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][1] However, I think the method applied to your problem.

Here is an reference: [Newton's Method for the Matrix Square Root][2]

There are also papers for p-th root and inverse p-th root: 
[A Schur-Newton method for the matrix pth root][3]

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 $y_k^{-p}A\to I$

  [1]: https://mathoverflow.net/questions/162402/approximate-the-square-root-of-1-x-efficiently-through-nested-products
  [2]: http://www.ams.org/journals/mcom/1986-46-174/S0025-5718-1986-0829624-5/S0025-5718-1986-0829624-5.pdf
  [3]: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCsQFjAA&url=http://www.researchgate.net/publication/30045719_A_Schur-Newton_Method_for_the_Matrix_p%27th_Root_and_its_Inverse/file/60b7d52439183c6c7d.pdf&ei=8AGFU6zuHcrioAT83IDgAQ&usg=AFQjCNH_0myG-4_3eifWzvH90BYEEfCnKg&sig2=XTLoDT_IKf33PSkJpPmDRQ