Abstract
GMRES(n,k), a version of GMRES for the solution of large sparse linear systems, is introduced. A cycle of GMRES(n,k) consists of n Richardson iterations followed by k iterations of GMRES. Such cycles can be repeated until convergence is achieved. The advantage in this approach is in the opportunity to use moderate k, which results in time and memory saving. Because the number of inner products among the vectors of iteration is about k2/2, using a moderate k is particularly attractive on message-passing parallel architectures, where inner products require expensive global communication. The present analysis provides tight upper bounds for the convergence rates of GMRES(n,k) for problems with diagonalizable coefficient matrices whose spectra lie in an ellipse in 0. The advantage of GMRES(n,k) over GMRES(k) is illustrated numerically.
Similar content being viewed by others
References
M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series 55 (Government Printing Office, Washington, DC, 1964).
W.E. Arnoldi, The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quart. Appl. Math. 9 (1951) 17-29.
O. Axelsson, Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations, Linear Algebra Appl. 29 (1980) 1-16.
J. Baglama, D. Calvetti, G.H. Golub and L. Reichel, Technical Report SCCN-96-15, Stanford University, Stanford, CA (1996).
A. Brandt and I. Yavneh, Accelerated multigrid convergence and high Reynolds recirculating flows, SIAM J. Sci. Statist. Comput. 14 (1993) 607-626.
S. Cabay and L.W. Jackson, A polynomial extrapolation method for finding limits and antilimits of vector sequences, SIAM J. Numer. Anal. 13 (1976) 734-752.
E.W. Cheney, Introduction to Approximation Theory, 2nd ed. (Chelsea, New York, 1982).
P. Concus and G.H. Golub, A generalized conjugate gradient method for nonsymmetric systems of linear equations, in: Proc. of the 2nd Internat. Symp. on Computing Methods in Applied Sciences and Engineering, eds. R. Glowinski and J.L. Lions, IRIA, Paris (December 1975), Lecture Notes in Economics and Mathematical Systems 134 (Springer, Berlin, 1976) pp. 56-65.
E. De Sturler, Nested Krylov methods based on GCR, J. Comput. Appl. Math. 67 (1996) 15-41.
R.P. Eddy, Extrapolating to the limit of a vector sequence, in: Information Linkage between Applied Mathematics and Industry, ed. P.C.C. Wang (Academic Press, New York, 1979) pp. 387-396.
S.C. Eisenstat, H.C. Elman and M.H. Schultz, Variational iterative methods for nonsymmetric systems of linear equations, SIAM J. Numer. Anal. 20 (1983) 345-357.
R. Freund and S. Ruscheweyh, On a class of Chebyshev approximation problems which arise in connection with a conjugate gradient type method, Numer. Math. 48 (1986) 525-542.
W. Gander, G.H. Golub and D. Gruntz, Solving linear equations by extrapolation, Manuscript NA-89-11, Stanford University, Stanford, CA (October 1989).
M. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. N.B.S. 49 (1952) 409-436.
S. Kaniel and J. Stein, Least-square acceleration of iterative methods for linear equations, J. Optim. Theory Appl. 14 (1974) 431-437.
S.A. Kharchenko and A.Y. Yeremin, Eigenvalue translation based preconditioners for the GMRES(k) method, Numer. Linear Algebra Appl. 2 (1995) 51-77.
G.G. Lorentz, Approximation by incomplete polynomials (problems and results), in: Padé and Rational Approximations: Theory and Applications, eds. E.B. Saff and R.S. Varga (Academic Press, New York, 1977) pp. 289-302.
M. Mešina, Convergence acceleration for the iterative solution of the equations X = AX + f, Comput. Methods Appl. Mech. Engrg. 10 (1977) 165-173.
R.B. Morgan, A restarted GMRES method augmented with eigenvectors, SIAM J. Matrix Anal. Appl. 16 (1995) 1154-1171.
Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comp. 37 (1981) 105-126.
Y. Saad and M.H. Schultz, A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986) 856-869.
E.B. Saff and R.S. Varga, On incomplete polynomials, in: Numerische Methoden der Approximationstheorie, Band 4, International Series of Numerical Mathematics, Vol. 42, eds. L. Collatz, G. Meinardus and H. Werner (Birkhäuser, Basel, 1978) pp. 281-298.
E.B. Saff and R.S. Varga, Uniform approximation by incomplete polynomials, Internat. J. Math. Math. Sci. 1 (1978) 407-420.
E.B. Saff and R.S. Varga, The sharpness of Lorentz’s theorem on incomplete polynomials, Trans. Amer. Math. Soc. 249 (1979) 163-186.
E.B. Saff and R.S. Varga, Incomplete polynomials II, Pacific J. Math. 92 (1981) 161-172.
A. Sidi, Convergence and stability properties of minimal polynomial and reduced rank extrapolation algorithms, SIAM J. Numer. Anal. 23 (1986) 197-209.
A. Sidi, Extrapolation vs. projection methods for linear systems of equations, J. Comput. Appl. Math. 22 (1988) 71-88.
A. Sidi, Efficient implementation of minimal polynomial and reduced rank extrapolation methods, J. Comput. Appl. Math. 36 (1991) 305-337.
A. Sidi, Convergence of intermediate rows of minimal polynomial and reduced rank extrapolation tables, Numer. Algorithms 6 (1994) 229-244.
A. Sidi and J. Bridger, Convergence and stability analyses for some vector extrapolation methods in the presence of defective iteration matrices, J. Comput. Appl. Math. 22 (1988) 35-61.
A. Sidi and Y. Shapira, Upper bounds for convergence rates of vector extrapolation methods on linear systems with initial iterations, TR # 701, Computer Science Department, Technion (December 1991). Also NASA TM 105608, ICOMP-92-09 (July 1992). For an extended abstract, see Proc. of the Cornelius Lanczos Internat. Centenary Conf., eds. J.D. Brown, M.T. Chu, D.C. Ellison and R.J. Plemmons (1994) pp. 285-287.
Y. Shapira, Convergence criteria for diagonalizable non-normal equations, Technical Report LAUR-98-1993, Los Alamos National Laboratory (1998).
E.L. Stiefel, Relaxationsmethoden bester Strategie zur Lösung linearer Gleichungssystems, Comment. Math. Helv. 29 (1955) 157-179.
G. Szegö, Orthogonal Polynomials, Vol. 23 (Amer. Math. Soc., Providence, RI, 1939).
R.S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1962).
P.K.W. Vinsome, Orthomin, an iterative method for solving sparse sets of simultaneous linear equations, in: Proc. 4th Symp. on Reservoir Simulation (Society of Petroleum Engineers of AIME, 1976) pp. 149-159.
O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal. 15 (1978) 801-812.
D.M. Young and K.C. Jea, Generalized conjugate gradient acceleration of nonsymmetrizable iterative methods, Linear Algebra Appl. 34 (1980) 159-194.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sidi, A., Shapira, Y. Upper bounds for convergence rates of acceleration methods with initial iterations. Numerical Algorithms 18, 113–132 (1998). https://doi.org/10.1023/A:1019113314010
Issue Date:
DOI: https://doi.org/10.1023/A:1019113314010