Adaptive coefficients iterative method for computing matrix inverse

Abstract

In this paper, we construct the new iterative method of the form X<inf>k+1</inf>=X<inf>k</inf>P<inf>k</inf>(AX<inf>k</inf>) where P<inf>k</inf> is polynomial, for computing the inverse A⁻¹ of a given invertible matrix A∈R^n×n. Coefficients of the polynomial P<inf>k</inf> in k-th iteration are variable and determined in a way to minimize the Frobenius norm of the error matrix I−AX<inf>k+1</inf>. The convergence of the new method is investigated, where several theoretical results are proven. The method is compared to the existing iterative methods of the similar type, on a various numerical examples. The results show that the new method outperforms the existing ones for almost all test matrices. Moreover, they suggest that the new method posses almost global convergence.