Abstract
The block changing minimal residual method based on the Hessenberg reduction algorithm (in short BCMRH) is a recent block Krylov method that can solve large linear systems with multiple right-hand sides. This method uses the block Hessenberg process with pivoting strategy to construct a trapezoidal Krylov basis and minimizes a quasi-residual norm by solving a least squares problem. In this paper, we describe the simpler BCMRH method which is a new variant that avoids the QR factorization to solve the least-squares problem. Another major difference between the classical and simpler variants of BCMRH is that the simpler one allows to check the convergence within each cycle of the block Hessenberg process by using a recursive relation that updates the residual at each iteration. This is not possible with the classical BCMRH where we can only compute an estimate of the residual norm. Experiments are described to compare the behavior of the new proposed method with that of the classical and simpler versions of the block GMRES method. These numerical experiments show the good performances of the simpler BCMRH method.
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References
Addam, M., Elbouyahyaoui, L., Heyouni, M.: On Hessenberg type methods for low-rank Lyapunov matrix equations. Applicationes Mathematicae 45, 255–273 (2018)
Addam, M., Heyouni, M., Sadok, H.: The block Hessenberg process for matrix equations. Electron. Trans. Numer. Anal. 46, 460–473 (2017)
Amini, S., Toutounian, F., Gachpazan, M.: The block CMRH method for solving nonsymmetric linear systems with multiple right-hand sides. J. Comput. Appl. Math. 337, 166–174 (2018)
Ballani, J., Grasedyck, L.: A projection method to solve linear systems in tensor format. Numerical Linear Algebra with Applications 20(1), 27–43 (2013)
Beik, F.P.A., Saberi-Movahed, F., Ahmadi-Asl, S.: On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations. Numerical Linear Algebra with Applications 23(3), 444–466 (2016)
Datta, B.N., Saad, Y.: Arnoldi methods for large Sylvester-like observer matrix equations, and an associated algorithm for partial spectrum assignment. Linear Algebra Appl. 154-156, 225–244 (1991)
Davis, T.A., Hu, Y.: The University of Florida sparse matrix collection. ACM Trans. Math. Softw. 38(1), 1,1–1,25 (2011)
Duminil, S., Heyouni, M., Marion, P., Sadok, H.: Algorithms for the CMRH method for dense linear systems. Numerical Algorithms 71(2), 383–394 (2016)
El Guennouni, A., Jbilou, K., Sadok, H.: A block version of BiCGSTAB for linear systems with multiple right-hand sides. Electron. Trans. Numer. Anal. 16, 129–142 (2003)
El Guennouni, A., Jbilou, K., Sadok, H.: The block Lanczos method for linear systems with multiple right-hand sides. Appl. Numer. Math. 51(2), 243–256 (2004)
Freund, R.W., Malhotra, M.: A block QMR algorithm for non-hermitian linear systems with multiple right-hand sides. Linear Algebra and its Applications 254(1), 119–157 (1997). Proceeding of the Fifth Conference of the International Linear Algebra Society
Freund, R.W., Nachtigal, N.M.: QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60(1), 315–339 (1991)
Golub, G.H., Van Loan, C.F.: Matrix computations, 4th edn. The Johns Hopkins University Press, Baltimore (2013)
Gu, X.-M., Huang, T.-Z., Carpentieri, B., Imakura, A., Zhang, K., Du, L.: Effecient variants of the CMRH method for solving multi-shifted non-Hermitian linear systems. arXiv (2018)
Gu, X.-M., Huang, T.-Z., Yin, G., Carpentieri, B., Wen, C., Du, L.: Restarted Hessenberg method for solving shifted nonsymmetric linear systems. J. Comput. Appl. Math. 331, 166–177 (2018)
Gutknecht, M.H.: Block Krylov Space Methods for Linear Systems with Multiple Right-Hand Sides: An Introduction (2006)
