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Hyperpower iteration is a powerful family of iterative methods for finding outer inverses with arbitrary order of convergence p ≥ 2. In this paper, we present several systematic algorithms for factorizations of the hyperpower iterative family of arbitrary orders with a view to reduce the necessary number of multiplications in each iterative step. Additionally, effective heuristics for factoring arbitrary higher orders hyperpower iteration are presented. The new formulations of the hyperpower iterative steps are convergent with higher computational efficiency indices.

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Some iterative methods are introduced and demonstrated for finding the matrix sign function. It is analytically shown that the new schemes are asymptotically stable. Convergence analysis along with the error bounds of the main proposed method is established. Different numerical experiments are employed to compare the behavior of the new schemes with the existing matrix iterations of the same type.