An Implicitly Restarted Lanczos Bidiagonalization Method With Refined Vector For Computing Smallest Singular Triplets

Many scientific applications lead to large-scale singular triplets problems, where typically only a few smallest singular values are of interest. For such promblems, we always use LBD to change the matrix to bidiagonal matrix, then find the SVD of the bidiagonal matrix by the Golub-Kahan SVD algorithm[9], by doing this we solve the problem. LBD first devised by Golub[10], later Sorensen introduces the implicitly restarted Arnold mehthod[25]. The implicitly restarted techniques was generalized to LBD by Bjock, then Jia[13] presents "refined algorithm" by augmenting "refined vector" . Recently Kokiopoulou generalized LBD by augmenting refined vecor for finding the smallest singular triplets[15].Our main work is the development of an implicitly restartde Lanczos bidig-onalization method with refined vector for computing smallest singular triplets. Our main point is the smallest singular triplets, so we use the hamonriz ritz as the shift which can get singular triplet more efficiently. In the algorithm we also use refined singular vector, the refined residual, refined Rayleigh equotient and the deflation technique. Our numerical experiment illustraste that our algorithm can solve the large-scale singular triplets problem efficiently and it converges rather fast.In chapter 1, we give an overview of the background of the singular problems and the research history of solving singular problem. In chapter 2, we firstly give the LBD algorithm, the implicitly LBD then the shift selection stategies. We prove some conclusion of ODT and present the refined vector, refined residual and the refined Rayleigh equotient In chapter 3. In chapter 4, we develop the whole algorithm, Lastly we illustrate the numerical experiments.