Gutknecht, M.H., Schmelzer, T.: The block grade of a block Krylov space. Linear Algebra Appl. 430(1), 174–185 (2009)
Heyouni, M., Sadok, H.: A new implementation of the CMRH method for solving dense linear systems. J. Comput. Appl. Math. 213(2), 387–399 (2008)
Jiránek, P., Rozloznik, M., Gutknecht, M.: How to make simpler GMRES and GCR more stable. SIAM Journal on Matrix Analysis and Applications 30(4), 1483–1499 (2009)
Jiránek, P., Rozožní, M.: Adaptive version of simpler GMRES. Numerical Algorithms 53(1), 93 (2009)
Kress, R.: Linear Integral Equations, 3rd edn. Springer, New York (1989)
Liu, H., Zhong, B.: Simpler Block GMRES for nonsymmetric systems with multiple right-hand sides. Electron. Trans. Numer. Anal. 30, 1–9 (2008)
Morgan, R.B.: Restarted block-GMRES with deflation of eigenvalues. Appl. Numer. Math. 54(2), 222–236 (2005)
O’Leary, D.P.: The block conjugate gradient algorithm and related methods. Linear Algebra Appl. 29, 293–322 (1980)
Saad, Y.: Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, second edition (2003)
Sadok, H.: CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm. Numerical Algorithms 20(4), 303–321 (1999)
Sadok, H., Szyld, D. B.: A new look at CMRH and its relation to GMRES. BIT Numer. Math. 52(2), 485–501 (2012)
Simoncini, V., Gallopoulos, E.: An iterative method for nonsymmetric systems with multiple right-hand sides. SIAM J. Sci. Comput. 16(4), 917–933 (1995)
Simoncini, V., Gallopoulos, E.: Convergence properties of block GMRES and matrix polynomials. Linear Algebra Appl. 247, 97–119 (1996)
Soodhalter, K.: Block Krylov subspace recycling for shifted systems with unrelated right-hand sides. SIAM J. Sci. Comput. 38(1), A302–A324 (2016)
Sun, D.-L., Carpentieri, B., Huang, T.-Z., Jing, Y.-F.: A spectrally preconditioned and initially deflated variant of the restarted block GMRES method for solving multiple right-hand sides linear systems. Int. J. Mech. Sci. 144, 775–787 (2018)
Sun, D.-L., Huang, T.-Z., Carpentieri, B., Jing, Y.-F.: A new shifted block GMRES method with inexact breakdowns for solving multi-shifted and multiple right-hand sides linear systems. J. Sci. Comput. 78(2), 746–769 (2019)
Sun, D.-L., Huang, T.-Z., Jing, Y.-F., Carpentieri, B.: A block GMRES method with deflated restarting for solving linear systems with multiple shifts and multiple right-hand sides. Numerical Linear Algebra with Applications 25(5), e2148 (2018 ). e2148 nla.2148
Vital, B.: Etude de quelques méthodes de résolution de problèmes linéaires de grande taille sur multiprocesseur. PhD thesis, Université, Rennes, 1 (1990)
Walker, H.F., Zhou, L.: A simpler GMRES. Numerical Linear Algebra with Applications 1(6), 571–581 (1994)
Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford University Press, Inc., New York (1988)
Xiu Zhong, H., Wu, G., Liang Chen, G.: A flexible and adaptive simpler block GMRES with deflated restarting for linear systems with multiple right-hand sides. J. Comput. Appl. Math. 282, 139–156 (2015)
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The authors are grateful to the anonymous referees for their valuable comments and suggestions which helped to improve the quality of this paper.
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Appendix
Appendix
In this section, we give Algorithms 6 and 7 a brief description of the restarted standard block GMRES and restarted simpler block GMRES methods.
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Abdaoui, I., Elbouyahyaoui, L. & Heyouni, M. The simpler block CMRH method for linear systems. Numer Algor 84, 1265–1293 (2020). https://doi.org/10.1007/s11075-019-00814-7
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DOI: https://doi.org/10.1007/s11075-019-00814-7