Distribution of the largest root of a matrix for Roy's test in multivariate analysis of variance

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 ${\bf A}$ and ${\bf B}$ 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.