We consider the approximation of the inverse square root of regularly accretive operators in Hilbert
spaces. The approximation is of rational type and comes from the use of the Gauss–Legendre rule applied to
a special integral formulation of the fractional power. We derive sharp error estimates, based on the use of the
numerical range, and provide some numerical experiments. For practical purposes, the finite-dimensional
case is also considered. In this setting, the convergence is shown to be of exponential type. The method is
also tested for the computation of a generic fractional power.
