Algorithms And Perturbation Analysis For Solving Consistant And Inconsistant Singular Linear Equations

The present Ph.D. thesis is concerned with the theoretical analysis and numerical algorithms for a class of singular linear equations.As we know not like with nonsingular linear equations, some singular linear equations are inconsistant. So we will deal with the two circumstances. Among so many singular linear equations we are very interested in the equations whose coefficient matrices are range Hermitian (EP matrices).First we will see the generalized inverse of EP matrix has so many qualities as same as the normal inverse, and the solution of a EP linear equation also has some good qualities that the common singular linear equations do not have.As the perturbation analysis , we make some results on the singular linear equations'condition number concerned with the generalized inverse AT,S,(2) which extend the results on the nonsingular linear equations' condition number, then we give the perturbation bound for the solutions of EP linear equations.At last we do some numerical algotithms, we will see the linear equations from the discreteness of Poisson equation and Navier-Stokes equation are all EP linear equations. When the equations are inconsistant we will make a good tolerance through the minimal || . ||M norm solution. Particularly for Navier-Stokes equation, we will compare the GMRES methods with different preconditioned. We make some variances from the HSS preconditioner ([15]), that are Ma1-Ma4 in our paper. Comparing with block triangular preconditioner we will find that the CPU time for Ma1 is the smallest. We also make some experiments for the perturbation theory of EP linear equations.