Block modifications of the disturbed Aitken-Steffensen method

Abstract

NEW modifications of the disturbed analogue of the block-type Aitken-Steffensen method are described for solving non-linear operator equations. In them, the inverse operator is evaluated after an arbitrary number of steps, while at intermediate steps the inverse operators are replaced by the partial sums of a special series. We show how to optimize the choice among such processes, in order to achieve a given accuracy while minimizing the number of computational operations. Consider the equation x = ϑ( x), (1) where ϑ( x) is a continuous non-linear operator, mapping Banach space X into itself. One of the authors described in [1] a second-order iterative method for solving Eq. (1), which has certain advantages over the Newton-Kantorovich method [2] and the chord method [3]: x n = X n−1 + Γ n−1 ε n−1 , n = 1, 2, …, (2) or alternatively, x n= x ̄ n−1 + Γ n−1 g ̄ 3 n−1, n = 1, 2, … (3) (where Γ n−1 = [I − Ψ ( x ̄ n−1, x n−1)] −1, Ψ ( x ̄ n−1, x n−1) is the first divided difference of the operator ϑ (x), x ̄ n−1 = x n−1 + με n−1, ε n−1 = ϑ(x n−1) − x n−1, g ̄ 3 n−1 = ϑ( x ̄ n−1) − x ̄ n−1 , and I is the identity operator in X, 0 < μ ⩽ 1), and simple modifications of the basic method were outlined, the transition to which is made after k steps, in accordance with Eqs. (2) or (3). From the point of view of optimizing the process for solving Eq. (1), it is worth considering alternative, and in a sense more general, modifications of the method (2), (3), in which iterations of the basic method alternate with several iterations of simpler kinds.