An efficient computation of generalized inverse of a matrix

We propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves 18th order of convergence by using only seven matrix multiplications per iteration loop. This yields a high efficiency index for that computational task. The algorithm has a relatively mild numerical instability, and we stabilize it at the price of adding two extra matrix multiplications per iteration loop. This implies an efficiency index that exceeds the known record for numerically stable iterations for this task, which means substantial acceleration of the long standing algorithms for an important problem of numerical linear algebra. Our numerical tests cover a variety of examples in the category of generalized inverses, such as Drazin case, rectangular case, and preconditioning of linear systems. The test results are in good accordance with our formal study.