On the semigroup property for some structured iterations

Abstract

Nonlinear matrix equations play a crucial role in science and engineering problems. However, solutions of nonlinear matrix equations cannot, in general, be given analytically. One standard way of solving nonlinear matrix equations is to apply the fixed-point iteration with usually only the linear convergence rate. To advance the existing methods, we exploit in this work one type of semigroup property and use this property to propose a technique for solving the equations with the speed of convergence of any desired order. We realize our way by starting with examples of solving the scalar equations and, also, connect this method with some well-known equations including, but not limited to, the Stein matrix equation, the generalized eigenvalue problem, the generalized nonlinear matrix equation, the discrete-time algebraic Riccati equations to express the capacity of this method.

The convergence analysis of an accelerated iteration for solving algebraic Riccati equations

“…In this section, for any integer r > 1, we first show that an accelerated of fixed-point iteration (referred as AFPI) with R-superlinear convergence order r is capable of computing the minimal positive semidefinite solution of equation (1). It has been proved in [13,3] that if ρ(T X ) < 1 the AFPI has convergence rate of any desired order r. We verify that the convergence speed remains invariant even ρ(T X ) ≥ 1. It is worth mentioning that AFPI includes SDA as a special r = 2 case [13], so that the quadratic convergence of SDA when ρ(T X ) ≥ 1 still holds and this acts as a complementary to the existing results on the convergence of SDA.…”

“…It has been proved in [13,3] that if ρ(T X ) < 1 the AFPI has convergence rate of any desired order r. We verify that the convergence speed remains invariant even ρ(T X ) ≥ 1. It is worth mentioning that AFPI includes SDA as a special r = 2 case [13], so that the quadratic convergence of SDA when ρ(T X ) ≥ 1 still holds and this acts as a complementary to the existing results on the convergence of SDA.…”

“…Here * stands the complex conjugate transpose. We provide new theoretical supports to the stability properties of solutions to the DARE (1) and new convergence results for an iteration method, proposed recently in [12,13].…”

“…Definition 4.1. [13] Let D ⊆ C n×m and F : D × D → D be a binary matrix operator. We call that an iteration…”

“…Recently, the semigroup property for some binary matrix operations is investigated in [13] and has been applied to the construction of iterations for solving DARE. More precisely, applying the semigroup property to a fixed-point iteration X k+1 = F (X k , X 1 ), one can obtain an accelerated iteration(AFPI) with at least the R-convergence rate of any desired order r. Moreover, this iterative method can be reduced to SDA when r = 2.…”