I have to numerically calculate the ratio of two bilinear forms:
$\frac{x_1}{x_2} = \frac{v^T U_1 v}{v^T U_2 v}$,
where $U_1$ and $U_2$ are unitary matrices. Both bilinear forms $x_1$ and $x_2$ are very small, the ratio however could be finite, which is why the numerical stability of this procedure is very low. However, ignoring the rules of calculus for a moment I am dreaming of a series that starts like this,
$\frac{x_1}{x_2} = \frac{v^T U_1 v}{v^T U_2 v} = v^T (U_1 / U_2) v + \dots $.
the vector $v$ is complex (i.e. v^T is the hermitian adjoint) and of unit length $v^T v = 1$
Is anyone aware of a series description that comes close to this?