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The hyperpower iteration revisited. (English) Zbl 0848.65021

This paper discusses an extension of the hyperpower method [cf. A. Ben-Israel, Math. Comput. 19, 452-455 (1965; Zbl 0136.12703)], which may be used for iterative computation of generalised inverses for example. The hyperpower method uses a basic iteration ${X}_{k+1}={X}_{k}\left(I+{R}_{k}+\cdots +{R}_{k}^{q-1}\right)$, $q\ge 2$, where $A$ and ${X}_{0}$ are arbitrary complex matrices and ${R}_{k}$ is the residual $I-A{X}_{k}$. The authors examine the method with residual modified to $P\left(I-A{X}_{k}\right)$, with $P$ idempotent.

The main thrust of the paper is analysis of the convergence of ${B}^{{q}^{k}}$ for some $B\in {ℂ}^{n×n}$, where $B$ will be related to the matrices defined previously. If the basic iteration converges, an appropriate $P$ and limit $L$ have to be found and the paper discusses such possibilities.

##### MSC:
 65F20 Overdetermined systems, pseudoinverses (numerical linear algebra) 65F10 Iterative methods for linear systems 15A09 Matrix inversion, generalized inverses