A MODIFIED NEWTON METHOD FOR SOLVING NON-LINEAR ALGEBRAIC EQUATIONS

38 Journal of Marne Scence and Technology, Vol. 7, No. 3, pp. 38-47 (9) A MODIFIED NEWTON METHOD FOR SOLVING NON-LINEAR ALGEBRAIC EQUATIONS Satya N. Atlur*, Chen-Shan Lu**, and Chung-Lun Kuo*** Key words: nonlnear algebrac equatons, teratve method, ordnary dfferental equatons, fcttous tme ntegraton method (FTIM), modfed Newton method (MNM). ABSTRACT The Newton algorthm based on the contnuaton method may be wrtten as beng governed by the equaton () t + B F( ) =, where F ( j ) =,, j =, n are nonlnear j j algebrac equatons (NAEs) to be solved, and B j = F / j s the correspondng Jacoban matr. It s nown that the Newton s algorthm s quadratcally convergent; however, t has some drawbacs, such as beng senstve to the ntal guess of soluton, and beng epensve n the computaton of the nverse of B j at each teratve step. How to preserve the convergence speed, and to remove the drawbacs s a very mportant ssue n the solutons of NAEs. In ths paper we dscretze the above equaton beng wrtten as Bj j () t + F( j) =, by a bacward dfference scheme n a new tme scale of s = e t, and an ODEs system s derved by ntroducng a fcttous tme-le varable. The new algorthm s obtaned by applyng a numercal ntegraton scheme to the resultant ODEs. The new algorthm does not need the nverse of B j, and s thus resultng n a sgnfcant reducton n computatonal tme than the Newton s algorthm. A smlar technque s also used to modfy the homotopy method. Numercal eamples gven confrm that the modfed Newton method s hghly effcent, nsenstve to the ntal condton, to fnd the solutons wth a very small the resdual error. I. INTRODUCTION The numercal soluton of nonlnear algebrac equatons s one of the man aspects of computatonal mathematcs. Usually t s hard to solve a large system of hghly-nonlnear algebrac equatons. Although a lot of pror research has been Author for correspondence: Chen-Shan Lu (e-mal: lucs@ntu.edu.tw). *Center for Aerospace Research & Educaton, Unversty of Calforna, Irvne. **Department of Cvl Engneerng, Natonal Tawan Unversty, Tape, Tawan, R.O.C. ***Department of Systems Engneerng and Naval Archtecture, Natonal Tawan Ocean Unversty, Keelung, Tawan, R.O.C. j conducted n ths area, we stll lac an effcent and relable algorthm to solve ths dffcult problem. In many practcal nonlnear engneerng problems, the methods such as the fnte element method, boundary element method, fnte volume method, the meshless method, etc., eventually lead to a system of nonlnear algebrac equatons (NAEs). Many numercal methods used n computatonal mechancs, as demonstrated by Zhu, Zhang and Atlur [48], Atlur and Zhu [8], Atlur [5], Atlur and Shen [7], and Atlur, Lu and Han [6] lead to the soluton of a system of lnear algebrac equatons for a lnear problem, and of an NAEs system for a nonlnear problem. Collocaton methods, such as those used by Lu [7-3] for the modfed Trefftz method of Laplace equaton also need to solve a large system of algebrac equatons. Over the past forty years two mportant contrbutons have been made towards the numercal solutons of NAEs. One of the methods has been called the predctor-corrector or pseudoarclength contnuaton method. Ths method has ts hstorcal roots n the embeddng and ncremental loadng methods whch have been successfully used for several decades by engneers to mprove the convergence propertes, when an adequate startng value for an teratve method s not avalable. Another s the so-called smplcal or pecewse lnear method. The monographs by Allgower and Georg [] and Deuflhard [8] are devoted to the contnuaton methods for solvng NAEs. The Newton s method and ts mprovements are etensvely used nowadays; however, those algorthms fal f the ntal guess of soluton s mproper. In general, t s dffcult to choose a good ntal condton for most large systems of NAEs. Thus, t s necessary to develop an effcent algorthm, whch s nsenstve to the ntal guess of the soluton, and whch converges fast. Ths paper s arranged as follows. In the net secton we ntroduce an evoluton from a dscretzed method to a contnuous method, where an artfcal tme s ntroduced for wrtng the NAEs n the form of ODEs. In Secton III the man algorthms are ntroduced, where the novel feature s a sutable combnaton of the fcttous tme ntegraton method (FTIM) wth the Newton method. In Secton IV we gve some numercal eamples to evaluate the new algorthms of the modfed Newton method (MNM) and the modfed homotopy method (MHM). Fnally, we draw conclusons n Secton V.

