In this work, we present an algorithm which converts the Integration-by-Parts (IBP) equations into single-variable recurrence relations. We refer to this procedure as {}. Diagonalized IBP equations are indispensable for reducing loop integrals with extremely high numerator powers to master integrals, and for solving IBP identities in closed analytic form. A prime example is provided by multivariate Mellin representations of loop amplitudes and cross sections. The extension of these methods to other multivariate recurrence relations is also discussed. As a by-product of our diagonalization procedure, we show how the IBP equations can be cast into an efficient, fully triangular form that is well suited for computer implementation. Several complicated topologies have been computed.