Iterative Methods and Applications 2014View this Special Issue
Research Article | Open Access
Jen-Yuan Chen, David R. Kincaid, Yu-Chien Li, "More on Generalizations and Modifications of Iterative Methods for Solving Large Sparse Indefinite Linear Systems", Journal of Applied Mathematics, vol. 2014, Article ID 628069, 15 pages, 2014. https://doi.org/10.1155/2014/628069
More on Generalizations and Modifications of Iterative Methods for Solving Large Sparse Indefinite Linear Systems
Continuing from the works of Li et al. (2014), Li (2007), and Kincaid et al. (2000), we present more generalizations and modifications of iterative methods for solving large sparse symmetric and nonsymmetric indefinite systems of linear equations. We discuss a variety of iterative methods such as GMRES, MGMRES, MINRES, LQ-MINRES, QR MINRES, MMINRES, MGRES, and others.
When solving large sparse linear systems of the form in which the coefficient matrix is indefinite, there are basis methods and a variety of generalizations and modifications of them. For example, basic iterative methods for symmetric indefinite linear systems are the MINRES method and the SYMMLQ method, while a basic method for nonsymmetric linear systems is the GMRES method. (See, e.g., Lanczos , Golub and Van Loan , Paige and Saunders , Saad , and Saad and Schultz .)
In Section 2, we review the Arnoldi process and present background material. In Sections 3 and 4, we describe the LQ-MINRES and the QR-MINRES methods, respectively, as well as discussing their relationship in Section 5. In Section 6, we take a closer look at the QR-MINRES method and the SYMMQR method. In Sections 7 and 8, we describe the modified MINRES (MMINRES) method and the generalized QR-MINRES method, respectively. In Section 9, we review the GMRES method. Finally, we discuss the differences between the modified MINRES (MMINRES) method and the modified GMRES (MGMRES) method, in Section 10.
2. Arnoldi Process
First, we assume that matrix is symmetric. In [6, 7], we use a short term recurrence to generate orthonormal vectors as follows: Here we assume that and , for all . If we let , then the subspace, is equivalent to the Krylov subspace Consequently, we have the following properties, for (, ): as well as these matrix equations where
Example. We illustrate (6), for the case .
3. LQ-MINRES Method
We choose such that . Hence, we have For the MINRES method , we let then Letting , we can minimize by solving this linear system for
First, using (6), (7), and (5), we expand the coefficient matrix on the left-hand side of linear system (18) since is symmetric. Second, we exam the right-hand side vector in linear system (18) since , , and .
Here . We obtain where Here a Givens rotation matrix iswith .
Here, we repeatedly apply Givens rotations to the right-hand side of , (10), in order to zero out the -diagonal above the main diagonal and change the tridiagonal matrix into a lower tridiagonal matrix .
Then, from (22), we have Since , we obtain Since is symmetric, we have Thus, we find that the coefficient matrix (26) can be written as Consequently, we are now interested in solving this linear system
Then, we have where Consequently, we obtain Then, we let where So we have since . Consequently, we have
From (24) and (36) we have the right-hand side vector in linear system (28) Since is nonsingular, and from (41) and (43), linear system (28) is which reduces to Thus, we obtain this equation for the th iteration of the LQ-MINRES method
4. QR-MINRES Method
First, we repeatedly apply Givens rotations to the left-hand side of where Here, we use Givens rotations applied on the left-hand side of , (10), to transform a tridiagonal matrix into an upper tridiagonal matrix , (21).
Then, we obtain Since is symmetric, we obtain Thus, the coefficient matrix in linear system (48) can be written as Consequently, we are interested in solving this linear system
Let where We have
Then, we have where So we have since . Consequently, we obtain
Since we have right-hand side vector in the linear system (54) Since is nonsingular, we obtain the linear system (54) which reduces to Thus, we obtain the equation for the th iteration of the QR-MINRES method
5. Relation between LQ-MINRES and QR-MINRES
Now, we show that the LQ-MINRES method and the QR-MINRES method are essentially the same. In the LQ-MINRES method (46), we have In the QR-MINRES method (72), we have For the LQ-MINRES method (73), we have In the QR-MINRES method (46), we have From the computation and by induction, we have the following relations between the LQ-MINRES method and the QR-MINRES method: By induction, we have Hence, we obtain Moreover, if we let then Thus, we obtain
6. A Closer Look at QR-MINRES and SYMMQR
The first estimated solution is where is the solution of this linear system with the right-hand side vector being From these equations, we have Thus, we have where is the solution of this least square problem
The second estimated solution is where is the solution of this linear system
7. Modified MINRES (MMINRES) Method
Next, we assume that matrix is nonsymmetric. In [6, 7], we use a long term recurrence to generate orthonormal vectors as follows: Consequently, we obtain the following matrix equations: where Since the matrix is a full upper Hessenberg matrix, the LQ-MINRES method is not a practical procedure. Hence, we discuss only a generalization of the QR-MINRES method.
8. Generalized QR-MINRES Method
Since the matrix is nonsymmetric, to minimize an expression such as this we choose to satisfy First, from the Arnoldi process and from the left-hand side of linear system (96), we can write the coefficient matrix of this linear system as Second, from the right-hand side vector of the linear system (96), we have Consequently, instead of solving (96), we solve this linear system
First, we repeatedly apply Givens rotations to the left-hand side of By defining we obtain so that Thus, the coefficient matrix in linear system (99) has the following form by using (103) and (104): Moreover, by using (102) and (104), the right-hand side vector in linear system (99) is where where and . Thus, we obtain . Then, we have
Let Then, we have and we obtain We have
Using , (112), and (114) we find that the coefficient matrix in linear system (104) is and the right-hand side vector is Since is nonsingular, we obtain this linear system which reduces to this linear system Thus, we obtain the th iteration of the generalized QR-MINRES method
9. GMRES Method
In the GMRES method, we let Multiplying by , we have Then For minimizing , we need to solve by using Givens rotation, which is the GMRES method.
In Saad’s book , there is a relation between the FOM method and the GMRES method. For the FOM method, we impose the Galerkin condition and we solve this linear system for For the ()st iteration, we solve this linear system for Between the th iteration and the ()st iteration of the FOM method, we obtain the solution of the this least squares problem which is the same as in the GMRES method.
10. Differences between MMINRES and MGMRES
In this section, we assume that matrices and are nonsingular symmetric matrices (but not necessarily positive definite). In [6, 7], we use a short term recurrence to generate orthonormal vectors as follows: We obtain these properties, for (, ), where Consequently, we obtain these matrix equations