S. N. Atlur et al.: A Modfed Newton Method for Solvng Non-Lnear Algebrac Equatons 39 II. FROM DISCRETE TO CONTINUOUS METHODS For the followng algebrac equatons: the Newton method s gven by F(,, ) =, =,, n, () n B F () = + [ ( )] ( ), where we use : = (,, n ) T and F: = (F,, F n ) T to represent the vectors, and B s an n n Jacoban matr wth ts j-th component gven by B j = F / j. Startng from an ntal guess of soluton by, Eq. () can be used to generate a sequence of, =,,. When are convergent under a specfed convergent crteron, the solutons of () are obtaned. The Newton method has a great advantage that t s quadratcally convergent. However, t stll has some drawbacs of not beng easy to guess the ntal pont, and the computatonal burden of [B( ))]. Some quas-newton methods are developed to overcome these defects of the Newton method; see the dscussons by Broyden [], Denns [5], Denns and More [6, 7], and Spedcato and Huang [46]. Hrsch and Smale [9] and many others have derved a contnuous Newton method governed by the followng dfferental equaton: ( t) = B ( ) F( ), (3) () = a, (4) where t s an artfcal tme, and a s an ntal guess of. It can be seen that the ODEs n (3) are dffcult to calculate, because they nclude an nverse matr. The correspondng dynamcs of (3) has been studed by several researchers, such as, Alber [], Boggs and Denns [], Smale [45], Chu [3], Maruster [4], and Ascher, Huang and van den Doel [3]. Presently, ths artfcal tme embeddng technque does not brng out any practcally useful result pertanng to the Newton s algorthm. Below we wll develop a new ODEs system, whch s equvalent to (3). Then, a natural embeddng technque from the NAEs nto the ODEs as developed by Lu and Atlur [37] wll be combned wth a new contnuous form of (3). Correspondng to the artfcal embeddng technque, whch s not yet proven to be useful, our embeddng technque by transformng the contnuous form n (3) nto a space, whch s one-dmenson hgher, may fnd to be very useful. III. MODIFIED METHODS. A Novel Technque Lu and Atlur [37] have ntroduced a novel contnuaton method, by embeddng the NAEs nto a system of nonautonomous frst-order ODEs. For the later requrement, we consder a sngle NAE: F( ) =. (5) The above equaton only has an ndependent varable. We may transform t nto a frst-order ODE by ntroducng a fcttous tme-le varable τ n the followng transformaton of varable from to y: y( τ) = ( + τ). (6) Here, τ s a varable whch s ndependent of ; hence, y' = dy/dτ =. If ν, Eq. (5) s equvalent to Addng the equaton y' = to (7) we obtan By usng (6) we can derve = ν F( ). (7) y = ν F( ). (8) y y y = ν F. + τ + τ Ths s a frst-order ODE for y(τ). The ntal condton for the above equaton s y() =, whch s however an unnown and requres a guess. Multplyng (9) by an ntegratng factor of /( + τ ) we can obtan d y ν y F. dτ = + τ + τ + τ Further usng y/( + τ ) =, leads to (9) () ν = F( ). () + τ The roots of F() = are the fed ponts of the above equaton. We should stress that the factor ν/( + τ) before F() s mportant.. Modfed Newton Method When one apples the forward Euler method to (3) wth a tme stepsze equal to, Eq. () s obtaned. If a sutable ntal condton s chosen, when tme ncreases to a large value, we may epect the sequence to converge to a true soluton. However, the Newton method s very tme consumng n the calculaton of B and s not easy to choose a sutable ntal condton.

