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XXV.—On Bernoulli's Numerical Solution of Algebraic Equations

Published online by Cambridge University Press:  15 September 2014

A. C. Aitken
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Extract

The aim of the present paper is to extend Daniel Bernoulli's method of approximating to the numerically greatest root of an algebraic equation. On the basis of the extension here given it now becomes possible to make Bernoulli's method a means of evaluating not merely the greatest root, but all the roots of an equation, whether real, complex, or repeated, by an arithmetical process well adapted to mechanical computation, and without any preliminary determination of the nature or position of the roots. In particular, the evaluation of complex roots is extremely simple, whatever the number of pairs of such roots. There is also a way of deriving from a sequence of approximations to a root successive sequences of ever-increasing rapidity of convergence.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 46 , 1927 , pp. 289 - 305
Copyright
Copyright © Royal Society of Edinburgh 1927

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References

page 289 note * Commentarii Acad. Sc. Petropol., III (1732)Google Scholar ; cf. Euler, Introductio in Analysin Infinitorum, I, Cap. XVII; Lagrange, Résolution des équations numériques, Note VI.

page 291 note * Cf. Borchardt, , J. für Math., 30 (1845), p. 38Google Scholar.

page 292 note * Cf. Euler and Lagrange, loc. cit., with reference to two equal greatest roots.

page 295 note * Cf. Euler, loc. cit., §§ 351, 352.

page 301 note * Nägelsbach, in the course of a very detailed investigation of Fürstenau's method of solving equations, obtains the formulæ (8.2) and (8.4), but only incidentally. Cf. Archiv d. Math. u. Phys., 59 (1876), pp. 147–192Google Scholar ; 61 (1877), pp. 19–85, and especially pp. 22, 31.

page 303 note * The choice of initial values of f(t), … 0, 0, 0, 1, is equivalent to this.

page 303 note † Darstellung der reellen Würzeln algebraischer Gleichungen durch Determinanten der Coefficienten, Marburg, 1860.

page 305 note * De functionibus alternantibus, … J.für Math., 22 (1841), pp. 370–371Google ScholarPubMed.

page 305 note † Sch.-Programm, Zweibrücken, 1871. Also “Studien zu Fürstenau's neuer Methode,” … Archiv d. Math. u. Phys., 59 (1876), pp. 150–151Google Scholar. Cf. Muir, , History of the Theory of Determinants, vol. iii, pp. 144, 154Google Scholar.