ..., X m ) (P(n)) m solution to the following system of nonlinear matrix equations:…”

Solutions of matrix equations in problems of mechanics and control

“…[23,28,32,[45][46][47]. This is because this equation, as already mentioned, is a special case of the Riccati equation, which plays a key role in various problems of synthesis of control systems [16, 39, 41, 43, etc.].…”

“…(1.4). As a rule, such algorithms are iterative, which leads to the additional task of choosing the initial approximation [28]. In what follows, we will show that algorithms based on LMI-solution procedures can be used in some cases to find the maximum and minimum solutions of Eq.…”

“…The following iterative procedures [28] can be used to refine the matrices X and Y obtained by solving problems (4.3) and (4.8):…”

Positive Definite Solutions of the Matrix EquationXr-∑i=1mA

“…During the last few years, many researchers worked to develop the theory and numerical approaches for positive definite solutions to the nonlinear matrix equations of the form (1) [6][7][8][9]. Recently, Duan et al [10] showed that the equation − ∑ =1 * = (0 < | | < 1) always has a unique positive definite solution by using the fixed point theorem of mixed monotone operators.…”