Several Iterative Methods For Structured Sparse Linear Systems

Many problems in scientific computing and engineering application can be solved by transforming them into large structured sparse linear systems,but these are often indefinite,ill-conditioned and even singular.Therefore,the research of effcient numerical algorithms for these problems has important theoretical significance and application value.In this paper,we propose four kinds of fast and effective iterative algorithms and the corresponding preconditioning techniques for the real valued 2 × 2block structured forms resulted from complex symmetric linear systems,where main contents are as follows:Firstly,we study the Double-parameters Symmetric Block Triangle Splitting(DSBTS)iterative algorithm.Its convergence conditions are given,and the numerical experiments shows that the algorithm is effective when the parameters are proper.Secondly,a New Block Triangular(NBT)preconditioner is devised to transform non-singular symmetric indefinite linear systems into symmetric positive definite linear systems,which are solved by using Preconditioned Conjugate Gradient(PCG)algorithm.The numerical results show that the PCG algorithm with NBT preconditioner is more efficient compared with several common preconditioners.Thirdly,we use the Successive Over-Relaxation(SOR)method to accelerate the Simplified Modified Skew-Normal Splitting(SMSNS)algorithm.The corresponding SMSNSSOR iterative algorithm is established,the convergence theory and optimal parameters are given,and the validity of the algorithm is verified by numerical experiments.Fourth,we construct a preconditioner which is very close to the original coefficient matrix for the structured linear systems with indefinite sub-block,and the numerical validity of the corresponding preconditioned Generalized Minimal Residual(GMRES)algorithm is verified.