  Let ${\bf X, Y} $ denote two independent real Gaussian $\mathsf{p} \times
\mathsf{m}$ and $\mathsf{p} \times \mathsf{n}$ matrices with $\mathsf{m},
\mathsf{n} \geq \mathsf{p}$, each constituted by zero mean i.i.d. columns with
common covariance. The Roy's largest root criterion, used in multivariate
analysis of variance (MANOVA), is based on the statistic of the largest
eigenvalue, $\Theta_1$, of ${\bf{(A+B)}}^{-1} \bf{B}$, where ${\bf A =X X}^T$
and ${\bf B =Y Y}^T$ are independent central Wishart matrices. We derive a new
expression and efficient recursive formulas for the exact distribution of
$\Theta_1$. The expression can be easily calculated even for large parameters,
eliminating the need of pre-calculated tables for the application of the Roy's
test.
