On the implementation of some residual minimizing Krylov space methods
- First Online:
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 ATA-variant of GMRES is proposed and the efficient implementation of the algorithm is discussed.
Unable to display preview. Download preview PDF.
- 1.Björck, Å.: Solving linear least squares problems by Gram-Schmidt orthogonalization. BIT 7 (1967) 1–21Google Scholar
- 2.Björck, A., Paige, C.C.: Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm SIAM J. Matrix Anal. Appl. 13, 1 (1992) 176–190Google Scholar
- 3.Freund, R.W., Golub G.H., Nachtigal, N.M.: Iterative solution of linear systems. Acta Numerica 1 (1992) 1–44Google Scholar
- 4.Rozložník, M., Strakoš, Z.: Variants of the residual minimizing Krylov space methods. Research Report 592, ICS AS CR, Prague 1994, 1–26Google Scholar
- 5.Saad, Y., Schultz, M.H.: GMRES: A Generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7 (1986) 856–869Google Scholar
- 6.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, 540–565Google Scholar
- 7.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, 35–53Google Scholar
- 8.Walker, H.F., Zhou Lu: A Simpler GMRES. Research Report 10/92/54, Dept. of Mathematics and Statistics, Utah State University, Logan, 1992Google Scholar
- 9.Young, D.M., Jea, K.C.: Generalized conjugate gradient acceleration of nonsymmetrizable iterative methods. Linear Algebra Appl. 34 (1980) 159–194Google Scholar