[PDF] [PDF] 3 Approximating a function by a Taylor series

[PDF] Newtons Method - Philadelphia University

Context Newton's (or the Newton-Raphson) method is one of the most powerful The Taylor series derivation of Newton's method points out the importance of 

[PDF] Analytic derivation of the Newton-Raphson method

Let p be a root of the function f ∈ C2[a, b] (i e f(p)=0), and p0 be an approximation to p If p0 is su ciently close to p, the expansion of f(p) as a Taylor series in 

[PDF] 3 Approximating a function by a Taylor series

dx2 ) and f(k)(x) is the kth derivative of f evaluated at x As we have function f : R R by a simpler function is to use the Taylor series representation for This method is also known as the Newton-Raphson method and is based on the approx-

[PDF] Derivation of the Newton-Raphson Method A - Jon Ernstberger

Performance of Numerical Optimization Routines Derivation of the Newton- Raphson Method • The Taylor polynomial for f(x) is • As the function approaches a 

[PDF] 62 THE NEWTON-RAPHSON METHOD

Newton-Raphson method may also be developed from the Taylor series expansion This alternative derivation is useful in that it also provides insight into the 

[PDF] The Newton-Raphson Method - UBC Math

the geometry is far less clear, but linear approximation still makes sense 2 3 The Convergence of the Newton Method The argument that led to Equation 1 used 

[PDF] Second Order Newton Iteration Method and Its - Hindawicom

Keywords: Numerical Algorithm, Newton Method, Nonlinear Equation, Second Order Iterative Scheme, MOS Modeling Newton-Raphson method is one of the most effective methods in series Up to thesecond degree in the Taylor expan-

[PDF] 94 Newton-Raphson Method Using Derivative

Newton-Raphson formula consists geometrically of extending the tangent line at familiar Taylor series expansion of a function in the neighborhood of a point,

[PDF] Generalized Newton Raphsons method free from second derivative

3 New iterative methods Let f : X → R, X ⊂ R is a scalar function then by using Taylor series expansion one can obtain generalized Newton Raphson's method:

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Whatdoesth ismeanforcomput ation?Innearly allourcomput ations, wewillre- placeexactform ulationswithapp roximations.Forexample,weapprox imatecontinuou s quantitieswithdiscretequanti ties,wearelim itedinsizebyfloatingpointre presentation ofnu mberswhichoftennecess itatesroundingort runcationandint roduces(hopefully) smallerrors.I fwearenotcareful,thes eerror willbe amplifie dwith inanill-conditioned problemanproduceinac cur ateresults. Thisleadsusto considerthestabilityandaccuracyofour algorith ms. Analgor ithmisstableifther esultisre lativelyuna↵ectedbyperturb ationsdu ringcom- putation.Thisissimilart otheideaofcon ditioni ngofproblems. Analgor ithmisaccurateisthec ompetedsol utionisclosetothetrues olutionofthe problem.Notethatastablealgor ithmc ouldbeinacc urat e. Weseek todesignandappl ystabl eandaccuratealgorithm stofin daccuratesoluti onsto well-posedproblems(ortofindway softransformingorapproxim atingil l-posed problems bywell -posedones).

3Appr oximatingafunctionbyaTaylorserie s

First,alittlenotati on.A real-valuedfunctionf:R!Risafun cti onfthattakesa realnumber argument,x2R,andreturn sarealnumber,f(x)2R.Wewritef 0 (x) todenot ethederivativeoff,sof 0 (x)= df dx ,and f 00 (x)is thesec ondderivati ve(so f 0 (x)= d dx df dx d 2 f dx 2 )and f (k) (x)isthekthderi vativeoffevaluatedatx. Aswe haveju stseen,simpl yevaluatingar ealvaluedfunctionc anbepronetoinstabili ty (ifthefu nctionhasal argeconditionnumber).Fort hatreason ,app roximation sof functionsareoftenused.Them ostcomm onmethodofappro ximatingthereal- valued functionf:R!Rbyasim ple rfunctionistousetheTa ylorseriesrepresentati onfor f. TheTaylor serieshastheformof apolynomialwherethec oe cientsofthepolynomial arethed erivatives offevaluatedatapoint.Solongas alld eri vativesofthefunc tion existsatthepointx=a,f(x)can beexp ressed intermsofofthevalueofthefuncti on andit's derivativesataas: f(x)=f(a)+(xa)f 0 (a)+ (xa) 2 2! f 00 (a)+...+ (xa) k k! f (k) (a)+...

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f(x)= 1 X k=0 (xa) k k! f (k) (a), wheref (0) =fand0!=1b yde fini tion . Thisisknown astheTaylorseriesforfabouta.I tisvali dforx"close"toa(strictly, withinthe"radiusofconv ergence"of theseries).When a=0,t he Taylorseri esisknown asaMaclaurinseries. 4 Thisisaninfini teseri es( thesumcontainsinfinitel ymanyterms)socannotb edirectly computed.Inpractice,wetru ncateth eseriesafterntermstogettheTaylorpolynomial ofdegre encentredata,wh ichwedenote f n (x;a): f(x)⇡ fquotesdbs_dbs2.pdfusesText_2