Application Of Preserving-Structure Algorithm In Solving Linear Systems

In the paper,numerical iterative methods for linear systems are studied based on structure-preserving algorithm and dynamic system method.A numerical method is called structure-preserving if it exactly preserves one or more physical or geometric properties owned by the systems.The dynamic system method is such methods that use the connections between iterative numerical methods and continuous dynamic systems to construct numerical iterative methods for linear system.Dynamical systems have some superiors to discrete ones as they can obtain the convergence results more easily under weaker restrictions on the convergence theorems.This thesis derived two type of iterative methods for linear systems as follows:A new iterative refinement for ill-conditioned linear systems is derived based on discrete gradient methods for gradient systems.It is proved that the new method is convergent for any initial values irrespective of the choice of the stepsize h.It is shown that the condition number of the coefficient matrix in the linear system to be solved in every step can be improved significantly compared with Wilkinson's iterative refinement.The ill-conditioning of the linear system is avoided to some extent.The numerical experiments illustrate that the new method is more effective and efficient than Wilkinson iterative refinement.Based on the exponential integrator for semi-linear differential equations,an exponential Jacobi-type iterative method is proposed for solving linear system.Since the Jacobi iterative method is suitable for massive parallelization,it is still widely used in the numerical computation of linear system.The convergence of the traditional Jacobi iterative method follows immediately from the convergence of the new Jacobi-type methods.The convergence and comparison theorems of the exponential Jacobi-type method are established for linear system with different type of coefficient matrices: M-matrix or nonnegative matrix.For the linear system with coefficient matrix that is M-matrix or nonnegative matrix,the new iterative method is convergent.The spectral radius of iteration matrix for new method is much smaller than traditional Jacobi iterative method for the case of nonnegative coefficient matrix.Numerical experiments are carried out to show the effectiveness of the new method compared with traditional Jacobi iterative method.