On Symmetric Eigenproblems Induced by the Bidiagonal SVD
Benedikt Großer, Bruno Lang · SIAM Journal on Matrix Analysis and Applications · 2005
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 B T B , BB T , or the Golub--Kahan matrix T GK . 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.