Computing the Generalized Singular Value Decomposition

An algorithm is described for computing the generalized singular value decomposition of $A(m \times n)$ and $B(p \times n)$. Unitary matrices U, V and Q are developed so that $U^H AQ$ and $V^H BQ$ have as many nonzero parallel rows as possible, and these correspond to the common row space of the two matrices. The algorithm consists of an iterative sequence of cycles where each cycle is made up of the serial application of $2 \times 2$ generalized singular value decompositions. Convergence appears to be at least quadratic. With the correct choice of ordering the algorithm can be implemented using systolic array processors (Gentleman,personal communication). The algorithm can also be used to compute any CS decomposition of a unitary matrix.