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Cetakan Pertama, 2OO9ISBN No. 97 8-602-95533-0-7

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Midpoint Newton's Method for Simple and Multiple Roots *

M. Imran

mimran@unri.ac.id

Laboratorium Matematika Terapan, Jurusan MatematikaFakultas Matematika dan IImu Pengetahuan Alam Universitas Riau

Kampus Binawidya Pekanbaru (28293)

AbstractIn this paper we propose a new modification of Newton's method based on midpoint rule forsolving nonlinear equations. Analysis of convergence shows that the new method is cubicallyconvergent for a simple root and linearly for multiple roots. The method require one functionevaluation and two of its first derivative, but no evaluations of its second derivative. We verifythe theoretical results on relevant numerical problems and compare the behavior of the proposemethod with some existing ones.

I(eywords: Modi,fi,ed Newton's Method, Midpoi,nt rule, trapezoi,dal rule

Introduction){ewton's method, which is quadratically convergent, is the most popular method to find a root ofa nonlinear equation,

f(r):0, f:DclR-+R' (1)

\[any researcher are interested in modifying the method to obtain a higher order method. Thefirst third order method resulted on the modifying the Newton's method appear in Wall [8]. Thismodification requires the second derivative of /(r).

Weerakoon and Fernando [7] have suggested an improvement to the iteration of Nernton's methodwithout requiring the second derivative of f (r) . They have approximated the indefinite integralusing trapezoidal rule.

In this study we suggest a modification of the iteration of Newton's method by approximatingthe indefinite integral using a midpoint rule. The modified method need one functional and two firstderivative evaluations for each iteration.

2 Preliminary ResultsDefinition 1 (See [1]) A sequence of iterates {nn: n > 0} is said to converge with order p> I toa point a if

lo- *.+rl I cla - rnlp, n) 0

for some c ) 0. If p :1, the sequence is said to converge linearly to o. In that case, we requirec < 1; the constant c is called the rate of linear convergence of r, to o.

Let en : nn a be the error in the nth iterate of the method which produce the sequence {r,}.Then the relation

an*t : cee. + O (ee*+') : o ("X)

is called the error equation. The value ofp is called the order of convergence and c is known as theasymptotic error constant of this method.

*Presented at the 14th National Conference in Mathematics, and the Congress of Indonesian Mathemd,tical Society,held at the Sriwijaya University in Palembang, J:uly 24-27,2OO8

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Definition 2 (See [7]) Let a be a root of the function / and suppose that nn-ttnntrn+l are closerto the root o. Then the computational order of convergence p carl be approximated using the formula

3 Some Modified Newton's MethodsNewton's method (NM) for computing the root o of the nonlinear equation (1) is to start with initialestimate rs sufficiently close to the root o and to use the one point iteration

f(".)Xln*t:*.- f,(rr)

where r, is the n-th approximation of a. We may also view frn*l os the root of the two-term Taylorexpansion or linear model of / about rn,l51,

M("): f (*.) + f'(**)(* - rn).

By integrating by part this local model can be viewed as the following obvious identity,[2],

f l,): f @.) + [" 7'1110, (4)J"^

Newton approximates {i^f'(")ds in (4) using the left Riemann-sum for one interval, resulted in

1,"_r'G)0, x ft(r.)(r - rn).

On substituting (5) into (4), setting f (") : 0 rearranging the terms of the resulting equation, we

end up with the equation (3).Weerakoon and Fernando [7] approximate the indefinite integral involved in ( ) by the trapezoidal

rule,

[' 1'$1as = 1l'@") J f'k))('-',),Jt^"-'--- \ 2 /'

and then by some algebra they end up with the following scheme (TNM)

_ _ In l(*,+, - a)l@. - a)lu- 1"11^-"1J1**r_ *11

2f (*.)rw,:"- il*^)+f\*h*)- f (r")frn+r: *. - f,(r.)

(2)

(3)

(5)

(6)

(7)

They prove that the scheme is a third order convergence. The way they choose rl*1 as in (7) was

introduced for the first time by Wall in [8].Now from the approximation error of trapezoidal rule and midpoint rule, that is

ur ,: fu f @)dr - f;tro+ /(b))) - -(b -r;)" I"{o. € € (a, b)andf e czia.bl,Jo

and

Epa :: fu f ,*ro* - (b - "\f (+) :9-9t1"1q1. ( e (a, b)and/ € c2la,b],Jo,

we see that the approximation error of the midpoint rule needs the same smoothness as the trapezeidal rule. By comparing the constant in front of the derivative of /, the absolute value of midpointapproximation error is slightly smaller than the trapezoidal rule. From this view we may use themidpoint rule to approximate the indefinite integral involved in (4), that is

Then, by some algebra we propose the scheme (MNM)1,"_f'G)0"

o (r - ";f'(ry)f (r.)frntr:*.-

f,@h),* f (r.)tn+t: *. - zf\r,).

