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.

联合双角化（JBD）过程是用于计算矩阵对的广义奇异值分解（GSVD）的有用算法。然而，它总是遭受舍入误差，这导致Lanczos向量失去相互的正交性。为了保持一定程度的正交性，我们提出了一种半正交化策略。我们的舍入误差分析表明，采用半正交化策略的JBD过程可以确保计算量的收敛不受舍入误差的影响，并且最终精度足够高。基于半正交化策略，我们开发了具有部分正交化（JBDPRO）的联合双角化过程。在JBDPRO算法中，仅在必要时才进行正交化，从而节省了大量的正交化工作，与完整的正交化策略相比。数值实验说明了我们的理论和算法。