Efficient algorithms for the inclusion of the inverse matrix using error-bounds for hyperpower methods

Abstract

--ZusammenfassungEfficient Algorithms for the Inclusion of the Inverse Matrix Using Error-Bounds for Hyperpower Methods.By exploiting generalized error-bounds for the well-known hyperpower methods for approximating the inverse of a matrix we derive inclusion methods for the inverse matrix. These methods make use of interval operations in order to give guaranteed inclusions whenever the convergence of the applied hyperpower method can be shown. The efficiency index of some of the new methods is greater than that of the optimal methods in [2] or [5]. A numerical example is given.

On the two classes of high‐order convergent methods of approximate inverse preconditioners for solving linear systems

“…is proposed by Amat et al, 19 which has at least third order of convergence. 21 Based on Neumann's series, methods (7)-(9) can be generalized to a larger class of high-order methods called the hyperpower method, or the pth-order method [15][16][17][18] for p ≥ 2, performing p matrix by matrix multiplications for each iteration as follows:…”

“…Class 2: (18) are used, then R m+1 = R̃q k m ,q k = 5 * 2 k − 1. The necessary and sufficient condition for the convergence of Equations 17 and 18 to A −1 is that (R 0 ) < 1 holds, where is spectral radius, R 0 = I − AV 0 , and V 0 is the initial approximation.…”

“…is valid, then for the sequence {V m } m=∞ m=1 obtained by using Class 1 methods (17) or Class 2 methods (18),…”

“…where V m is the approximate inverse of A obtained at the mth iteration using the proposed classes of methods (17) and (18). Thus, R m = AE m , and…”

“…11 In recent years, there has been great interest in algebraic methods of rapid convergence rate to compute matrix inversion, where algorithms are defined by matrix multiplications and additions. Some of the approaches that have been published to propose and investigate iterative methods for computing the generalized inverses and approximate inverse of a square matrix are by Schulz, 12 Ben-Israel et al, 13 Soderstorm et al, 14 Householder, 15 Isaacson et al, 16 Stickel, 17 Herzberger, 18 Amat et al, 19 Li et al, 20,21 Krishnamurthy et al, 22 Soleymani,23,24 and Haghani et al 25 (see also references therein).…”