Many estimators and tests are of the form of a ratio of quadratic
forms in normal variables. Excepting a few very special cases little
is known about the density or distribution of these ratios,
particularly if we allow for noncentrality in the quadratic forms.
This paper assumes this generality and derives saddlepoint
approximations for this class of statistics. We first derive and
prove the existence of an exact inversion based on the joint
characteristic function. Then the saddlepoint algorithm is applied
and the leading term found, and analytic justification of the
asymptotic nature of the approximation is given. As an illustration
we consider the calculation of sizes and powers of
F-tests, where a new exact result is found.