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Mathematics of Computation

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The Graeffe process as applied to power series

Author: D. H. Lehmer
Journal: Math. Comp. 1 (1945), 377-383
MSC: Primary 65.0X
Corrigendum: Math. Comp. 3 (1948), 227.
MathSciNet review: 0012913
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    For a typical account of this method, together with some historical remarks, see E. T. Whittaker & G. Robinson, The Calculus of Observations, third ed., London, 1940, p. 106-120. References to other accounts are given in C. A. Hutchinson, “On Graeffe’s method for the numerical solution of algebraic equations,” Am. Math. Mo., v. 42, 1935, p. 149-161, and L. L. Cronvich, “On the Graeffe method of solution of equations,” Am. Math. Mo., v. 46, 1939, p. 185-190. An early exponent of this method, who did much to make it well known, was the astronomer J. F. Encke (J.f. d. reine u. angew. Math., v. 22, 1841, p. 193f). He attributed the method to Graeffe. He was also responsible for an utterly useless scheme of changing the signs of all the roots. Many writers on the subject still use this confusing notion of the “Encke roots” of an equation. Graeffe’s method of solutions of numerical equations was really due to Dandelin (1794-1847), Mem. Acad. Brussels, v. 3, 1826, p. 46, or to Waring (1734-1798), Meditationes Analyticae, 1776, p. 311. It was published by C. H. Graeffe (1799-1873) in a prize paper, Die Auflösung der höheren numerischen Gleichungen, Zürich, 1837. Other accounts of the history of the method will be found in A. M. Ostrowski, “Recherches sur la méthode de Graeffe et les zéros des polynomes et des séries de Laurent,” Acta Math., v. 72, 1940, p. 99-155, and E. Carvallo, Méthode pratique pour la Résolution numérique complète des équations algébriques ou transcendantes, Paris diss., 1890. L. Euler, Akad. Nauk, S.S.S.R., Leningrad, Acta, 1781, part 1, 1784, p. 170f. G. Pólya, “Über das Graeffesche Verfahren,” Z. Math. Phys., v. 63, 1915, p. 275-290. S. Brodetsky & G. Smeal, “On Graeffe’s method for complex roots of algebraic equations,” Camb. Phil. So., Proc., v. 22, 1924, p. 83f. A. C. Aitken has proposed a root-cubing procedure in which real roots are determined together with their signs; see Math. Gazette, v. 15, 1931, p. 490-491.

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Article copyright: © Copyright 1945 American Mathematical Society