4 Journal of Marne Scence and Technology, Vol. 7, No. 3 (9) Frst, we propose a varable transformaton s = e t and wrte (3) as ( s) B ( ) ( s) + F=, () where the prme denotes the dervatve of (s) wth respect to s. Now the nterval of s s s [, ), when t [, ). We dvde the nterval of [, ) nto m subntervals wth s = /m, and appromate the above equaton by a bacward fnte dfference: ( s ) B( ) + F( ) =, =,, m, (3) s = a, (4) where = (s ) wth s = s, and now = a s a boundary condton, nstead of the ntal condton n (3). Agan, Eq. (3) s a coupled system of NAEs, wth m vectoral-varables, =,, m. When m s solved from (3), the soluton of NAEs s found. Now, we can apply the technque n () to (3), obtanng d ν = ( s ) B ( ) + F ( ), =,, m. (5) dτ + τ s We fnd that the present formulaton s nsenstve to the condton of = a, because a s just a boundary value of the many ODEs n (5) wth m-unnown vectors; hence, we may set a =. It deserves to note that there are two advantages to transformng (3) nto (): frst, the doman length of s s such that we can use a small nteger m to dvde the whole nterval nto some subntervals by usng s = /m, and second, we no longer need to use the nverse of B. In (3), because we need to ntegrate the ODE along the t-drecton, the nverse of B s requred; however, n (5) we only ntegrate the ODEs along the τ-drecton, and the nverse of B s not requred any more. Eq. (5) s a new equaton, whch s a combnaton of the contnuous form of the Newton s algorthm wth the fcttous tme ntegraton form. We wll use ths equaton to solve the NAEs. It s nterestng that when we tae m =, s =, s =, the followng term ( ) ( ) s B drops out, and (5) s reduced to s d ν = F ( ), (6) dτ + τ where we replace by. Ths equaton has been used by Lu and Atlur [37] to solve the NAEs, and the new method s called a fcttous tme ntegraton method (FTIM). As reported by Lu and Atlur [37], when the technque of FTIM s used to solve a large system of NAEs, hgh performance can be acheved. The above dea of ntroducng a fcttous tme coordnate τ nto the governng equaton was frst proposed by Lu [3] to treat an nverse Sturm-Louvlle problem by transformng an ODE nto a PDE. Then, Lu [3-34], and Lu, Chang, Chang and Chen [4] etended ths dea to develop new methods for estmatng parameters n the nverse vbraton problems. More recently, Lu [35] has used the FTIM technque to solve the nonlnear complementarty problems, whose numercal results are very well. Then, Lu [36] used the FTIM to solve the boundary value problems of ellptc type partal dfferental equatons. Lu and Atlur [38] also employed ths technque of FTIM to solve the med-complementarty problems and optmzaton problems. Then, Lu and Atlur [39] usng the technque of FTIM solved the nverse Sturm-Louvlle problem, for specfed egenvalues. 3. Modfed Homotopy Method Davdeno [4] was the frst who developed a new dea of homotopy method to solve () by numercally ntegratng H H (7) () t = t (,), t () = a, (8) where H s a homotopc vector functon gven by H = ( t)( a) + tf( ), (9) and H and H t are respectvely the partal dervatves of H wth respect to and t. The soluton (t) of (7) forms a homotopy path for t. One then solves a sequence of problems H(t) = for values of t ncreasng from to, where for each such problem a good ntal guess from prevous steps s at hand. Ths powerful dea has been around for a whle; see Watson, Sosonna, Melvlle, Morgan and Waler [47] for a general pacage, Nocedal and Wrght [43] for a dscusson n the contet of optmzaton, and Ascher, Matthej and Russell [4] for boundary value ODEs. The homotopy theory was later refned by Kellogg, L and Yore [], Chow, Mallet-Paret and Yore [], L and Yore [3], and L [4]. For some hghly complcated NAEs, a contnuaton approach of the homotopc method may yeld the only practcal route for a soluton algorthm. Wth the use of (9), the homotopc ODEs n (7) can be wrtten as sb+ ( s) I n ( s) + a + F =. () Here we use s to replace t n order to be consstent wth the notaton s used n ().