(8)

(e)

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4 Asymptotic Error AnalysisTheorema 3 Let / : D - IR for an open interval D. Assume that / has sufficiently differentiablefunction on the interval D. It f has a simple root a € D and 16 is sufficiently close to a, thep theMNM defined by (8) and (9) satisfies the following error equation:

whereer:rn-6-and

Proof: See Imran [4]

where

Moreover we have

an-.r : @i +]"r1"'. + O(ef;),

1{i)@)",: ffi, j:2,J.

Theorema 4 Let f : D ---+ lR. for an open interval D. Assume that / has sufficiently differentiablefunctionontheintervalD,and/hasamultiplerootaofmultiplicitym)l€D.Ifxsissufficientlyclose to o, then the MNM defined by (8) and (9) satisfies the following error equation:

€n*t : (, - #), " - (W ","?-

+ ct 1"'.1),

wheree, :r,n-r-t "r:ffi,for

j-2,3, K:lform:1 andK= (r- *)*-'form>7'

Proof:Wefollowthetechnicalproof from[6]. Letobearootof multiplicitym,(i.e.f(a):f'(!r):i":7{^) (a): 0, and 7@+r) @) I 0. We expand /(r,) about , : a using Taylor expansion, that is

f (*.) : #o- - d)* +ffi ,, n - o)**t . ffi(*n - o)*+,

. Stn+t\ (a) ,+ ffi (*n - o)*+' -10((r* - o) +n)

: "21", "f

(1 + c2en * cae2n -t cae3. + o(ei)), (10)

f' (,,) - #X(rn - a)*-t . ry(*n - o)*

. ##(*n - o)*+' + 0((r* - o)*+')

_ f@) (o) .^-.1 ( ,r - (m + l) f (*+r) (a) ": t, _ ry.." \, i f (*)(a)mOn + t)'n

-m"i+ofi1): m"T,Q*@#,,,^*@,J?.cael+o("il)' (11)

Computing f (r.)/(2ft(r,)) from (10) and (11) and recalling

7@+i-t) (t) j =2,3,4.cj:7@) @)(m + L)(m + 2). . . (m + j - t)'

(t + z)-1 : I - r I n2 - 13 + 14 - 15 +... -l O(r^)

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we obtain afber some algebra

f("")2f '(*.)

{:}P"TG t czen * csel-t cael + o(ef))z ffi"r-'(r + @!)" ze. * v#cae'z. + o(el))

fi{". * c2el * cse3. + o (e*))

* (r + @jJ)""' * @#",". + +ctl"'-1)-'

fiA" * c2e?, * caes. + o(el))

' [, - (t + !)r".* ( - ", -2:2 +,* * "Z*#)"?,* oa,.))

: *1"_ - fi"r" * ccar.)1.

On substituting (12) into (8) and simplifying yields

r1+t :,, - ;;1". - Z"? + ot"il:,+

[(r - *)""* #"1*ot"i,))or

ni,+t -" : (t - *)"" * #"'. + o(e3*).

Expanding f (r* +r) about z : a, noting (14) and simplifying, we end up with

r'@h+): #*ft- *).-+ ffi"?.+o1d,1)*-',.

[r + @*!",rc- *)"-+ ffi",,*or"il)* @#,,((r - #), .+ ffi"?-* <t("il)' * oa,.)l

: ## ft - *) "' + ffi "?' + o("11)*-'

r r2m2 -l- m. - 1rx ft + \ 2*, 1"'*, 1Qm+2), (4*3 +4m2 -7m+2)'\.4rn3

4m3

_ I(^)(o) Kom_.1, _ r2m2* m - I r

@ - ly.^e; lt + \ ,* )cze'

, 1Qm+2) , (4m3t4m2-7m+2)-\ +-, cz'tT

": ((, - *)+fi",+{){",,1)--1

(12)

(13)

(14)

"r).?. + CCA'"))

"r).?-+ rcA')1, (15)

where

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F\rithermore, using (10) and (15), and recalling (4), we compute f (")lf,@L*r), that is

'*a.T G * czen * cte?* * cn"3^ + o ("*))l@*) _f '(ri+r)

(16)

1 (en*cze?n*cze3*+tc(.*)):*K

: L=@* * cze?^ + ctel + o(ei))

" [r + g#).,"* * (W-. 9t fiiw ",)"?* + or"il]

: #(^*ff",.'^+ct1"l1).On substituting (16) into (9), we obtain

f('")frn*r:"_- f\*;+j

en*l: (r- #)""- (W"2"?n+oa'-))This end the proof.

5 Numerical SimulationsThe modified methods and Newton's method are tested using some functions and initial points,which have been used [7], [3] and [1]. We compute the computational order of convergen"i, (COC;using formula (2). We stop the program using the following criteria

E--*'- xnl - -l'; "'lf(r-+r)l < r,

lrn+r-al 1e,

where e :2.22e- 10 and o is the root. All programs are written in Matlab 7 and run on WindowsPC with Intel Processor at 2.4 GHz. The computational results for the case of simple roots are givenin Table 1 and for the multiple roots are given in Table 2 .

