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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?

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  • $\begingroup$ Is the vector $v$ real or complex? $\endgroup$ – jobe Mar 1 at 14:04
  • $\begingroup$ 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. $\endgroup$ – greg Mar 2 at 16:51

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