Let $A,B \in \mathbb{C}^{n \times n}$ be two given matrices. Under the assumption that at least one of the matrices $A$ or $B$ is nonsingular, we introduce an order reduction method for computing the generalized eigenvalues of the regular matrix pencil $A-zB$. This method is based on the SVD of the matrix $B$ (resp. $A$) if the matrix $A$ (resp. $B$) is nonsingular. When both $A$ and $B$ are nonsingular, the method is based on the SVD of $A$ or $B$ exclusively. The performance of this algorithm is studied and the accuracy of the computed eigenvalues is demonstrated by comparing them with those computed by the QZ method in Matlab.