S. N. Atlur et al.: A Modfed Newton Method for Solvng Non-Lnear Algebrac Equatons 4 Smlarly, by a dscretzaton of the above equaton we can obtan a new algebrac equaton: It s nterestng that (4) and (7) can be combned together nto a smple matr equaton: s ( ) ( s) B + In + a + F( ) =, s =,, m, () = a. () f (, t) n n d =. dt (, t) f (8) Agan, applyng the technque n () to () we can obtan d ν = s ( ) + ( s) n + + ( ), dτ + τ B I a F s =,, m. (3) We wll ntegrate (5) and (3) by usng the group preservng scheme ntroduced n the net secton. 4. The GPS for Dfferental Equatons System We develop a stable group preservng scheme (GPS) as follows. We can wrte a vector form of ODEs by n = f(, t), R, t >. (4) A GPS can preserve the nternal symmetry group of the consdered ODEs system. Although we do not now prevously the symmetry group of dfferental equatons system, Lu [5] has embedded t nto an augmented dfferental equatons system, whch concerns wth not only the evoluton of state varables themselves but also the evoluton of the magntude of the state varables vector. We note that It s obvous that the frst row n (8) s the same as the orgnal equaton (4), but the ncluson of the second row n (8) gves us a Mnowsan structure of the augmented state varables of X: = ( T, ) T, whch satsfes the cone condton: where X gx =, (9) In n g = (3) n s a Mnows metrc, and I n s the dentty matr of order n. In terms of (, ), Eq. (9) becomes XgX= = =. (3) It follows from the defnton gven n (5), and thus (9) s a natural result. Consequently, we have an n + -dmensonal augmented dfferental equatons system: wth a constrant (9), where X = AX (3) = =, (5) where the dot between two n-dmensonal vectors denotes ther nner product. Tang the dervatves of both the sdes of (5) wth respect to t, we have satsfyng f (, t) n n A : =, f (, t) (33) d =. dt Then, by usng (4) and (5) we can derve d f =. dt (6) (7) Ag+ ga=, (34) s a Le algebra so(n, ) of the proper orthochronous Lorentz group SO o (n, ). Ths fact prompts us to devse the GPS, whose dscretzed mappng G must eactly preserve the followng propertes: GgG= g, (35)

4 Journal of Marne Scence and Technology, Vol. 7, No. 3 (9) Table. Comparson of MNM and MHM for Eample. Method m h ν ε IN (, y) (F, F ) MNM 5.5 5 79 (.683,.683) ( 6.75 6, 6.75 6 ) MNM 5.5-5 94 (.683,.683) ( 6.948 6, 6.948 6 ) MHM 4.7. 4 738 (.67965,.67965) (.539 4,.539 4 ) MHM..5 4 3 (.68,.68) ( 8.74 4, 8.74 4 ) det G =, (36) G >, (37) b f + ( a ) f η : =. f (44) where G s the -th component of G. Although the dmenson of the new system s rased by one more, t has been shown that the new system permts a GPS gven as follows [5]: X G ( ) X, (38) = + where X denotes the numercal value of X at t, and G() SO o (n, ) s the group value of G at t. If G() satsfes the propertes n (35)-(37), then X satsfes the cone condton n (9). The Le group can be generated from A so(n, ) by an eponental mappng, where ( a ) bf In + f f ( ) ep [ h ( ) f f G = A ] =, b f a f h f a: = cosh, h f b: = snh. Substtutng (39) for G() nto (38), we obtan where, (39) (4) (4) = +η f (4) + b a f + = +, f (43) The group propertes are preserved n ths scheme for all h >, and s called a group- preservng scheme. 5. Numercal Procedure Startng from an ntal value of (), we may employ the above GPS to ntegrate (5) or (3) from τ = to a selected fnal tme τ f. In the numercal ntegraton process we can chec the resdual norm by / n m [ F ( )] ε, = (45) where ε s a gven convergent crteron. If at a tme τ τ f the above crteron s satsfed, then the soluton of s obtaned, and thus m gves the soluton of (). IV. NUMERICAL TESTS. Eample We frst consder two smple algebrac equatons: F y y F y y (, ) = =, (, ) = =. (46) The roots are (, ), (, ), (( + 5) /, ( + 5) / ) and (( 5) /, ( 5) / ). In the computatons by usng the modfed Newton method (MNM) and modfed homotopy method (MHM) we requre to specfy the values of m, h used n the GPS, ν, ε, and some ntal condtons; however, we let a =. We calculate ths eample by MNM and MHM, of whch the thrd root (( + 5)/, (+ 5)/) (.6834,.6834) and the fourth root (( 5) /, ( 5) / ) (.6834,.6834) are calculated, and the values of these parameters are recorded n Table, where IN s a shorthand of the teraton number spent n the calculaton.

