Research Of Several Iterative Algorithms For Solving Large Sparse Linear Systems And Application

With the rapid development of computer technology, how to effectively solve thelarge sparse linear systems has become the core problem in scientific and engineeringcomputation, numerical simulation, and financial optimization. Since the time spent insolving the linear system often occupies a large proportion of the total computing time, tosolve the large sparse linear systems efficiently can improve the solution efficiency of theproblem largely. This thesis focuses on Krylov subspace methods based on lanczos processfor solving the large linear systems, The main work is as follows:At first, a discription of Conjugate Residual-Like methods proposed resently isprovided, which is based on the CR method, such as: BiCR, BiCRSTAB, CRS, etc. Andthe solving thought and algorithm of these methods is analysed briefly, then somenumerical examples are used to compare the convergence rate, stability and executionefficiency.Secondly, on the one hand, an adaptive preconditioned BiCRSTAB methods based onBiCRSTAB algorithm is put forward, which is combined with an polynomialpreconditioner constructed implicitly, and several steps of GMRES are inserted inBiCRSTAB algorithm. Numerical experiments illustrate this method can reduce theiterative steps and computation time effectively; on the other hand, we try to useA-Lanczos biorthogonal process instead of the Lanczos biorthogonal process. At the sametime, the linear combination of the approximate solution and the wasted basis vector isused as a new approximate solution of the algorithm, and the residual norm of newapproximate solution satisfy a one-dimensional minimization problem, so as to get amodified QMR algorithm based on the A-Lanczos biorthogonal process. The numericalexperiments show that the new algorithm convergences faster than the original QMRalgorithm for some large sparse linear systems.Finally, MQMRA method proposed above is applied to solve the Navier-Stokesequation in fluid dynamics, taking the sudden enlargement in parallel pipe as an example, the feasibility and efficiency of MQMRA algorithm is verified in the problem, and theresults are compared with CR-like methods.