Numerical Linear Algebra Problems

The contents of the article include two parts. In the first part, we study three kinds of pre-conditioners -- block-triangular preconditioner, constraint preconditioner and block-diagonalpreconditioner respective1y and fast iterative so1vers fOr indefinite 2 by 2 b1ock linear systemswith zero (2,2) block. The spectrum and fOrm of eigenvalues of each preconditioned nlatrix arepresented. All of the three kinds of preconditioned linear syStems are so1ved by GMRES(20)and theiI efficiencies are cornpared. NumericaJ results are presented by a maJss of tables andfigures. In the second part, we propose direct perturbation methods of eigensensitivity ana1ysisa defective nlatrix fOr three different cases. The first case is that all of the first-order pertur-batioxl coefficients of a defective eigenvalue associated with its Jordan bIocks with same orderare distinct and nonzero. In this case, we give the formulas to caIculate first- to third-orderperturbation coefficients of the eigenvalues and first- to second-order perturbation coefficientsof the eigenvectors. 1n second case where the eigenprobIem for the first-order perturbationcoefficients of a defective eigenvalue haJs repeated eigenvalues, we give the fOrmulas to calculatethe first- to third-order perturbation coefficients of the eigenvalues and first order perturbationcoefficients of the eigenvectors. The third case is an extension of the first case, where one of thefirst-order perturbation coefficients of the eigenvalues associated with the lowest-order Jordanblocks is zero. In this case, we give the formulaJs to calculate first- to third--order perturbationcoefficients of the eigenva1ues and eigenvectors. The nun1erical examples show the validity oftllese methods.