A Rapid Numerical Algorithm to Compute Matrix Inversion

The aim of the present work is to suggest and establish a numerical algorithm based on matrix multiplications for computing approximate inverses. It is shown theoretically that the scheme possesses seventh-order convergence, and thus it rapidly converges. Some discussions on the choice of the initial value to preserve the convergence rate are given, and it is also shown in numerical examples that the proposed scheme can easily be taken into account to provide robust preconditioners

Combined high-order algorithms in robust least-squares estimation with harmonic regressor and strictly diagonally dominant information matrix

This article describes new high-order algorithms in the least-squares problem with harmonic regressor and strictly diagonally dominant information matrix. Estimation accuracy and the number of steps to achieve this accuracy are controllable in these algorithms. Simplified forms of the high-order matrix inversion algorithms and the high-order algorithms of direct calculation of the parameter vector are found. The algorithms are presented as recursive procedures driven by estimation errors multiplied by the gain matrices, which can be seen as preconditioners. A simple and recursive (with respect to order) algorithm for update of the gain matrix, which is associated with Neumann series, is found. It is shown that the limiting form of the algorithm (algorithm of infinite order) provides perfect estimation. A new form of the gain matrix is also a basis for unification method of high-order algorithms. New combined and fast convergent high-order algorithms of recursive matrix inversion and algorithms of direct calculation of the parameter vector are presented. The stability of algorithms is proved and explicit transient bound on estimation error is calculated. New algorithms are simple, fast and robust with respect to round-off error accumulation