Theory And Perturbation Analysis For A Class Of Singular Linear Equations

The present Ph.D.thesis is concerned with the theoretical analysis for a class of singular linear equations. 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.For solving the singular linear equations,we present the iterative methods based on proper splittings and give the convergence analysis.In the third chapter we make some results of Sylvester displacement rank on the displacement structer of generalized inverse. As the perturbation analysis.we make results on the condition number with the generalized inverse and weighted generalized inverse,which extend the results on the nonsingular linear equations' condition number,then we give the perturbation.Finally, we will see the equations from the discreteness of Navier-Stokes equation are all EP linear equations.Saddle point problems are induced from Navier-Stokes equation,Oseen equation and Stokes equation.There are many methods for solving them.Among them we mention direct solver,Uzawa type algorithms and Krylov subspace methods.We first review all the existing Uzawa methods,and then apply Zulehner's unified approach in [82]to analyze the stabilized saddle point problems.