Quartic Rational Iterations for the Matrix Sign with Applications to Principal Square Roots and Inverse Square Roots

This paper develops a new rational fixed-point iteration for sign ( M ) whose scalar prototype enjoys half-plane global convergence and a quartic local error contraction. By embedding M into a block matrix and exploiting the classical sign–square root identity, the proposed iteration yields M 1 / 2 and M − 1 / 2 within a single iterative run under standard spectral admissibility conditions. A norm-balancing scaling strategy is also incorporated to accelerate the initial phase without changing the asymptotic order. Numerical experiments in a unified implementation demonstrate that the method reduces iteration counts and time compared with Newton-type and Padé-based competitors.