Western Washington University Western CEDAR Mathematics College of Science and Engineering 1995 Historical Development of the Newton-Raphson Method Tjalling Ypma Western Washington University, [email protected] Follow this and additional works at: https://cedar.wwu.edu/math_facpubs Part of the Mathematics Commons Recommended Citation Ypma, Tjalling, "Historical Development of the Newton-Raphson Method" (1995). Mathematics. 93. https://cedar.wwu.edu/math_facpubs/93 This Article is brought to you for free and open access by the College of Science and Engineering at Western CEDAR. It has been accepted for inclusion in Mathematics by an authorized administrator of Western CEDAR. For more information, please contact [email protected]. Society for Industrial and Applied Mathematics Historical Development of the Newton-Raphson Method Author(s): Tjalling J. Ypma Source: SIAM Review, Vol. 37, No. 4 (Dec., 1995), pp. 531-551 Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/2132904 Accessed: 21-10-2015 17:15 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extend access to SIAM Review. http://www.jstor.org This content downloaded from 140.160.178.72 on Wed, 21 Oct 2015 17:15:03 UTC All use subject to JSTOR Terms and Conditions SIAM REVIEW ( 1995Society for Industrial and AppliedMathematics Vol.37, No. 4, pp. 531-551,December 1995 003 HISTORICAL DEVELOPMENT OF THE NEWTON-RAPHSON METHOD* TJALLINGJ. YPMAt Abstract.This expository paper traces the development of theNewton-Raphson method for solving nonlinear algebraicequations through the extant notes, letters, and publications of Isaac Newton,Joseph Raphson, and Thomas Simpson.It is shownhow Newton's formulation differed from the iterative process of Raphson,and thatSimpson was thefirst to givea generalformulation, in termsof fluxionalcalculus, applicable to nonpolynomialequations. Simpson'sextension of the method to systemsof equationsis exhibited. Key words. nonlinearequations, iteration, Newton-Raphson method, Isaac Newton,Joseph Raphson, Thomas Simpson AMS subjectclassifications. 01A45, 65H105, 65H10 1. Introduction.The iterativealgorithm (1. 1) x Xi= (xi)/f-(xi)-f forsolving a nonlinearalgebraic equation f(x) = 0 is generallycalled Newton'smethod. Occasionallyit is referredto as theNewton-Raphson method. The method(1.1), and its extensionto thesolution of systemsof nonlinearequations, forms the basis forthe most frequentlyused techniques for solving nonlinear algebraic equations. In thisexpository paper we tracethe development of themethod (1.1) by exhibitingand analyzingrelevant extracts fromthe preserved notes, letters, and publicationsof Isaac Newton,Joseph Raphson, and ThomasSimpson. Much of thesequence of eventsrecounted here is familiarto historians, and thematerials on whichthis paper is based are fairlyreadily available; hence we make no claimsto originality.Our purposeis simplyto providea comprehensiveaccount of the historicalroots of the ubiquitous process (1.1), assemblinga numberof previouslypublished accountsinto a readilyaccessible whole. In ??2 and3 we showthat methods which may be viewedas replacingthe term f'(xi) in (1.1) bya finitedifference approximation of the form (1.2) f'(xi) ; ht1[f (xi + hi)-f(xi-], andalso thesecant method in which (1.3) f'(xi) f(Xi)-f Xi-Xi-l wereprecursors to the method (1.1). Froma modemperspective each of the methods described by(1. 1)-(1.3) arisesnaturally from a linearizationof the equation f (x) = 0. In ?4 we review Newton'soriginal presentation of his method,contrasting this with the current formulation (1.1) andwith Raphson's iterative formulation for polynomial equations discussed in ?7. There is no clearevidence that Newton used anyfluxional calculus in derivinghis method,though we showin ?6 thatin thePrincipia Mathematica Newton applied his technique in an iterative mannerto a nonpolynomialequation. Simpson's general formulation for nonlinear equations interms of the fluxional calculus is presentedin ?8,and we discussthere Simpson's extension ofthe process to systemsof nonlinear equations. *Receivedby the editors October 6, 1993; acceptedfor publication (in revisedform) January 12, 1995. tDepartmentof Mathematics,Western Washington University, Bellingham, Washington 98225 (t j ypma@ nessie.cc.wwu.edu). 