On the Error Computation for Polynomial Based Iteration Methods
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 = ||x − x 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.
KeywordsOrthogonal Polynomial Conjugate Gradient Method Continue Fraction Positive Definite Matrix Initial Error
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- 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
- 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
- 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
- G. H. Golub and M. D. Kent, Estimates of eigenvalues for iterative methods, Math. Comp., 53 (89), pp. 619–626.Google Scholar
- 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