The authors propose two algorithms that avoid matrix inversion for every iteration, called inversion free variant of the basic point iteration. The first algorithm computes the maximal positive definite solution
of the nonlinear equation
is a nonsingular matrix,
is a Hermitian positive definite matrix and
The second algorithm computes the minimal positive definite solution
of the same nonlinear equation with the case
. Convergence theorems are provided. Some numerical examples are added to illustrate the convergence features.