I asked a question on similar topic a while ago. The answer I got is using newton's method.

My original question was ""Approximate the square root of (1-X) efficiently through (nested) products However, I think the method applied to your problem.

Here is an reference: Newton's Method for the Matrix Square Root

There are also papers for p-th root and inverse p-th root: A Schur-Newton method for the matrix pth root

The general idea is that 1) we need to scale your matrix, so that its eigenvalues fall into a specific circle. 2)then, apply Newton's method with initial value $y_0=I$.

The update rule for p-th root is

$$ y_{k+1}=\frac{1}{p}[(p-1)y_k+y_k^{1-p}A] $$

Then we will have $y_k^{-p}A\to I$

I asked a question on similar topic a while ago. The answer I got is using newton's method.

My original question was ""Approximate the square root of (1-X) efficiently through (nested) products However, I think the method applied to your problem.

Here is an reference: Newton's Method for the Matrix Square Root

There are also papers for p-th root and inverse p-th root: A Schur-Newton method for the matrix pth root

The general idea is that 1) we need to scale your matrix, so that its eigenvalues fall into a specific circle. 2)then, apply Newton's method with initial value $y_0=I$.

The update rule for p-th root is

$$ y_{k+1}=\frac{1}{p}[(p-1)y_k+y_k^{1-p}A] $$

Then we will have $y_k^{-p}A\to I$