Successive matrix squaring algorithm for computing the Drazin inverse1

摘要

This paper derives a successive matrix squaring (SMS) algorithm to approximate the Drazin inverse which can be expressed in the form of successive squaring of a composite matrix T. Given an n by n matrix A, the study shows that the Drazin inverse of A can be computed in parallel time ranging from O(logn) to O(log2n) provided that there are enough processors to support matrix multiplication in time O(logn).The SMS algorithm is generalized to higher-order schemes, where the composite matrix is repeatedly raised to an integer power l⩾2. This form of expression leads to a simplified notation compared to that of earlier methods, we argue that there is no obvious advantage in choosing l other than 2. Our derived error bound for the approximation of AD is new.