This paper studies the solution of nonsymmetric linear systems by preconditioned Krylov methods based on the normal equations, LSQR in particular. On some examples, preconditioned LSQR is seen to produce errors many orders of magnitude larger than classical direct methods; this paper demonstrates that the attainable accuracy of preconditioned LSQR can be greatly improved by applying iterative refinement or restarting when the accuracy stalls. This observation is supported by rigorous backward error analysis. This paper also provides a discussion of the relative merits of GMRES and LSQR for solving nonsymmetric linear systems, demonstrates stability for left-preconditioned LSQR without iterative refinement, and shows that iterative refinement can also improve the accuracy of preconditioned conjugate gradient.

翻译：本文研究基于法方程的预条件Krylov方法（特别是LSQR）求解非对称线性系统的解。在某些示例中，预条件LSQR产生的误差比经典直接方法大数个数量级；本文证明，当精度停滞时，通过应用迭代精化或重启策略，可显著提升预条件LSQR的可达精度。这一观察得到了严格的后向误差分析的支持。本文还讨论了GMRES与LSQR在求解非对称线性系统时的相对优劣，证明了无迭代精化的左预条件LSQR的稳定性，并表明迭代精化同样能提升预条件共轭梯度法的精度。