If A is a symmetric positive definite matrix, the
function SelfAdjointEigenSolver::operatorInverseSqrt()
uses the eigendecomposition to compute the
inverse square root of A.

My question is the following: would it not be
computationally more efficient (for symmetric
positive definite matrices) to compute the U
matrix (using LLT) and then backsolve ("invert")
that matrix? Using a high level programming
language (R) I find that this second approach
is 8-10 times faster depending on the dimensions
of A.

but that does not give you the same, in one case you have: A^-1 = R * R and in the other: A^-1 = R' * R...