A strengthened Newton-Kantorovich method with approximation of the inverse operator

Abstract

THE convergence of the method of solving the equation P( x) = 0, indicated in the title, with replacement of the operator [ P′( x n )] −1 by some approximation of it, is investigated. Many iterative methods of solving the equation (1) P( x) = 0 are constructed in such a way that to find x it is necessary to calculate [ P′( x n )] −1 on some element y n . The impossibility in the majority of cases of an accurate calculation of [ P′( x n )] −1 y n leads to the need to replace [ P′( x n )] −1 by some approximation A n . Therefore, if the initial method has the form (2) X n+1 = X n + Q( X n ,[ P′( X n )] −1), instead of (2) we consider its analog X n+1 = X n + Q( x n , A n ). In the present note we calculate A n by using one step of the method of the ( k + 1)-th order inversion of the operator considered in [1], and as initial approximation we take the operator A n−1 . We consider in detail the corresponding analogs of the Newton-Kantorovich method, the third-order Chebyshev method and the method studied in [2]. It is shown that the analog of Chebyshev's method cannot have third-order convergence if k < 5.