Western Washington University
Western CEDAR
Mathematics College of Science and Engineering
1995
Historical Development of the Newton-Raphson
Method
Tjalling Ypma
Western Washington University,
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 .
, 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 .
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 ( 1995SocietyforIndustrial
andAppliedMathematics
Vol.37, No. 4, pp. 531-551,December1995 003
HISTORICAL DEVELOPMENT OF THE NEWTON-RAPHSON METHOD*
TJALLINGJ. YPMAt
Abstract.Thisexpository papertracesthedevelopment oftheNewton-Raphson methodforsolvingnonlinear
algebraicequationsthroughtheextant andpublications
notes,letters, ofIsaac Newton,JosephRaphson,andThomas
Simpson.It is shownhowNewton'sformulation fromtheiterative
differed processof Raphson,and thatSimpson
was thefirst in termsof fluxional
to givea generalformulation, calculus,applicableto nonpolynomial
equations.
Simpson'sextension ofthemethodtosystemsofequationsis exhibited.
Key words.nonlinear
equations,iteration,
Newton-Raphson Isaac Newton,JosephRaphson,Thomas
method,
Simpson
AMS subjectclassifications. 65H10
01A45,65H105,
1. Introduction.The iterative
algorithm
(1.1) x -f
Xi= (xi)/f-(xi)
forsolvinga nonlinear algebraicequationf(x) = 0 is generallycalledNewton'smethod.
Occasionallyit is referredto as theNewton-Raphson method.The method(1.1), and its
extensionto thesolutionof systemsof nonlinear equations,formsthebasis forthemost
frequentlyusedtechniques forsolvingnonlinear algebraicequations.In thisexpositorypaper
we tracethedevelopment of themethod(1.1) by exhibiting and analyzingrelevantextracts
fromthepreserved notes,letters,and publications of Isaac Newton,JosephRaphson,and
ThomasSimpson.Muchof thesequenceof eventsrecounted hereis familiar
to historians,
and thematerialson whichthispaperis based are fairlyreadilyavailable;hencewe make
no claimsto originality.Ourpurposeis simplyto providea comprehensive accountof the
historical
rootsoftheubiquitous process(1.1), assembling a number ofpreviouslypublished
accountsintoa readilyaccessiblewhole.
In ??2 and3 we showthatmethods whichmaybe viewedas replacing thetermf'(xi) in
(1.1) bya finite
differenceapproximation oftheform
(1.2) f'(xi) ; ht1[f(xi + hi)-f
(xi-],
andalso thesecantmethodin which
(1.3) f'(xi) f(Xi)-f
Xi-Xi-l
wereprecursors tothemethod(1.1). Froma modemperspective eachofthemethods described
by(1.1)-(1.3) arisesnaturally
froma linearizationoftheequationf (x) = 0. In ?4 we review
Newton'soriginalpresentation of his method,contrastingthiswiththecurrent formulation
(1.1) andwithRaphson'siterativeformulationforpolynomial equationsdiscussedin?7. There
is no clearevidencethatNewtonusedanyfluxional calculusin deriving
hismethod, though
we showin ?6 thatinthePrincipiaMathematica Newtonappliedhistechnique inan iterative
manner toa nonpolynomial equation.Simpson'sgeneralformulation fornonlinear equations
intermsofthefluxional in ?8,andwe discussthereSimpson'sextension
calculusis presented
oftheprocessto systems ofnonlinear equations.
*ReceivedbytheeditorsOctober6, 1993;acceptedforpublication
(in revisedform)January
12, 1995.
tDepartmentof Mathematics,WesternWashington 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
Western CEDAR
Mathematics College of Science and Engineering
1995
Historical Development of the Newton-Raphson
Method
Tjalling Ypma
Western Washington University,
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 .
, 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 .
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 ( 1995SocietyforIndustrial
andAppliedMathematics
Vol.37, No. 4, pp. 531-551,December1995 003
HISTORICAL DEVELOPMENT OF THE NEWTON-RAPHSON METHOD*
TJALLINGJ. YPMAt
Abstract.Thisexpository papertracesthedevelopment oftheNewton-Raphson methodforsolvingnonlinear
algebraicequationsthroughtheextant andpublications
notes,letters, ofIsaac Newton,JosephRaphson,andThomas
Simpson.It is shownhowNewton'sformulation fromtheiterative
differed processof Raphson,and thatSimpson
was thefirst in termsof fluxional
to givea generalformulation, calculus,applicableto nonpolynomial
equations.
Simpson'sextension ofthemethodtosystemsofequationsis exhibited.
Key words.nonlinear
equations,iteration,
Newton-Raphson Isaac Newton,JosephRaphson,Thomas
method,
Simpson
AMS subjectclassifications. 65H10
01A45,65H105,
1. Introduction.The iterative
algorithm
(1.1) x -f
Xi= (xi)/f-(xi)
forsolvinga nonlinear algebraicequationf(x) = 0 is generallycalledNewton'smethod.
Occasionallyit is referredto as theNewton-Raphson method.The method(1.1), and its
extensionto thesolutionof systemsof nonlinear equations,formsthebasis forthemost
frequentlyusedtechniques forsolvingnonlinear algebraicequations.In thisexpositorypaper
we tracethedevelopment of themethod(1.1) by exhibiting and analyzingrelevantextracts
fromthepreserved notes,letters,and publications of Isaac Newton,JosephRaphson,and
ThomasSimpson.Muchof thesequenceof eventsrecounted hereis familiar
to historians,
and thematerialson whichthispaperis based are fairlyreadilyavailable;hencewe make
no claimsto originality.Ourpurposeis simplyto providea comprehensive accountof the
historical
rootsoftheubiquitous process(1.1), assembling a number ofpreviouslypublished
accountsintoa readilyaccessiblewhole.
In ??2 and3 we showthatmethods whichmaybe viewedas replacing thetermf'(xi) in
(1.1) bya finite
differenceapproximation oftheform
(1.2) f'(xi) ; ht1[f(xi + hi)-f
(xi-],
andalso thesecantmethodin which
(1.3) f'(xi) f(Xi)-f
Xi-Xi-l
wereprecursors tothemethod(1.1). Froma modemperspective eachofthemethods described
by(1.1)-(1.3) arisesnaturally
froma linearizationoftheequationf (x) = 0. In ?4 we review
Newton'soriginalpresentation of his method,contrastingthiswiththecurrent formulation
(1.1) andwithRaphson'siterativeformulationforpolynomial equationsdiscussedin?7. There
is no clearevidencethatNewtonusedanyfluxional calculusin deriving
hismethod, though
we showin ?6 thatinthePrincipiaMathematica Newtonappliedhistechnique inan iterative
manner toa nonpolynomial equation.Simpson'sgeneralformulation fornonlinear equations
intermsofthefluxional in ?8,andwe discussthereSimpson'sextension
calculusis presented
oftheprocessto systems ofnonlinear equations.
*ReceivedbytheeditorsOctober6, 1993;acceptedforpublication
(in revisedform)January
12, 1995.
tDepartmentof Mathematics,WesternWashington 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
