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The hyperpower iteration revisited. (English) Zbl 0848.65021
This paper discusses an extension of the hyperpower method [cf. {\it 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 $X_{k+1} = X_k (I + R_k + \dots + R^{q-1}_k)$, $q \geq 2$, where $A$ and $X_0$ are arbitrary complex matrices and $R_k$ is the residual $I - AX_k$. The authors examine the method with residual modified to $P(I - AX_k)$, with $P$ idempotent. The main thrust of the paper is analysis of the convergence of $B^{q^k}$ for some $B \in \bbfC^{n \times n}$, 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.

65F20Overdetermined systems, pseudoinverses (numerical linear algebra)
65F10Iterative methods for linear systems
15A09Matrix inversion, generalized inverses
Full Text: DOI
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