Recursive generalized pseudoinverse

Question

Let $A\in\mathbb{R}^{\text{n$\times$m}}$ be a rectangular matrix, and $n,m\in\mathbb{Z}^{+}$. The generalized pseudo-inverse $A^g$ is defined as $AA^gA = A$. The computation complexity of the generalized pseudo-inverse would grow rapidly as dimension $n,m$ grow. I wonder if there is any recursive method to calculate $A^g$ from matrix $A$ without involving inversion. Maybe at a form of $$A^g_{k} = f_1(A,A^g_{k-1}) + f_2(A,A^g_{k-1})$$ where $f_1,f_2$ are some non-complex operations given some initialization $A^g_{0}$. Thank you!