Alkiviadis G. Akritas
ABSTRACT
In this paper an attempt is made to correct the misconception of several authors [1] that there exists a method by Upensky (based on Vincent's theorem) for the isolation of the real roots of a polynomial equation with rational coefficients. Despite Uspensky's claim, in the preface of his book [2], that he invented this method, we show that what Upensky actually did was to take Vincent's method and double its computing time. Upensky must not have understood Vincent's method probably because he was not aware of Budan's theorem [3]. In view of the above, it is historically incorrect to attribute Vincent's method to Upensky.
- 1.Collins, G.E. and R. Loos: Real Zeros of Polynomials. In: Computer Algebra, Symbolic and Algebraic Computations. Edited by B. Buehberger, G.E. Collins and R. Loos, Springer Verlag, Wie~, New York, 83-95, 1982. Google Scholar
Digital Library
- 2.Uspensky, J.V.: Theory of Equations. McGraw- Hill Co., New York 1948.Google Scholar
- 3.Akritas, A.G.: Reflections on a pair of theorems by Budan and Fourier. Mathematics Magazin% Vol. 55, No. 5, 292-298, 1982.Google Scholar
- 4.Vincent, A.J.H.: Sur la r~solution des ~quations num~riques. Journal de Math~matiques Pures et Appliqu~es, Vol. i, 341-372, 1836Google Scholar
- 5.Akritas, A.G.: An implementation of Vincent's theorem. Numerische Mathematik, Vol. 36, 53-62, 1980.Google ScholarDigital Library
- 6.Cajori, F.: A history of the arithmetical methods of approximation to the roots of numerical equations of one unknown quantity. Colorado College Publications, General Series No. 51, Science Series Vo. XII, no. 7, Colorado Springs, CO., 171-215, 1910.Google Scholar
- 7.Akritas, A.G. and S.D. Danielopoulos: On the forgotten theorem of Mr. Vincent. Historia Mathematica, Vol. 5, 427-435, 1978.Google ScholarCross Ref
- 8.Akritas, A.G.: There is no "Uspensky's Method". TR-86-10, University of Kansas, Department of Computer Science, Lawrence Ks 66045, 1986.Google Scholar
- 9.Collins, G.E. and A.G. Akritas: Polynomial real root isolation using Descartes' rule of signs. Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computations. Yorktown Heights, N.Y., 272-275, 1976. Google ScholarDigital Library
- 10.Akritas, A.G.: A remark on the proposed Syllabus for an AMS short course on Computer Algebra. ACM - SIGSAM Bulletin, Vol. 14, No. 2, 24-25, 1980. Google ScholarDigital Library
Index Terms
- There is no “Uspensky's method.”
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Reviews
Harvey Cohn
The continued fraction methods for solving a polynomial equation consist essentially of transforming a polynomial by using x = a + 1/ x?, (for instance, to isolate the roots by reducing the number of sign variations). Thus, in stages, a continued fraction expansion for a root emerges. The author strongly advocates the recognition of the priority of A. J. H. Vincent [1] over J. V. Uspensky [2], and he shows the computational and theoretical superiority of Vincent's work, incidentally offering improvements of his own [3].
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