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Justas fixed-pointiterationcan be usedto solve systems nonlinearequations, other of open root location methodssuchas the Newton-Raphson methodcan be usedfor the sanepurposeRecall that the Newton-Raphson methodwas predicated employingthe derivative on (ie, the slope) of a function to estimateits intercept with the axis of the independent variable-that is, the root In Chap6 we useda graphicalderivationto computethisestimateAn alternativeis to derive it fiom a first-orderTavlor seriesexpansion: ( x r * r ) : f ( x i ) * ( x i + r- x , \ f ' \ x i ) (128) "f wherex; is the initial guessat the root andx;11 is the point at which the slopeintercepts the r axisAt this intercept, (ri*r) by definitionequals zeroandEq (128)canbe reananged I to yield
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form of the Newton-Raphson which is the single-equation method The multiequationfonn is derived in an identical fashionHowever, a multivariable variable Taylor seriesmust be usedto accountfor the fact that more than one independent contributesto the determination the root For the two-variablecase,a first-order of Taylor seriescan be written for eachnonlinearequationas f t iv t : f i i - | ( t 1 , ; * , ",,,1H ' - ' r 0fuI (x21,1- r" ; ) xz (xzi+r
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(12l}a)
f z i + t: , f 2 , * ( x r i + r * r , l Y f
( 121 0i)
Justas for the single-equation version,the root estimate corresponds the valuesof r1 and to where/y ;11and [111eQual zeroFor this situation, ( 12l0) canberearranged give d2, Eq to (12ltal
(t2ttu
NONLINEAR SYSTEMS
to with i 's areknown (they correspond the latestguessor apBecause valuessubscripted all ( i p r o x i m a t i o n ) , t h e o n l y u n k n o w n s o r e " r l , ; a1 -d x 2 , i a 1 T h u s , E q 1 2 1 1 ) s a s e t o f t w o -n1 linear equations with two unknowns Consequently,algebraic manipulations (eg, Cramer'srule) can be employedto solvefor
t, J)t n
3f2, OX2
t^,JZ,r
\J\,i
{ti+t :
-tli -
\fi,i \fz,i _ \fi,i \fzl 0x10x2 3x2'dx1 3f ,,, ]fz,i " "
t^_ Jlt _
(1212a1
x2,i+t : x2,i -
t,,_ Jt,r
r,Xt
3fi,i 0fzl * 3fi,i 3fzi
\xt \xz 3x2 \xy
(1212b)
The denominator of eachof theseequationsis formally referred to as the determinantof the Jacobian of the system methodAs in Equation (1212) is the two-equationversion of the Newton-Raphson the following example,it can be employediterativelyto home in on the rootsof two simultaneous equations E X A M P1 2 3 LE N e w t o n - R o p h s o f o r o N o n l i n e q rS y s t e m n method to determine Newton-Raphson Problem Stotement Use the multiple-equation with guesses x1 : 15 andxz:35 of the rootsof Eq (126)Initiate computation of them at the initial guesses Solution First computethe partial derivativesand evaluate -r and y:
: 2xr xz:2(15t* 5 65 3 : +
:rr : 15 : 325 : | * 6xtxz:I + 6( r 5) ( 35)
ofz'o -3xi 2 : 3(35) : 36i5 3xr
is for Thusthedeterminant theJacobian thefirst iteration of : 65(32s) 15(3675) 156125 as at ofthe functions be evaluated theinitialguesses can Thevalues - ) 1 0: - 2 5 : + fro (lS)2 15(35 5 f z : t3 5+ 3 (1 5 ) ( 3 5-) 2 l : 1 6 2 5 , into to values be substituted Eq (1212) give can These
-2s(32s) r62s (rs)
-^- <_
203603 284388
- (-2s) s) (367 r62s(6s) :
156t25
Thus, the resultsare convergingto the true valuesofxl : 2 and12 : 3 The computation accuracyis obtained until an acceptable can be repeated
ITERATIVE METHODS
When the multiequation Newton-Raphson works,it exhibitsthe samespeedy quadratic as versionHowever,just as with successive convergence the single-equation subsfitution it can diverge if the initial guesses are not sufficiently close to the true rootsWhereas graphicalmethodscould be employedto derive good guesses the single-equation for case, no such simple procedureis availablefbr the multiequationversionAlthough there are for someadvanced approaches obtainingacceptable first estimates, often the initial guesses must be obtainedon the basisof trial and error and knowledgeof the physicalsystem being modeled The two-equationNewton-Raphson approachcan be generalized solve n simultato To neousequations do this, Eq ( I 2I I ) can be written for the ftth equationas
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