【摘要】 <正> 1.引 言 在大量的科学和工程计算中,如整体最小二乘问题、矩阵数值秩的确定、因子分析、回归分析、图象处理等,需要求解如下的 问题1.计算一个大规模矩阵A∈RM×N的k个最大(最小)的奇异值及其对应的奇异子空间,其中k要比M和N要小的多.更多还原

【Abstract】 The singular value decomposition problem is mathematically equivalent to the eigenproblem of an argumented matrix. Golub et al. give a bidiagonalization Lanczos method for computing a number of largest or smallest singular values and corresponding singular vertors, but the method may encounter some convergence problems. In this paper we analyse the convergence of the method and show why it may fail to converge. To correct this possible nonconvergence, we propose a refined bidiagonalization Lanczos method and apply the implicitly restarting technique to it, and we then present an implicitly restarted bidiagonalization Lanczos algorithm (IRBL) and an implicitly restarted refined bidiagonalization Lanczos algorithm (IRRBL). A new implicitly restarting scheme and a reliable and efficient algorithm for computing refined shifts are developed for this special structure eigenproblem. Theoretical analysis and numerical experiments show that IRRBL performs much better than IRBL.更多还原