My supervisor and I were discussing a specific optimisation problem this afternoon.
To be simple: solve for $R$ in the equation $Rx=y$, where $x$, $y$ are made of samples in two difference coordination frames, with the constraint that $R$ is a rotational matrix, i.e. unit orthogonal matrix.
I know there is a paper [Horn1987] solving the problem, which gives an analytic solution.
But my supervisor argues that we should solve it iteratively for a ROBUST and STABLE solution by asserting that the iterative way is more ROBUST and STABLE than analytic one.
I think this concept is ridiculous, but I don't know how to point out his error, since I search with google but could not find some hard proof related.
So, my question is:
In this problem ($Rx=y$), is the solution given by Horn better than a direct minimisation method, for example solving $\hat{R} = \mathrm{argmin}\,(|Rx-y|^2)$ with COBYLA algorithm.
Generally speaking, is an iterative solution for a minimisation problem alway more stable and robust than a analytic solution if the latter exists?
[Horn1987] B. K. P. Horn, “Closed-form solution of absolute orientation using unit quaternions,” Journal of the Optical Society of America, vol. 4, no. 4, pp. 629–642, April 1987