# Convergent series description of ratio of two bilinear forms

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,

$$\frac{x_1}{x_2} = \frac{v^T U_1 v}{v^T U_2 v} = v^T (U_1 / U_2) v + \dots$$.

Is anyone aware of a series description that comes close to this?

• Is the vector $v$ real or complex? – jobe Mar 1 at 14:04
• The problem is likely the magnitude of the $v$ vector, since the ratio is independent of the length of the vector. You should lengthen it until $(x_1, x_2)$ are no longer small. – greg Mar 2 at 16:51