Summary: 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 . Given an by matrix , the study shows that the Drazin inverse of can be computed in parallel time ranging from to provided that there are enough processors to support matrix multiplication in time .
The SMS algorithm is generalized to higher-order schemes, where the composite matrix is repeatedly raised to an integer power . 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 other than 2. Our derived error bound for the approximation of is new.