具有部分再正交化的联合双对角化过程
The joint bidiagonalization process with partial reorthogonalization
摘要：
联合双对角化（JBD）过程是计算矩阵对广义奇异值分解（GSVD）的一种有效算法。然而，它总是遭受舍入误差，这导致Lanczos向量失去它们的相互正交性。为了保持一定程度的正交性，我们提出了一个半正交化策略。舍入误差分析表明，采用半正交化策略的JBD过程可以保证计算量的收敛性不受舍入误差的影响，最终精度足够高。在半正交化策略的基础上，提出了部分再正交化的联合双对角化过程（JBDPRO）。JBDPRO算法只在必要时进行重正交化，与完全重正交化策略相比，节省了大量的重正交化工作。数值实验验证了我们的理论和算法。
英文摘要：
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 loss their mutual orthogonality. In order to maintain some level of orthongonality, 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.
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