I am trying to calculate

$$Y = A^{\frac 12} X$$

where $A$ is a very large and sparse positive definite matrix, say, $10^4 \times 10^4$. Matrix $X$ is known and, say, $10^4 \times 100$. Is there any method that can compute or approximate $A^{\frac{1}{2}}$ efficiently?

I noticed that matrix $A$ can be written as $D + B$, where $D=\mbox{diag}(A)$ and $B=A-D$, which is very sparse. But I still do not know how to compute the square root of the sum of matrices, $(D + B)^{\frac{1}{2}}$.

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