zbMATH — the first resource for mathematics

An efficient reliable algorithm for the approximation of all polynomial roots based on the method of D. Bernoulli. (English) Zbl 1369.30004
Proc. Steklov Inst. Math. 280, Suppl. 2, S43-S55 (2013) and Sovrem. Probl. Mat. 16, 52-65 (2012).
From the text: The design of the complete algorithm is based on eight theorems. Since most of them share the same conditions and abbreviations, we first state these specifications. Then we summarize the contents of this paper and put them in order with the efficient reliable polynomial root finding methods already known.
30C10 Polynomials and rational functions of one complex variable
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
Maple; SPAViews
Full Text: DOI
[1] H. Möller, Algorithmische Lineare Algebra (Vieweg, Braunschweig, 1997); http://www.math.unimuenster.de/u/mollerh .
[2] M. A. Jenkins and J. F. Traub, ”Principles for testing polynomial zerofinding programs,” ACM Trans. Math. Software 1(1), 26–34 (1975). · Zbl 0311.65039 · doi:10.1145/355626.355632
[3] D. V. Chudnovsky and G. V. Chudnovsky, ”Computer algebra in the service of mathematical physics and number theory,” in Computers in Mathematics (Marcel Dekker, New York, 1990), pp. 109–232. · Zbl 0712.11078
[4] M. A. Jenkins and J. F. Traub, ”A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration,” Numer. Math. 14(3), 252–263 (1970). · Zbl 0176.13701 · doi:10.1007/BF02163334
[5] T. E. Hull and R. Mathon, ”The mathematical basis and a prototype implementation of a new polynomial rootfinder with quadratic convergence,” ACM Trans. Math. Software 22(3), 261–280 (1996). · Zbl 0884.65042 · doi:10.1145/232826.232830
[6] D. Loewenthal, ”Improvements on the Lehmer-Schur root detection method,” J. Comput. Phys. 109(2), 164–168 (1993). · Zbl 0803.65067 · doi:10.1006/jcph.1993.1209
[7] A. Schönhage, ”Equation solving in terms of computational complexity,” in Proceedings of the International Congress of Mathematicians (Berkeley, Calif., USA, 1986), (Amer. Math. Soc., Providence, R.I., 1987), Vol. 1, pp. 131–153.
[8] E. Laguerre,OEuvres (Villars, Paris, 1898), Vol. 1 [in French].
[9] H. Möller, SPAview.mws, Maple worksheet; http://www.math.uni-muenster.de/u/mollerh .
[10] P. Turán, On a New Method of Analysis and Its Applications (Wiley, New York, 1978).
[11] L. Euler, Introduction to Analysis of the Infinite (Bousquet, Lausanne, 1748; Springer-Verlag, New York, 1988), Book 1. · JFM 17.0200.01
[12] H. Möller, Visualization of the First Stage of the SPA, Report of [9], http://www.math.unimuenster.de/u/mollerh .
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.