Published online by Cambridge University Press: 15 September 2014
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.
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 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.