Backward error analysis of the Lanczos bidiagonalization with reorthogonalization

TL;DR

This paper provides a backward error analysis of the Lanczos bidiagonalization with reorthogonalization, showing how orthogonality levels affect stability and perturbations, with implications for SVD and LSQR algorithms.

Contribution

It introduces a backward error analysis framework for LBRO, linking orthogonality levels to stability and perturbations, and applies it to SVD and LSQR algorithms.

Findings

LBRO's computed bidiagonalization corresponds to an exact one of a perturbed matrix.

Orthogonality levels control the difference between actual and ideal orthonormal matrices.

LBRO is stable if orthogonality of $U_{k+1}$ and $V_{k+1}$ is maintained.

Abstract

The $k$-step Lanczos bidiagonalization reduces a matrix $A\in\mathbb{R}^{m\times n}$ into a bidiagonal form $B_k\in\mathbb{R}^{(k+1)\times k}$ while generates two orthonormal matrices $U_{k+1}\in\mathbb{R}^{m\times (k+1)}$ and $V_{k+1}\in\mathbb{R}^{n\times {(k+1)}}$. However, any practical implementation of the algorithm suffers from loss of orthogonality of $U_{k+1}$ and $V_{k+1}$ due to the presence of rounding errors, and several reorthogonalization strategies are proposed to maintain some level of orthogonality. In this paper, by writing various reorthogonalization strategies in a general form we make a backward error analysis of the Lanczos bidiagonalization with reorthogonalization (LBRO). Our results show that the computed $B_k$ by the $k$-step LBRO of $A$ with starting vector $b$ is the exact one generated by the $k$-step Lanczos bidiagonalization of $A+E$ with starting vector $b+\delta_{b}$ (denoted by LB($A+E,b+\delta_{b}$)), where the 2-norm of perturbation vector/matrix $\delta_{b}$ and $E$ depend on the roundoff unit and orthogonality levels of $U_{k+1}$ and $V_{k+1}$. The results also show that the 2-norm of $U_{k+1}-\bar{U}_{k+1}$ and $V_{k+1}-\bar{V}_{k+1}$ are controlled by the orthogonality levels of $U_{k+1}$ and $V_{k+1}$, respectively, where $\bar{U}_{k+1}$ and $\bar{V}_{k+1}$ are the two orthonormal matrices generated by the $k$-step LB($A+E,b+\delta_{b}$) in exact arithmetic. Thus the $k$-step LBRO is mixed forward-backward stable as long as the orthogonality of $U_{k+1}$ and $V_{k+1}$ are good enough. We use this result to investigate the backward stability of LBRO based SVD computation algorithm and LSQR algorithm. Numerical experiments are made to confirm our results.