Abstract
For the parallel conjugate gradient algorithm polynomial preconditioners are more suitable than the more common incomplete Cholesky preconditioner. In this paper we examine the Chebyshev polynomial preconditioner. This preconditioner is based on an interval which approximately contains the eigenvalues of the matrix. If we know the extreme eigenvalues of the matrix then the preconditioner based on this interval minimises the condition number of the preconditioned matrix. Unfortunately this does not minimise the number of conjugate gradient iterations. We propose an adaptive procedure to find the interval which gives optimal rate of convergence. We demonstrate the success of this adaptive procedure on three matrices from the Harwell-Boeing collection.
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Ashby, S. F., Manteuffel, T. A., Otto, J. S.: Adaptive polynomial preconditioning for HPD linear systems. In R. Glowinski and A. Lichnewsky, editors, Proc. Ninth International Conference on Computing Methods in Applied Sciences and Engineering, pages 3–23, 1990.
Ashby, S. F., Manteuffel, T. A., Otto, J. S.: A comparison of adaptive Chebyshev and least squares polynomial preconditioning for Hermitian positive definite linear systems. SIAM Journal of Scientific and Statistical Computing 13 (1992) 1–29
Axelsson, O.: Iterative Solution Methods. Cambridge University Press, 1994.
Brent, R. P.: Algorithms for Minimisation without Derivatives, Chapter 5. Englewood Cliffs, NJ: Prentice-Hall, 1973.
Duff, I. S., Grimes, R. G., Lewis, J. G.: Sparse matrix test problems. ACM Transactions on Mathematical Software 15 (1989) 1–14
Hestenes, M. R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards 49 (1952) 409–435
Johnson, O. G., Micchelli, C. A., Paul, G.: Polynomial preconditioning for conjugate gradient calculations. SIAM Journal of Numerical Analysis 20 (1983) 362–376
Meijerink, J. A., van der Vorst, H. A.: An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Mathematics of Computation 31 (1977) 148–162
Reid, J. K.: On the method of conjugate gradients for the solution of large sparse systems of linear equations. In Proc. Conference on Large Sparse Sets of Linear Equations. Academic Press, New York, 1971.
Saad, Y.: Practical use of polynomial preconditionings for the conjugate gradient method. SIAM Journal of Scientific and Statistical Computing, 6 (1985) 865–881
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© 1996 Springer-Verlag Berlin Heidelberg
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Field, M.R. (1996). Adaptive polynomial preconditioning for the conjugate gradient algorithm. In: Dongarra, J., Madsen, K., Waśniewski, J. (eds) Applied Parallel Computing Computations in Physics, Chemistry and Engineering Science. PARA 1995. Lecture Notes in Computer Science, vol 1041. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60902-4_22
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DOI: https://doi.org/10.1007/3-540-60902-4_22
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