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 \(X_{k+1} = X_k (I + R_k + \dots + R^{q-1}_k)\), \(q \geq 2\), 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 \(B \in \mathbb{C}^{n \times n}\), 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.


65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F10 Iterative numerical methods for linear systems
15A09 Theory of matrix inversion and generalized inverses


Zbl 0136.12703
Full Text: DOI


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