Research On Parallel Jacobi Method For SVD Problem

The computation of the Singular Value Decomposition of a matrix is important in manyfields, such as meteorology, quantum mechanic, biology, medical science and so on. In thiscomputation, performance and accuracy are two basic problems. In order to improve theperformance of the computation with higher relative accuracy, Jacobi methods get moreattractions. There are two important aspects in the parallel Jacobi methods: Jacobi planerotation and ordering. In this paper, we talk about the JRS method and the dynamic orderingrespectively. The main research of the work is as follows:Firstly, this paper makes a deep research on the fundamental theory of JRS method,and then gives a new plane rotation called JRS2 method which can sort the column norms inordering without many oscillations in the process of the computation. Secondly, the paperstudies the e?ect of the relaxation parameter in the JRS2 method and a new method calledJVRS algorithm is designed to further decrease the number of sweeps of the computation.In the JVRS algorithm, the relaxation parameter could be adjusted according to the state ofthe column norms. Finally, the paper introduces the dynamic ordering which employed inthe two-sided Jacobi firstly into the one-sided Jacobi algorithm. According to the featuresof the one-sided Jacobi method, the paper adopts the spectral norm to give a view of theorthogonality among the matrix columns, and then a parallel power method is proposed toget all the spectral norms of sub-blocks used in the calculation of the dynamic ordering.Our research has significance on the Jacobi method for the computation of SingularValue Decomposition of the matrix. The paper expands the study of the Jacobi plane rotationand gives a deeper explanation of the whole process of Jacobi method. At the same time,the dynamic ordering is introduced into the One-Sided Jacobi method, which provides somereference for the application of the dynamic ordering in Jacobi method.