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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… (More)
Some semi-discrete analogous of well known one-point family of iterative methods for solving nonlinear scalar equations dependent on an arbitrary constant are proposed. The new families give multi-point iterative processes with the same or higher order of convergence. The convergence analysis and numerical examples are presented.
In this paper we consider the positive definite solutions of nonlinear matrix equation X + A ૽ X −δ A = Q, where δ ∈ (0, 1], which appears for the first time in [ On an iteration methods for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl. 23 (2001) 632–645]. The necessary and sufficient conditions for the existence of a solution… (More)
The two matrix equations X + A * X −2 A = I and X − A * X −2 A = I are studied. We construct iterative methods for obtaining positive definite solutions of these equations. Sufficient conditions for the existence of two different solutions of the equation X + A * X −2 A = I are derived. Sufficient conditions for the existence of positive definite solutions… (More)
We propose and analyze a generalization of the known Newton-Secant iterative method in case of solving system of nonlinear equations, which is essentially of third order. One modification of the presented method is considered, also. Convergence analysis and numerical examples are included.