Abstract and Applied Analysis

Volume 2014 (2014), Article ID 237808, 11 pages

http://dx.doi.org/10.1155/2014/237808

Research Article

Some Generalizations and Modifications of Iterative Methods for Solving Large Sparse Symmetric Indefinite Linear Systems

Yu-Chien Li,¹ Jen-Yuan Chen,¹ and David R. Kincaid²

¹Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan

²Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA

Received 27 November 2013; Revised 10 January 2014; Accepted 4 February 2014; Published 3 April 2014

Academic Editor: Chi-Ming Chen

Copyright © 2014 Yu-Chien Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We discuss a variety of iterative methods that are based on the Arnoldi process for solving large sparse symmetric indefinite linear systems. We describe the SYMMLQ and SYMMQR methods, as well as generalizations and modifications of them. Then, we cover the Lanczos/MSYMMLQ and Lanczos/MSYMMQR methods, which arise from a double linear system. We present pseudocodes for these algorithms.

The authors dedicate this paper to the memory of Professor David M. Young, Jr., for his pioneering research, inspirational teaching, and exceptional life

1. Introduction

Frequently, when computing numerical solutions of partial differential equations, one needs to solve systems of very large sparse linear algebraic equations of the form

where is an matrix, is an vector, and one seeks a numerical solution vector or a good approximation of it. Particularly for large linear systems arising from partial differential equations in three dimensions, well-known direct methods, such as Gaussian elimination, may become prohibitively expensive in terms of both computer storage and computer time. On the other hand, a variety of iterative methods may avoid these difficulties.

For linear systems involving symmetric positive definite (SPD) matrices, the conjugate gradient (CG) method (and variations of it) may work well. On the other hand, when solving linear systems, where the coefficient matrix is symmetric indefinite, the choice of a suitable iterative method is not at all clear. On the other hand, the SYMMLQ and MINRES methods have been shown to be useful in certain situations (see Paige and Saunders [1]). For nonsymmetric systems, Saad and Schultz [2] generalized the MINRES method to obtain the GMRES method.

In Section 2, we review the Arnoldi process. In Sections 3 and 4, we describe the SYMMLQ and SYMMQR methods. Then we can generalize them, in Section 5, and we outline the modified SYMMLQ method, in Section 6. Next, in Section 7, we discuss applying the MSYMMLQ and MSYMMQR methods applied to a double linear system. Finally, we present pseudocodes in Sections 8–11.

2. Arnoldi Process

We begin with a review of the Arnoldi process.

Theorem 1. Suppose that is an symmetric matrix. One can generate orthonormal vectors using this short-term recurrence

where

Here, one assumes that and , for all . Then the following properties hold, for (, ):

Proof. If we let , then the subspace

is equivalent to the Krylov subspace

We obtain

since .

From Theorem 1, in matrix form, it follows that

where

Example 2. We illustrate Theorem 1 for the case .

From (2) and (3), we have

Consequently, we obtain, since ,

So we obtain

3. SYMMLQ Method

We choose , such that . Hence, we have

Imposing the Galerkin condition , we obtain

We obtain

because

Instead of solving for directly from the triangular linear system (15), Paige and Saunders [1] factorize the matrix into a lower triangular matrix with bandwidth three (resulting in the SYMMLQ method). Also, we have

where is an orthogonal matrix, and

where . Since , we have

Letting

then

Next letting

we have

Defining

we have

where

We let

where

From (21) and (28), we have . Since

we have

If , then is nonsingular. We can find by solving

4. SYMMQR Method

We choose such that . Hence, we have

Imposing the Galerkin condition , as before, we obtain

Since

we have

Instead of solving for directly from the triangular system (35), Paige and Saunders [1] factorized the matrix into a lower triangular matrix with bandwidth three.

We can use a different factorization of to obtain a slightly different method, which is called the SYMMQR method. We multiply the matrix by an orthogonal matrix on the left-hand side instead of the right-hand side. We have

where

We obtain the matrix , where

with being the Givens rotation. Letting be the solution of

then we have

which satisfies the Galerkin condition , where . We note that is not always nonzero and, thus, might be singular. We assume that is nonsingular and then we define

where

We have

For the next iterate , we need to solve

where

Applying the Givens rotation to both sides of (45), we have

where .

To eliminate , we compute the th Given rotation by

By multiplying times and times , we have

where

Let

We define . Since , then and is nonsingular. We can solve for from

We discuss the case later.

Consider solving the least square problem involving minimizing , where

We have

Hence, the solution from minimizes and .

Let

where

We have

Since

we obtain

We note that is the estimated solution vector satisfying the Galerkin condition, while

with minimizing .

5. Generalized SYMMLQ and SYMMQR Methods

Now, we generalize the SYMMLQ and SYMMQR methods.

Theorem 3. Suppose that is an symmetric positive definite (SPD) matrix and is an symmetric matrix. One can generate orthonormal vectors using this short-term recurrence

where

Then the following properties hold, for (, ):

Proof. We obtain

Since .

As before, we let

Moreover, we have

where

As before, we let

Imposing the Galerkin condition again, we have

We obtain

because

Since is symmetric, we can apply the same techniques as in the SYMMLQ method. Also, if , the method reduces to the SYMMQR method.

6. Modified SYMMLQ Method

Next, we outline the modified SYMMLQ method.

Theorem 4. Suppose that is an symmetric (not necessary positive definite) matrix and is an symmetric matrix. One can generate orthonormal vectors using this short-term recurrence

where

Then the following properties hold, for :

and, for (, ),

From Theorem 4, in matrix form, we obtain

where

Moreover, from Theorem 4, we obtain

Then, we have

Here the second term on the right-hand side is the zero matrix!

In addition, we have

Imposing the Galerkin condition, , as we did before, we obtain

In other words, we use

We obtain

because

Here

We note that is symmetric, for :

7. Lanczos/MSYMMLQ Method

Next, we consider this double linear system:

We obtain the block symmetric matrices , , and , where

For example, the modified SYMMLQ method and the modified SYMMQR method can be applied to the double linear system (86). This leads us to the LAN/MSYMMLQ method and the LAN/MSYMMQR method. The pseudocodes for these methods are given in the following sections. For additional details, see Li [3]. See the books by Golub and Van Loan [4] and Saad [5], as well as the papers by Lanczos [6] and Kincaid et al. [7], among others.

8. MSYMMLQ Pseudocode

9. MSYMMQR Pseudocode

10. LAN/MSYMMLQ Pseudocode

11. LAN/MSYMMQR Pseudocode

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

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Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM Journal on Scientific and Statistical Computing, vol. 7, no. 3, pp. 856–869, 1986.

Y. Li, The Modified MINRES methods for solving large sparse non-symmetric linear systems [M.S. thesis], Department of Mathematics, National Kaohsiung Normal University, Kaohsiung, Taiwan, 2007.

G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, Md, USA, 3rd edition, 1996.

Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, 2nd edition, 2003.

C. Lanczos, “An iteration method for the solution of the eigenvalue problem of linear differential and integral operators,” Journal of Research of the National Bureau of Standards, vol. 45, no. 4, pp. 255–282, 1950.

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