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A Convergence Analysis of the MINRES Method for Some Hermitian Indefinite Systems

  • Ze-Jia Xie (a1), Xiao-Qing Jin (a2) and Zhi Zhao (a3)


Some convergence bounds of the minimal residual (MINRES) method are studied when the method is applied for solving Hermitian indefinite linear systems. The matrices of these linear systems are supposed to have some properties so that their spectra are all clustered around ±1. New convergence bounds depending on the spectrum of the coefficient matrix are presented. Some numerical experiments are shown to demonstrate our theoretical results.


Corresponding author

*Corresponding author. Email (Z.-J. Xie), (X.-Q. Jin), (Z. Zhao)


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[1] Axelsson, O., A class of iterative methods for finite element equations, Comput. Methods Appl. Mech. Engrg. 9, 123127 (1976).
[2] Axelsson, O., Lindskog, G., On the rate of convergence of the preconditioned conjugate gradient method, Numer. Math. 48, 499523 (1986).
[3] Axelsson, O., Solution of linear systems of equations: Iterative methods, In Barker, V., editor, Sparse Matrix Techniques, v. 572 of Lecture Notes in Mathematics, Springer, 1997, pp. 151.
[4] Bai, Z., Jin, X., Yao, T., Superoptimal preconditioners for functions of matrices, Numerical Mathematics: Theory, Methods and Applications 8, 515529 (2015).
[5] Beckermann, B., Kuijlaars, A., Superlinear convergence of conjugate gradients, SIAM J. Numer. Anal. 39, 300329 (2001).
[6] Beckermann, B., Discrete orthogonal polynomials and superlinear convergence of Krylov sub-space methods in numerical linear algebra, In Orthogonal Polynomials and Special Functions, Springer, 2006.
[7] Campbell, S., Ipsen, I., Kelley, C., Meyer, C., GMRES and the minimal polynomial, BIT 36, 664675 (1996).
[8] Chan, T., An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Stat. Comput. 9, 766771 (1988).
[9] Chan, R., Ng, M., Conjugate gradient methods for Toeplitz systems, SIAM Rev. 38, 427482 (1996).
[10] Chan, R., Potts, D., Steidl, G., Preconditioners for nondefinite Hermitian Toeplitz systems, SIAM J. Matrix Anal. Appl. 22, 647665 (2000).
[11] Chan, R., Jin, X., An Introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia, 2007.
[12] Elman, H., Silvester, D., Wathen, A., Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, Oxford University Press, Oxford, 2005.
[13] Fischer, B., Polynomial Based Iteration Methods for Symmetric Linear Systems, Willey and Teubner, Chichester, West Essex, England, and Stuttgart, 1996.
[14] Gergelits, T., Strakoš, Z., Composite convergence bounds based on Chebyshev polynomials and finite precision conjugate gradient computations, Numer. Algor. 65, 759782 (2014).
[15] Greenbaum, A., Iterative Methods for Solving Linear Systems, SIAM, Philadelphia, 1997.
[16] Herzog, R., Sachs, E., Superlinear convergence of Krylov subspace methods for self-adjoint problems in Hilbert space, SIAM J. Numer. Anal. 53, 13041324 (2015).
[17] Jennings, A., Influence of the eigenvalue spectrum on the convergence rate of the conjugate gradient method, J. Inst. Math. Appl. 20, 6172 (1977).
[18] Jin, X., Preconditioning Techniques for Toeplitz Systems, Higher Education Press, Beijing, 2010.
[19] Jin, X., Wei, Y., Zhao, Z., Numerical Linear Algebra and Its Applications, 2nd edition, Science Press, Beijing, 2015.
[20] Jin, X., Vong, S., An Introduction to Applied Matrix Analysis, Higher Education Press, Beijing, and World Scientific Publishing, Singapore, 2016.
[21] Lei, S., Sun, H., A circulant preconditioner for fractional diffusion equations, J. Comput. Phys. 242, 715725 (2013).
[22] Lin, F., Yang, S., Jin, X., Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys. 256, 109117 (2014).
[23] Moret, I., A note on the superlinear convergence of GMRES, SIAM J. Numer. Anal. 34, 513516 (1997).
[24] Ng, M., Iterative Methods for Toeplitz Systems, Oxford University Press, Oxford, UK, 2004.
[25] Olshanskii, M., Tyrtyshnikov, E., Iterative Methods for Linear Systems: Theory and Applications, SIAM, Philadelphia, 2014.
[26] Paige, C., Saunders, M., Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12, 617629 (1975).
[27] Pestana, J., Wathen, A., A preconditioned MINRES method for nonsymmetric Toeplitz matrices, SIAM J. Matrix Anal. Appl. 36, 273288 (2015).
[28] Simoncini, V., Szyld, D., On the occurrence of superlinear convergence of exact and inexact Krylov subspace methods, SIAM Rev. 47, 247272 (2005).
[29] Simoncini, V., Szyld, D., On the superlinear convergence of MINRES, In Numerical Mathematics and Advanced Applications 2011, Springer, 2013.
[30] Sleijpen, G., van der Sluis, A., Further results on the convergence behavior of conjugate-gradients and Ritz values, Linear Algebra Appl. 246, 233278 (1996).
[31] Strang, G., A proposal for Toeplitz matrix calculations, Stud. Appl. Math. 74, 171176 (1986).
[32] Tyrtyshnikov, E., Optimal and superoptimal circulant preconditioners, SIAMJ.Matrix Anal. Appl. 13, 459473 (1992).
[33] van der Sluis, A., van der Vorst, H., The rate of convergence of conjugate gradients, Numer. Math. 48, 543560 (1986).
[34] van der Vorst, H., Vuik, C., The superlinear convergence behaviour of GMRES, J. Comput. Appl. Math. 48, 327341 (1993).
[35] Wathen, A., Preconditioning, Acta Numer. 24, 329376 (2015).
[36] Winther, R., Some superlinear convergence results for the conjugate gradient method, SIAM J. Numer. Anal. 17, 1417 (1980).


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