Currently, I encountered a problem of approximating the following series:

$$ (I-X)^{-\frac{1}{2}}=I+\frac{1}{2}X+\frac{1\cdot3}{2\cdot4}X^{2}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6}X^{3}+\ldots $$

where $X$ is a diagonalizable matrix and the largest (in absolute value) eigenvalue is less than $1-\frac{1}{\kappa},\kappa>1$. What I am looking for is something like the following

$$ (1-X)^{-1}=I+X+X^{2}+X^{3}+\dots\approx\prod_{k=0}^{d}(1+X^{2^{k}}) $$

With $O(\log(\kappa\log(1/\epsilon)))$ multiplications and summations, we can achieve accuracy of $O(\epsilon)$.

We can assume the following representations are cheap to get (we assume that their computational cost is $1$):

$1+aX^{n},a\in(-1,1),n\in[-1,0,1,\dots)$

Multiplication between any two cheap representations.

Summation between any two cheap representations.

So how can we approximate $(I-X)^{-\frac{1}{2}}$ efficiently?

My initial idea is

$$ (I-X)^{-\frac{1}{2}}=I+\frac{1}{2}X(1+\frac{3}{4}X(1+\dots(1+\frac{2n-1}{2n}X(1-X)^{-1}))\dots) $$

But it is not efficient enough.