*(English)*Zbl 1022.65043

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 $T$. 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 $O\left({log}^{2}n\right)$ 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 $l\ge 2$. 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 ${A}^{D}$ is new.

##### MSC:

65F20 | Overdetermined systems, pseudoinverses (numerical linear algebra) |

65F30 | Other matrix algorithms |

65Y05 | Parallel computation (numerical methods) |

65Y20 | Complexity and performance of numerical algorithms |