Computing the Generalized Singular Value Decomposition

Zhaojun Bai, James Weldon Demmel

Abstract: A variation of Paige’s algorithm is presented for computing the generalized singular value decomposition (GSVD) of two matrices A and B. There are two innovations. The first is a new preprocessing step which reduces A and B to upper triangular forms satisfying certain rank conditions. The second is a new $2 \times 2$ triangular GSVD algorithm, which constitutes the inner loop of Paige’s algorithm. Proofs of stability and high accuracy of the $2 \times 2$ GSVD algorithm are presented and are demonstrated using examples on which all previous algorithms fail.