On stagnation of the DGMRES method

Document Type : Research Article

Author

F. Kyanfar

Department of Applied Mathematics, Shahid Bahonar University of Kerman, Iran.

10.22067/ijnao.2022.73913.1081

Abstract

Let $A$ be an $n$-by-$n$ matrix with index $\alpha>0$ and $b \in \mathbb{C}^n$.  In this paper, the problem of stagnation of the DGMRES method for the singular linear system $Ax=b$ is considered. We show that DGMRES$(A, b, \alpha)$ has partial stagnation of order at least $k$ if and only if  $(0, \ldots, 0)$ belongs to the the joint numerical range of matrices  ${B^{\alpha+1}, \ldots, B^{\alpha+k}},$ where $B$ is a compression of $A$ to the range of $A^{\alpha}.$ Also, we characterize nonsingular part of a matrices $A$ such that DGMRES$(A, b, \alpha)$ does not stagnate for all $b \in \mathbb{C}^n$.  Moreover, a sufficient condition for non-existence of real stagnation vectors $b \in \mathcal{R}(A^{\alpha}) $ for DGMRES method is presented and the DGMRES stagnation of special matrices are studied.

Keywords

Stagnation

DGMRES method

Singular systems

Subjects

• Numerical linear algebra and its applications

---

References

1. Greenbaum, A., Kyanfar F. and Salemi, A. On the convergence rate of DGMRES, Linear Algebra Appl., 552 (2018), 219–238.

2. Horn R.A. and Johnson, C.R. Matrix analysis, Second edition. Cam-bridge University Press, Cambridge, 2013.

3. Kyanfar, F., Mohseni Moghadam, M. and Salemi, A. Complete stagnation of GMRES for normal matrices, Comput. Appl. Math., 263 (2014), 417–422.

4. G. Meurant, The complete stagnation of GMRES for n ≤ 4, Electron. Trans. Numer. Anal. 39 (2012), 75–101.

5. Meurant, G. Necessary and sufficient conditions for GMRES complete and partial stagnation, Appl. Numer. Math., 75 (2014), 100–107

6. Saad Y. and Schultz, M.H. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986), 856–869.

7. Sidi, A. DGMRES: A GMRES-type algorithm for Drazin-inverse solu-tion of singular nonsymmetric linear systems, Linear Algebra Appl., 335 (2001), 189–204.

8. Toutounian, F. and Buzhabadi, R. New methods for computing the Drazin-inverse solution of singular linear systems, Appl. Math. Comput. 294 (2017), 343–352.

9. Williams, J.P. Similarity and the Numerical Range, J. Math. Anal. Appl. 26 (1969), 307–314.
10. Zavorin, I., O,Leary, D.P. and Elman, H. Complete stagnation of GMRES, Linear Algebra Appl. 367 (2003), 165–183.

11. Zhou J. and Wei, Y. Stagnation analysis of DGMRES, Appl. Math. Comput. 151 (2004), 27–39.