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Successive matrix squaring algorithm for computing the Drazin inverse. (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 $O\left(logn\right)$ to $O\left({log}^{2}n\right)$ provided that there are enough processors to support matrix multiplication in time $O\left(logn\right)$.

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