We establish two types of block triangular preconditioners applied to the linear saddle point problems with the singular (1,1) block. These preconditioners are based on the results presented in the paper of Rees and Greif (2007). We study the spectral characteristics of the preconditioners and show that all eigenvalues of the preconditioned matrices are strongly clustered. The choice of the parameter is involved. Furthermore, we give the optimal parameter in practical. Finally, numerical experiments are also reported for illustrating the efficiency of the presented preconditioners.

Consider the following saddle point linear system:

Recently, T. Rees and C. Greif explored a preconditioning technique applied to the problem of solving linear systems arising from primal-dual interior point algorithms and quadratic programming in [

The rest of this paper, two types of block triangular preconditioners are established for the saddle point systems with an ill-conditioned (1,1) block. Our methodology extends the recent work done by Greif and Schötzau [

This paper is organized as follows. In Section

For linear systems, the convergence of an applicable iterative method is determined by the distribution of the eigenvalues of the coefficient matrix. In particular, it is desirable that the number of distinct eigenvalues, or at least the number of clusters, is small, because in this case convergence will be rapid. To be more precise, if there are only a few distinct eigenvalues, then optimal methods like BiCGStab or GMRES will terminate (in exact arithmetic) after a small and precisely defined number of steps.

Rees and Greif [

The matrix

Suppose that

Let the vectors

Next, we consider the remaining

When the parameter

Let

The matrix

According to Theorem

From (

When the parameter

Let

The matrix

The proof is similar to the proof of Theorem

When the parameter

Let

The matrix

The proof is similar to the proof of Theorem

When the parameter

Let

The above theorems and corollaries illustrate the strong spectral clustering when the (1, 1) block of

It is clearly seen from Theorems

Similarly, the nonsymmetric saddle point linear systems can also obtain the above results.

All the numerical experiments were performed with MATLAB 7.0. The machine we have used is a PC-Intel(R), Core(TM)2 CPU T7200 2.0 GHz, 1024 M of RAM. The stopping criterion is

Our numerical experiments are similar to those in [

We construct the saddle point-type matrix

From the matrix

In our numerical experiments the matrix

In the following, we summarize the observations from Tables

Values of

Number and time of iterations of GMRES(10) with preconditioners

Time ( | Time | Time | ||||
---|---|---|---|---|---|---|

Number and time of iterations of GMRES(10) with preconditioners

Time ( | Time | Time | ||||
---|---|---|---|---|---|---|

Number and time of iterations of GMRES(10) with preconditioners

Time ( | Time | Time | ||||
---|---|---|---|---|---|---|

Number and time of iterations of GMRES(10) with preconditioners

Time ( | Time | Time | Time | |||||
---|---|---|---|---|---|---|---|---|

Number and time of iterations of GMRES(10) with preconditioners

Time ( | Time | Time | Time | |||||
---|---|---|---|---|---|---|---|---|

Number and time of iterations of GMRES(10) with preconditioners

Time ( | Time | Time | Time | |||||
---|---|---|---|---|---|---|---|---|

Convergence curve and total numbers of inner GMRES(10) iterations for different

Convergence curve and total numbers of inner GMRES(10) iterations for different

Convergence curve and total numbers of inner GMRES(10) iterations for different

From Tables

Number and time of iterations with the preconditioner

Number and time of iterations with the preconditioner

Number of iterations decreases but the computational cost of incomplete LU factorization increases with decreased

The eigenvalues of

We have proposed two types of block triangular preconditioners applied to the linear saddle point problems with the singular (1,1) block. The preconditioners have the attractive property of improved eigenvalues clustering with increasing ill-conditioned (1,1) block. The choice of the parameter is involved. Furthermore, according to Corollaries

In fact, our methodology can extend the unsymmetrical case; that is, the (1,2) block and the (2,1) block of the saddle point linear system are unsymmetrical.

Warm thanks are due to the anonymous referees and editor Professor Victoria Vampa who made much useful and detailed suggestions that helped the authors to correct some minor errors and improve the quality of the paper. This research was supported by 973 Program (2008CB317110), NSFC (60973015, 10771030), the Chinese Universities Specialized Research Fund for the Doctoral Program of Higher Education (20070614001), and the Project of National Defense Key Lab. (9140C6902030906).