Some Preconditioning Techniques For Generalized Saddle Point Problems And PageRank Problems

Preconditioning techniques for large sparse matrix problems axise in a wide variety of appli-cations throughout computational science and engineering.In this dissertation,we propose and examine preconditioning techniques for generalized saddle-point problems,continuous Sylvester equations and PageRank problems.Due to the indefiniteness and often poor spectral properties of large linear systems of saddle point type,such linear systems represent a significant challenge for solver developers.We present a class of generalized relaxed PSS preconditioner for general-ized saddle point problems.Given the complex square matrices A,B and the complex matrix C of conforming dimensions,we consider the linear matrix equation AX+ XB = C in the unknown matrix X.Our aim is to provide a preconditioned MHSS approach in the numerical solution of Sylvester equations.PageRank is a method for ranking Web pages whereby a page's importance(or rank)is determined according to the link structure of the Web.Due to the great size and sparsity of the matrix,iterative methods are used.Based on the inner-outer iteration,we present new iterative schemes for PageRank computation.Our main results are as follows:1.A class of generalized relaxed positive semi-definite and skew-Hermitian splitting(GRPSS)preconditioners for generalized saddle point problems are discussed.Properties of the preconditioned matrix are analyzed and compared with the closely related positive semi-definite and skew-Hermitian splitting(PSS)preconditioner introduced by Shen(2014)-2.On the basis of the MHSS iteration method,we present preconditioned MHSS(PMHSS)iteration and its inexact variant for solving large sparse continuous Sylvester equations with non-Hermitian and complex symmetric positive definite/semi-definite matrices.Under suitable conditions,we prove the convergence of the PMHSS iteration approach and discuss the spectral properties of the preconditioned matrix.Numerical results show that the two iteration approaches are efficient and robust solvers for this class of continuous Sylvester equations.3.By taking the advantage of both an inner-inner-outer(?O)iteration and the Arnoldi pro-cess,we derive a preconditioned Arnoldi-Inout approach for the computation of Pagerank vector.The implementation and convergence of the new algorithm are discussed in detail.To improve the efficiency of the Arnoldi-type algorithm,we also propose a preconditioned Arnoldi-type algorithm,which is based on a periodic combination of the ?O iteration with the Arnoldi-type algorithm.Properties of the new approach are analysed.Numerical experiments are presented to illustrate the effectiveness of our approaches.