Abstract
Several variants of the GMRES method for solving linear nonsingular systems of algebraic equations are described. These variants differ in building up different sets of orthonormalized vectors used for the construction of the approximate solution. A new A T A-variant of GMRES is proposed and the efficient implementation of the algorithm is discussed.
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Björck, Å.: Solving linear least squares problems by Gram-Schmidt orthogonalization. BIT 7 (1967) 1–21
Björck, A., Paige, C.C.: Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm SIAM J. Matrix Anal. Appl. 13, 1 (1992) 176–190
Freund, R.W., Golub G.H., Nachtigal, N.M.: Iterative solution of linear systems. Acta Numerica 1 (1992) 1–44
Rozložník, M., Strakoš, Z.: Variants of the residual minimizing Krylov space methods. Research Report 592, ICS AS CR, Prague 1994, 1–26
Saad, Y., Schultz, M.H.: GMRES: A Generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7 (1986) 856–869
Stoer, J.: Solution of large linear systems of equations by conjugate gradient type methods. In Mathematical Programming — The State of the Art (A. Bachern, M. Grotschel and B. Korte eds.), Springer, Berlin 1983, 540–565
Stoer, J., Freund, R.W.: On the solution of large indefinite systems of linear equations by conjugate gradients algorithm. In Computing Methods in Applied Sciences and Engineering V (R.Glowinski, J.L.Lions eds.), North Holland-INRIA, 1982, 35–53
Walker, H.F., Zhou Lu: A Simpler GMRES. Research Report 10/92/54, Dept. of Mathematics and Statistics, Utah State University, Logan, 1992
Young, D.M., Jea, K.C.: Generalized conjugate gradient acceleration of nonsymmetrizable iterative methods. Linear Algebra Appl. 34 (1980) 159–194
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© 1995 Springer-Verlag Berlin Heidelberg
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Rozložník, M., Strakoš, Z. (1995). On the implementation of some residual minimizing Krylov space methods. In: Bartosek, M., Staudek, J., Wiedermann, J. (eds) SOFSEM '95: Theory and Practice of Informatics. SOFSEM 1995. Lecture Notes in Computer Science, vol 1012. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60609-2_32
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DOI: https://doi.org/10.1007/3-540-60609-2_32
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