An accelerated technique for solving the positive definite solutions of a class of nonlinear matrix equations

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.…”

On the semigroup property for some structured iterations

“…From Theorem 3.2 and Theorem 4.1, we can apply Algorithm 3.1 to provide an accelerated method. More details can be found in [5].…”

“…For us, this work is to discover a characteristic to speed up the standard iterative ways. The possible real-life examples and numerical experiments are referred to the ones given in [19,20,3,12,4,5,13,14], while a much heavier demonstration is carried out. Unlike the existing results, we demonstrate these examples concerning the insight of the semigroup property.…”

“…The reader is referred to [5] for a thorough discussion of a class of NMEs (4.7), where A = B H . It can be proved that the limit of Q k is exactly equal to a solution of (4.7) under some additional assumptions on the coefficients Q, A and B.…”