An implicitly restarted refined bidiagonalization Lanczos method for computing a partial singular value decomposition

作者：Jia ZX^*; Niu DT

来源：SIAM Journal on Matrix Analysis and Applications, 2003, 25(1): 246-265.

DOI：10.1137/S0895479802404192

• singular value
• singular vector
• the bidiagonalization Lanczos method
• Ritz value
• Ritz vector
• refined Ritz vector
• the refined bidiagonalization Lanczos method
• implicit restart
• exact shifts
• refined shifts
• convergence

The bidiagonalization Lanczos method can be used for computing a few of the largest or smallest singular values and corresponding singular vectors of a large matrix, but the method may encounter some convergence problems. In this paper the convergence of the method is analyzed, showing why it may converge erratically and perhaps fail to converge. To correct this possible nonconvergence and improve the method, a refined bidiagonalization Lanczos method is proposed. The implicitly restarting technique due to Sorensen is applied to the method, and an implicitly restarted refined bidiagonalization Lanczos algorithm (IRRBL) is developed. A new selection of shifts is proposed for use within IRRBL, called refined shifts, and a reliable and efficient algorithm is developed for computing the refined shifts. Numerical experiments show that IRRBL can perform better than the implicitly restarted bidiagonalization Lanczos algorithm (IRBL) proposed by Larsen, in particular when the smallest singular triplets are desired.

• 出版日期2003
• 单位清华大学; 大连理工大学