svd - How to find the Takagi decomposition of a symmetric (unitary) matrix?

The Takagi decomposition is a special case of the singular value decomposition for symmetric matrices. More exactly:

Let $U$ be a symmetric matrix, then Takagi tells us there is a unitary
$V$ such that $U = VDV^T$ (with $D>0$ diagonal).

My question is basically: how to construct this $V$? Preferably I am looking for the `easiest'/most straight-forward way (which probably won't be the most efficient way!)

Note: For the case I am interested in, $U$ is in fact unitary (in which case Takagi gives $U = VV^T$). I'm happy to specialize to that special case if that makes the algorithm easier.