# On the Error Computation for Polynomial Based Iteration Methods

• Bernd Fischer
• Gene H. Golub
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 60)

## Abstract

In this note we investigate the Chebyshev iteration and the conjugate gradient method applied to the system of linear equations Ax = f where A is a symmetric, positive definite matrix. For both methods we present algorithms which approximate during the iteration process the kth error ε k = ||xx k || A . The algorithms are based on the theory of modified moments and Gaussian quadrature. The proposed schemes are also applicable for other polynomial iteration schemes. Several examples, illustrating the performance of the described methods, are presented.

## Keywords

Orthogonal Polynomial Conjugate Gradient Method Continue Fraction Positive Definite Matrix Initial Error
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
D. Calvetti and L. Reichel, Adaptive Richardson iteration based on Leja points, Tech. Rep. ICM-9210-42, Institute for Computational Mathematics, Kent State University, 1992.Google Scholar
2. [2]
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, 1978, pp. 154–172.Google Scholar
3. [3]
G. G. Dahlquist, S. C. Eisenstat, and G. H. Golub, Bounds for the error of linear systems of equations using the theory of modified moments, J. Math. Anal. Appl., 37 (1972), pp. 151–166.
4. [4]
B. Fischer and G. H. Golub, How to generate unknown orthogonal polynomials out of known orthogonal polynomials, J. Comp. Appl. Math., 43 (1992), pp. 99–115.
5. [5]
W. Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev., 9 (1967), pp. 24 - 82.
6. [6]
W. Gautschi, An algorithmic implementation of the generalized Christoffel theorem, in Numerical Integration, G. Hämmerlin, ed., Basel, 1982, Birkhäuser, pp. 89–106. Internat. Ser. Numer. Math., v. 57.Google Scholar
7. [7]
G. H. Golub and M. D. Kent, Estimates of eigenvalues for iterative methods, Math. Comp., 53 (89), pp. 619–626.Google Scholar
8. [8]
G. H. Golub and C. F. van Loan, Matrix computations, The Johns Hopkins University Press, Baltimore, second ed., 1989.
9. [9]
L. A. Hageman and D. M. Young, Applied Iterative Methods, Academic Press, New York, 1981.
10. [10]
R. A. Sack and A. F. Donovan, An algorithm for Gaussian quadrature given modified moments, Numer. Math., 18 (1971/72), pp. 465–478.Google Scholar
11. [11]
J. C. Wheeler, Modified moments and Gaussian quadrature, Rocky Mt. J. Math., 4 (1974), pp. 287–296.
12. [12]
H. Wilf, Mathematics for the Physical Sciences, Wiley, New York, 1962.