Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-2qt69 Total loading time: 0.222 Render date: 2022-08-09T04:31:37.033Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

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

Published online by Cambridge University Press:  15 September 2014

Get access

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
Copyright
Copyright © Royal Society of Edinburgh 1927

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 147192Google 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. 370371Google ScholarPubMed.

page 305 note † Sch.-Programm, Zweibrücken, 1871. Also “Studien zu Fürstenau's neuer Methode,” … Archiv d. Math. u. Phys., 59 (1876), pp. 150151Google Scholar. Cf. Muir, , History of the Theory of Determinants, vol. iii, pp. 144, 154Google Scholar.

310
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

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

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

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

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *