A rounding error analysis of the joint bidiagonalization process with applications to the GSVD computation

12/18/2019
by   Haibo Li, et al.
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The joint bidiagonalization(JBD) process is a useful algorithm for approximating some extreme generalized singular values and vectors of a large sparse or structured matrix pair A, L. We present a rounding error analysis of the JBD process in finite precision arithmetic. The analysis builds connections between the JBD process and the two joint Lanczos bidiagonalizations. We investigate the loss of orthogonalities of the three groups of Lanczos vectors computed by the JBD process and show that semiorthogonality of the Lanczos vectors is enough to guarantee the accuracy of the computed quantities, which is a guidance for designing an efficient semiorthogonalization strategy for the JBD process. Based on the results of rounding error analysis, we investigate the convergence and accuracy of the approximate generalized values and vectors of A, L. We also analyze the residual norm appeared in the GSVD computation and show that we can use the residual norm as a stopping criterion for approximating generalized singular values and vectors.

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