On Symmetric Eigenproblems Induced by the Bidiagonal SVD

The relatively robust representations (RRR) algorithm is the method of choice to compute highly accurate eigenvector approximations for symmetric tridiagonal matrices. The task of computing singular vector pairs for a bidiagonal matrix $B = U \Sigma V ^T$ is closely connected to the RRR algorithm regarding BTB, BBT, or the Golub--Kahan matrix TGK. Nevertheless, separate application of the RRR algorithm to these matrices leads to poor results regarding either numerical orthogonality or the residual $\| {B}{V} - {U}{\Sigma} \|$. It turns out that the coupling strategy proposed in [B. Grosser and B. Lang, Linear Algebra Appl., 358 (2003), pp. 45--70] resolves this problem. This article provides the corresponding perturbation theory: We compare the eigenvalues of the separate and coupled decompositions and explain why singular vector pairs approximated via couplings are of superior quality.

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