531 This content downloaded from 140.160.178.72 on Wed, 21 Oct 2015 17:15:03 UTC All use subject to JSTOR Terms and Conditions 532 TJALLINGJ. YPMA In theabsence of directaccess to sourcematerials we havebased thispaper on a recent translationof the relevant works of Viete [19], the invaluable volumes of Newton's mathemat- ical paperscollected and annotatedby Whiteside [21], Newton's correspondence reproduced byRigaud [14] andTumbull [18], the third edition of Newton's Principia Mathematica in the editionby Cajori [4] withcommentary by Koyre and Cohen [11], and copies of Raphson's text [12] and Simpson'sbook [16]. Forreferences to the work and influences of other mathemati- cianswe haverelied on thenotes in [13], [21] andrelated commentary inthe concise summary ofthis material by Goldstine [6, pp. 64-68]. In ourcomments on thework of Newton we are heavilyindebted to the notes in [21],while the recent papers [17] and [7] providemuch insight intothe contributions ofRaphson and Simpson,respectively. The references,particularly [6] and [21],provide detailed bibliographic references to relevantsource materials. The biographyby Westfall [20] includesincisive discussions of Newton's general math- ematicaldevelopment in additionto hismany significant achievements-scientific and other- wise. The fewverifiable facts concerning the life of JosephRaphson are presentedin [17]. The lifeof ThomasSimpson is describedin [5], whilea severelycritical commentary on his characterand achievements appears in [11]. 2. The methodof Viete. By late 1664,soon afterhis interesthad beendrawn to math- ematics,Isaac Newtonwas acquaintedwith the work of the French algebraist Fran9ois Viete (1540-1603) (oftenlatinized to "Vieta"). Viete'swork concerning the numerical solution of nonlinearalgebraic equations, De numerosapotestatum, published in Parisin 1600,sub- sequentlyreappeared in a collectionof Viete'sworks assembled and publishedas Francisci VietaeOpera Mathematicaby Fransvan Schootenin Leidenin 1646. A summaryof that material,incorporating notational simplifications and someadditional material, appeared in variouseditions of WilliamOughtred's Clavis Mathematicaefrom 1647 onward.A recent translationof therelevant work appears in [19], whichincludes a biographyof Viete. New- tonhad access to bothSchooten's collection and thethird Latin edition of Oughtred'sbook, publishedin Oxfordin 1652,and madeextensive notes from them. These notes constitute a firstsign of Newton's interest in thenumerical solution of nonlinearequations. Vieterestricted his attention to monicpolynomial equations. In modernfunctional nota- tionwe maywrite Viete's equations in theform (2.1) p(x) = N in whichthe constant term N appearson theright of theequation. Viete's technique can be regardedas yieldingindividual digits of the solution x*. of (2.1) oneby one as follows.Let the successivesignificant decimal digits of x* be ao, a,, a2,. ., so thatx* = a010k + a I0k- 1 + a2 10k-2+.. , and letthe ith estimate xi of x* be givenby xi = ao 10k+a, a 10k- + . +ai 10k-i. Assuming xi is given,we have xi+, = xi + ai+l 10k-i+1), and Vi&te's techniqueamounts to computingai+I as theinteger part of N - (2.2) .5 (p(xi + 10k-i-1) + p(X,)) p(X, + lOk-i-1) - p(X,) - 10(k-i-I)n' wheren is thedegree ofthe polynomial. Occasionally the quantity .5 (p(X, + 1ok-i-l)+p(X,)) inthe numerator of (2.2) is replacedby the value of p(Xi) . As notedby Maas [10],the integer aji+ mayin factbe negativeor haveseveral digits, thus permitting the correction of earlier estimatesxi ofx*. Rewriting(2.1) as f(x) = 0 wheref(x) = p(x) - N, theexpression (2.2) can be reformulatedas -.5(f(X, + 1Ok-i-1) + f(x,)) f(X; + lOk-i-1) - f(xi) - 10(k-i-I)n' This content downloaded from 140.160.178.72 on Wed, 21 Oct 2015 17:15:03 UTC All use subject to JSTOR Terms and Conditions HISTORY OF THE NEWTON-RAPHSONMETHOD 533 henceViete's method is almostequivalent to (2.3) xi+1 = xi1-1ioki--)f Lf(xi + f(x()] This closelyparallels the expression produced by substitutionof (1.2) into(1.1) withhi = 10k-i-I. In thissense themethod of Vieteis a forerunnerof thefinite difference scheme (1. 1)-(1.2), whichis oftenpresented as a modificationof (1.1). The subtra6tonof theterm 10(k-i-1)nin the denominator of (2.2) can be motivatedin the case thatp(x) = Xnby consid- eringthe binomial expansion of (xi + l0k-i-l),; formonic quadratic equations the resulting methodcorresponds exactly to theNewton-Raphson method (1.1), butin practicethis term is negligibleand was oftenomitted [21, I]. An interestingcomparison between this technique andthe Newton-Raphson method for the case p(x) = Xnis givenin [10]. Thistechnique was widelyused until supplanted by the Newton-Raphson