The joint bidiagonalization process with partial reorthogonalization

The joint bidiagonalization (JBD) process is a useful algorithm for the computation of the generalized singular value decomposition (GSVD) of a matrix pair. However, it always suffers from rounding errors, which causes the Lanczos vectors to lose their mutual orthogonality. In order to maintain some level of orthogonality, we present a semiorthogonalization strategy. Our rounding error analysis shows that the JBD process with the semiorthogonalization strategy can ensure that the convergence of the computed quantities is not affected by rounding errors and the final accuracy is high enough. Based on the semiorthogonalization strategy, we develop the joint bidiagonalization process with partial reorthogonalization (JBDPRO). In the JBDPRO algorithm, reorthogonalizations occur only when necessary, which saves a big amount of reorthogonalization work, compared with the full reorthogonalization strategy. Numerical experiments illustrate our theory and algorithm.