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Abstract and Applied Analysis

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

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

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

^{1}Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan^{2}Institute 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*

*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
HereWe note that is symmetric, for :
*

*7. Lanczos/MSYMMLQ Method*

*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*

*8. MSYMMLQ Pseudocode**9. MSYMMQR Pseudocode*

*9. MSYMMQR Pseudocode**10. LAN/MSYMMLQ Pseudocode*

*10. LAN/MSYMMLQ Pseudocode**11. LAN/MSYMMQR Pseudocode*

*11. LAN/MSYMMQR Pseudocode**Conflict of Interests*

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

*References*

*References*

- C. C. Paige and M. A. Saunders, “Solutions of sparse indefinite systems of linear equations,”
*SIAM Journal on Numerical Analysis*, vol. 12, no. 4, pp. 617–629, 1975. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at MathSciNet - 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. - D. R. Kincaid, D. M. Young, and J.-Y. Chen, “An overview of MGMRES and LAN/MGMRES methods for solving nonsymmetric linear systems,”
*Taiwanese Journal of Mathematics*, vol. 4, no. 3, pp. 385–396, 2000. View at Zentralblatt MATH · View at MathSciNet

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