Table 1: Comparisons of the number of iterations and the COC of the modified methods for thecase of simple roots

f(") fiONumber of Iterations COCNM MNM ,I'NM NM MNM TNM

cos(r) - e

-1.0 8 6 .) ND ND NDt.7 4 3 r) 2.006 ND Nl)2.0 i) 3 3 2.001 ND 2.99ti3.0 6 3 8 2.001 2.958 2.8854.0 29 4 6 ND ND 2.978

(r-1.0)3-1

2.5 5 3 3 1.998 2.985 2.9794.0 7 4 5 2.019 2.97t NL)-0.5 15 5 15 1.999 ND 3.U131.0 10 5 7 ND 2.993 3.018

-2.0 10 6 8 2.0t4 3.013 NDre@') -sin2(r)+3 cos(r) + 5.0

-3.0 13 18 I 2.000 3.019 NDr.2 r00+ 38 20 1.002 ND NI]

"(*2+zr-lo) -,3.3 8 5 6 2.000 2.99t NDJ.b 11 7 8 2.000 2.b39 ND

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Table 2: Comparisons of the number of iterations and the COC of the modified methods lbr thecase of multiple roots

f(") frgNumber of Iterations COCNM MNM TNM NM MNM TNM

12 -2.22r+t.2321

-i.0 18 11 11 1.000 1.000 1.0000.6 16 10 10 1.000 1.000 1.0002.2 L7 I 11 1.000 1.000 1.00010.0 20 13 13 1.000 1.000 1.000

rn - b.4r3 +10.56e2 -8.954n -t 2.7957

0.6 18 11 t2 1.000 1.000 1.0000.8 16 10 11 1.000 1.000 1.0001.4 15 10 10 1.000 1.000 1.0001.8 18 13 11 1.000 1.000 r.UUU

13 - b.b6r2 +9.1389r -4.68999

0.0 18 11 t2 1.000 1.000 1.000u.b L7 11 1.0(x) 1.000 1.0001.5 15 10 10 1.000 1.000 1.000

-2.0 20 13 13 1.000 1.000 1.000

*a _ 8r3 +24,.2 -32r*16

-2.5 25 16 t7 1.000 1.000 1.0000.0 22 t4 15 1.000 1.000 1.0004.0 22 t4 15 r.000 1.000 1.00010.0 27 t7 18 1.000 1.000 1.000

From the Table 1, we see that some of the COCs is not defined (ND). This occurs becausedivision by zero while applying the formula (2). The COC of all methods for the case of simple rootsmatches the theoretical results, both for Nern'ton's method and the modified methods. In general,the number of iterations produced by applying the MNM is fewer than the other methods, except atthe starting point re : -3.0 and rs :1.2 fot f (r) : *"{,2) - sin2(z) f 3cos(r) -1_ 5.0, where MNMmethod needs 18 and 38 number of iterations respectively.

From the Table 2, we see that the COC for the all methods for the case of multiple roots is one.This matches the theoretical result as stated in the Theorem 4 for the MNM. In general, the numberof iterations produced by applying the MNM is fewer than the other methods, except at the startingpoint zs:1.8 for f ("): ra -5.413 -1 10.5612 _ 8.gb4r+2.Tg57, where MNM method needs 13mrmber of iterations. In all experiment for the multiple roots, the iteration ends not because of testof error given fulfilled, but because the value of function satisfies the second stopping criteria. Thismatter represents especial constraint in approximating multiple roots.

References

[1] Atkinson, K. E. 1989. An Introduction to Numerical Analysis. Second Edition. JohnWilev & Son, New York.

[2] Hamming, R. H. 1973. Numerical Method for Scientists and .Engineers. McGraw-HillInc. New York. Republished by. Dover, New York.

[3] Hasanov, V.I., Ivanov, I. G. & Nedjibov, G. 2005. A new mod,i,fi,cation of Newton's Method,.preprint laboratorium of Mathematical Modelling, Shoumen university, Bulgaria.

[4] Imran, M. 2008. A Third Order of Mod,'ified, Neuton's Method,. Jurnal sains MIPA. U(l\ 57--61.

Kelley, C. T. 1995. Iterative Methods for Linear and Nonlinear Equations. Frontier inApplied Mathematics 16. SIAM, Philadelphia.

Lukic T. & Ralevic N. M. 2007. Geometri mean Newton's method for s,imple and, multiple roots.Applied Mathematics Letters. 21: 30 36.

[7] Weerakoon, S. & Fernando, T. G. I. 2000. A Variant ,if Newton's Method, W,ith Accelerated,Third-Order Conuergence. Applied Mathematics Letters. 18: 87 93.

[8] wall, H. s. 1948. A Modificati.on of Newton's Method.. Amer. Math. Monthly. bb: 90-94.

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