关键词：有限精度；兰索斯法； Krylov子空间；误差分析
摘 要：In this paper, we present, for the first time, a complete rounding error analysis of the $s$-step Lanczos method. Our methodology is analogous to Paige's rounding error analysis for the classical Lanczos method [\emph{IMA J. Appl. Math.}, 18(3):341--349, 1976]. Our analysis gives upper bounds on the loss of normality of and orthogonality between the computed Lanczos vectors, as well as a recurrence for the loss of orthogonality. The derived bounds are very similar to those of Paige for classical Lanczos, but with the addition of an amplification term which depends on the condition number of the Krylov bases computed every $s$-steps. Our results confirm theoretically what is well-known empirically: the conditioning of the Krylov bases plays a large role in determining finite precision behavior.