On the eigenvalue distribution of preconditioned nonsymmetric saddle point matrices

Abstract

In this paper, we derive bounds for the complex eigenvalues of a nonsymmetric saddle point matrix with a symmetric positive semidefinite .2, 2/ block, that extend the corresponding previous bounds obtained by Bergamaschi. For the nonsymmetric saddle point problem, we propose a block diagonal preconditioner for the conjugate gradient method in a nonstandard inner product. Numerical experiments are also included to test the performance of the presented preconditioner.A large variety of applications and numerical solution methods of (1) or (2) have been comprehensively reviewed by Benzi, Golub, and Liesen [1]. Recently, the eigenvalue distribution of the nonsymmetric saddle point matrix A has been deeply studied. On one hand, eigenvalue bounds for A can be used to analyze the spectral properties of preconditioners such as symmetric indefinite preconditioners, inexact constraint preconditioners and primal-based penalty preconditioners for (1); see, for example, [3][4][5][6]. On the other hand, eigenvalue estimates for A can give theoretical basis for the CG method for (2) in a nonstandard inner product; see, for example, [3,[6][7][8]. This paper aims to provide further bounds for the complex eigenvalues of A that extend the corresponding ones in [4].An efficient implementation of the CG method for (2) has been established by Liesen and Parlett [7]. The authors pointed out that preconditioning should be studied in practical applications,(1) If Im( )=0, then minf% 1 , & 1 g 6 6 maxf% n , m g.

A Priori Indicator of Convergence Rate for Power Method for Computing the Spectral Radius of Saddle Point Matrices

“…Saddle point problems arise from many scientific research fields and engineering applications, such as mixed finite element methods, constrained least square problems, image processing and so on [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], and usually generate the following linear system:…”

“…is called "saddle point matrix" [10][11][12][13][14][15][16]. Here the sub block ∈ R × is symmetric and positive definite, ∈ R × ( ≥ ) has full column rank, and is an n-order zero matrix.…”

“…Most of these techniques need eigenvalue information of matrix (or other related matrices). Therefore, the research on the properties of the key eigenvalues of saddle point matrix has attracted more attentions [9][10][11][12][13][14][15][16]. We should note that in practice the size of matrices are usually very large, and the computational complexity should be as small as possible.…”