Numerical Analysis: Stable numerical methods for obtaining the Chebyshev solution to an overdetermined system of equations

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Numerical Analysis: Stable numerical methods for obtaining the Chebyshev solution to an overdetermined system of equations

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Communications of the ACM archive
Volume 11 , Issue 6 (June 1968) table of contents

Pages: 401 - 406

Year of Publication: 1968

ISSN:0001-0782

Authors

Richard H. Bartels	Stanford Univ., Stanford, CA
Gene H. Golub	Stanford Univ., Stanford, CA

Publisher

ACM New York, NY, USA

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Downloads (6 Weeks): 5, Downloads (12 Months): 26, Citation Count: 2

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abstract references cited by index terms collaborative colleagues

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ABSTRACT

An implementation of Stiefel's exchange algorithm for determining a Chebyshev solution to an overdetermined system of linear equations is presented, that uses Gaussian LU decomposition with row interchanges. The implementation is computationally more stable than those usually given in the literature. A generalization of Stiefel's algorithm is developed which permits the occasional exchange of two equations simultaneously.

REFERENCES

Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.

	1	STIEFEL, EDUARD L. Uber diskrete und lineare Tschebyscheff-Approximation. Numer. Math. 1 (1959), 1-28.
	2	STIEFEL, EDUARD L. Note on Jordan elimination, linear programming and Tschebyscheff approximation. Numer. Math. 2 (1960), 1-17.
	3	James H. Wilkinson, Rounding Errors in Algebraic Processes, Dover Publications, Incorporated, 1994
	4	DAVIS, PHILLIP J. Interpolation and Approximation. Blaisdell, Waltham, Mass., 1903.
	5	Richard H. Bartels , Gene H. Golub, Computational considerations regarding the calculation of Chebyshev solutions for overdetermined linear eqauation systems by the exchange method, Stanford University, Stanford, CA, 1967
	6	David Moursund, Chebyshev Solution of n+1 Linear Equations in n + 1nUnknowns, Journal of the ACM (JACM), v.12 n.3, p.383-387, July 1965 [doi>10.1145/321281.321289]
	7	J. H. Wilkinson, Error Analysis of Direct Methods of Matrix Inversion, Journal of the ACM (JACM), v.8 n.3, p.281-330, July 1961 [doi>10.1145/321075.321076]
	8	WILKINSON, J. H. Rounding errors in algebraic processes. Proc. Int. Conf. on Inform. Process., UNESCO, 1959.
	9	CHENEY, E. W. Introduction to approximation theory. (International Series in Pure and Applied Mathematics.) McGraw-Hill, New York, 1966.
	10	MOLER, C. B. Accurate solution of linear algebraic systemsa survey. Proc, AFIPS 1967 Spring Joint Comput. Conf., Vol. 30. Thompson Book Co., Washington, D. C., 321-324.
	11	FORSYTHE, G. E. Today's computational methods of linear algebra. SIAM Rev. 14, (1967), 489-515.
	12	Cleve B. Moler, Iterative Refinement in Floating Point, Journal of the ACM (JACM), v.14 n.2, p.316-321, April 1967 [doi>10.1145/321386.321394]

CITED BY 2

	Richard H. Bartels , Gene H. Golub, The simplex method of linear programming using LU decomposition, Communications of the ACM, v.12 n.5, p.266-268, May 1969
	G. H. Golub , L. B. Smith, Algorithm 414: Chebyshev approximation of continuous functions by a Chebyshev system of functions, Communications of the ACM, v.14 n.11, p.737-746, Nov. 1971

INDEX TERMS

Keywords:
Chebyshev solutions, exchange algorithm, linear equation, overdetermined linear systems

Collaborative Colleagues:

Richard H. Bartels: colleagues

Gene H. Golub: colleagues

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