The role of the inner product in stopping criteria for conjugate gradient iterations

BIT Numerical Mathematics

Volumn 41, Issue 1, 2001, Pages 26-52

  Ashby, S F ^a   Holst, M J ^b   Manteuffel, T A ^c   Saylor, P E ^d  

^a LAWRENCE LIVERMORE NATIONAL LABORATORY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c UNIVERSITY OF COLORADO   (United States)

^d University of Illinois at Urbana Champaign   (United States)

Author keywords

B normal matrices; Conjugate gradient methods; Stopping criteria

Indexed keywords

EID: 0142038248     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1021961616621     Document Type: Article

References (32)

1. M. Arioli, I. S. Duff, and D. Ruiz, Stopping criteria for iterative solvers, Tech. Report RAL-91-057, Rutherford Appleton Laboratory, England.
2. M. Arioli, I. S. Duff, and D. Ruiz, Stopping criteria for iterative solvers, SIAM J. Matrix Anal. Appl., 13:1 (1992), pp. 138-144.
3. S. F. Ashby and M. H. Gutknecht, A matrix analysis of conjugate gradient algorithms, in Proc. Ninth Symposium on Preconditioned Conjugate Gradients (Parallel Processing for Scientific Computing), M. Natori and T. Nodera, eds., Keio University, 1993, pp. 32-47.
4. S. F. Ashby and M. H. Gutknecht, A matrix analysis of conjugate gradient algorithms, Tech. Report UCRL-JC-113560, Lawrence Livermore National Laboratory, CA, 1994.
5. S. F. Ashby, T. A. Manteuffel, and J. S. Otto, A comparison of adaptive Chebyshev and least squares polynomial preconditioning for Hermitian positive definite linear systems, SIAM J. Sci. Statist. Comput., 13:1 (1992), pp. 1-29.
6. S. F. Ashby, T. A. Manteuffel, and P.. E. Saylor, A taxonomy for conjugate gradient methods, SIAM J. Numer. Anal., 27:6 (1990), pp. 1542-1568.
7. G. Auchmuty, A posteriori error-estimates for linear equations, Numer. Math., 61:1 (1992), pp. 1-6.
8. P. Concus, G. H. Golub, and D. P. O'Leary, A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, New York, 1976, pp. 309-332.
9. E. J. Craig, The N-step iteration procedures, J. Math. Physics, 34:1 (1955), pp. 64-73.
10. G. Dahlquist, S. C. Eisenstat, and G. H. Golub, Bounds for the error of linear systems of equations using the theory of moments, J. Math. Anal. Applic., 37 (1972), pp. 151-166.
11. G. Dahlquist, G. H. Golub, and S. Nash, Bounds for the error in linear systems, in Proceedings of the Workshop on Semi-Infinite Programming, R. Hettich, ed., Springer-Verlag, New York, 1978, pp. 154-172.
12. H. C. Elman, Iterative Methods for Large, Sparse Nonsymmetric Systems of Linear Equations, PhD thesis, Tech. Report 229, Dept. of Computer Science, Yale University, CT, 1982.
13. D. J. Evans and C. Li, New stopping criteria for some iterative methods for a class of unsymmetric linear systems, SIAM J. Numer. Anal, 12:3 (1991), pp. 549-551.
14. V. Faber and T. A. Manteuffel, Necessary and sufficient conditions for the existence of a conjugate gradient method, SIAM J. Numer. Anal., 21:2 (1984), pp. 352-362.
15. B. Fischer and G. H. Golub, On the error computation for polynomial based iteration methods, in Recent Advances in Iterative Methods, Springer-Verlag, New York, 1993, pp. 59-69.
16. R. G. Gordon, Error bounds in equilibrium statistical mechanics, J. Math. Physics, 9:5 (1967), pp. 655-663.
17. R. G. Gordon, Error bounds in spectroscopy and nonequilibrium statistical mechanics, J. Math. Physics, 9:7 (1967), pp. 1087-1092.
18. L. A. Hageman and D. M. Young, Applied Iterative Methods, Academic Press, New York, 1981.
19. M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards, 49 (1952), pp. 409-435.
20. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 1991.
21. W. D. Joubert and D. M. Young, Necessary and sufficient conditions for the simplification of generalized conjugate gradient algorithms, Linear Algebra Appl., 88/89 (1987), pp. 449-485.
22. E. F. Kaaschieter, A practical termination criterion for the conjugate gradient method, BIT, 28 (1988), pp. 308-322.
23. C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM Frontiers Series Vol. 16, SIAM, Philadelphia, 1995.
24. D. R. Kincaid, J. R. Respess, D, M. Young, and R. G. Grimes, ITPACK 2C: A FORTRAN package for solving large sparse linear systems by adaptive accelerated iterative methods, ACM Trans. Math. Software, 8:3 (1982), pp. 302-322.
25. T. A. Manteuffel and J. S. Otto, On the roots of the orthogonal polynomials and residual polynomials associated with a conjugate gradient method, Numer. Linear Algebra Appl., 1:5 (1994), pp. 449-475.
26. W. Oettli and W. Prager, Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides, Numer. Math., 58 (1984), 343-361.
27. R. D. Skeel, Scaling for numerical stability in Gaussian elimination, SIAM J. Sci. Statist. Comput., 26 (1979), pp. 494-526.
28. R. D. Skeel, Iterative refinement implies numerical stability for Gaussian elimination, Math. Comp., 35 (1980), pp. 817-832.
29. C. Vuik, Solution of the discretized incompressible Navier-Stokes equations with the GMRES method, J. Assoc. Comp. Mach., 16 (1983), pp. 507-523.
30. K. F. Warnick, Continued fraction error bound for conjugate gradient method, Tech. Report No. TR-L100-98.3, Dept. of Electrical and Computer Engineering, Brigham Young University, Provo, UT.
31. D. M. Young and K. C. Jea, Generalized conjugate gradient acceleration of non-symmetrizable iterative methods, Linear Algebra Appl., 34 (1980), pp. 159-194.
32. D. M. Young and T. Mai, ITPACK 3A user's guide (preliminary version), Tech. Report CNA-197, Center for Numerical Analysis, University of Texas, Austin, TX, 1984.