S. N. Atlur et al.: A Modfed Newton Method for Solvng Non-Lnear Algebrac Equatons 43..5 MNM for 3rd root MHM for 3rd root MNM for 4th root MHM for 4th root 6 Resdual Norm..5 Resdual Norm 8 4. -.5 4 6 8 Fg.. Comparng the teratve resdual norms of Eample by MNM and MHM. -4 4 6 8 Fg.. The resdual norm of Eample. In Fg. we plot the varaton of the resdual norms for MNM and MHM wth respect to the number of teraton, denoted by. It can be seen that the MNM converges very fast, whch s much fast than that of the MHM. It s nterestng that the MNM can fnd two dfferent roots by merely changng the sgn of ν and usng the same ntal guess of = =.5. However, for the MHM ths s not worng, and we use dfferent ν and dfferent ntal guess of = =.5 for the thrd root, and = =.5 for the fourth root.. Eample We study the followng system of two algebrac equatons [46]: F y y F (, y) = ( y ) ( y ) + ( y ) =. (, ) = =, (47) Resdual Norm.E+7.E+6.E+5.E+4.E+3.E+.E+.E+.E-.E-.E-3.E-4.E-5.E-6 Problem Problem Problem 3 5 5 5 Fg. 3. The resdual norms of Eample 3. The two real roots are (, y) = (, ) and (, y) = (4, ). In ths test of the MNM we tae m =, h =., ν =. As shown n Fg. the resdual norm converges very fast wth only 67 steps for satsfyng the convergent crteron of ε = 3. We get the solutons (, y) = (.3,.) wth the resdual errors (F, F ) = ( 9.66 4,.7 6 ). 3. Eample 3 Then we consder a system of two algebrac equatons n two-varables [9]: F (, y) = 3 y + a ( + y) + b y + c + a y =, (48) F y y y a y y b c 3 3 (, ) = 3 (4 ) + + =. The parameters used n ths test are lsted n Table. For these problems the ntal guesses are respectvely (, y) = (, ), (, y) = (.,.), and (, y) = (,.). In Fg. 3 we dsplay the resdual errors for the above three problems. The thrd problem s hard to solve because there appears a much large coeffcent a than others. As reported by Hsu [], he could not calculate the thrd problem by usng the homotopc algorthm wth a Gordon-Shampne ntegrator, the L-Yore algorthm wth the Euler predctor and Newton corrector, and the L-Yore algorthm wth the Euler predctor and quas-newton corrector.

44 Journal of Marne Scence and Technology, Vol. 7, No. 3 (9) Table. The parameters and results for Eample 3. Problem Problem Problem 3 (a, b, c, a, b, c ) (5,,, 3, 4, 5) (5,,, 3, 4, 5) (,,, 3,, ) (m, ν, h, ε) (,.,., 6 ) (5,.5,., 6 ) (,.,., 6 ) IN 7 3 (, y) ( 5.397755,.8446) (.34,.88) ( 4.95897,.36) (F, F ) (9.96 7, 7.836 9 ) (.4 7, 8.93 7 ) (8.573 7,.38 ) Hrsch and Smale [9] used the contnuous Newton algorthm to calculate the above three problems. However, as pont out by Lu and Atlur [37], the results obtaned by Hrsch and Smale [9] are not accurate. Under ths stuaton we may say that the present modfed Newton method can offer more effcent and accurate solutons; and also the new MNM as compared wth the FTIM reported by Lu and Atlur [37] for calculatng ths eample s convergent fast than FTIM and can retan the same accuracy. 4. Eample 4 We consder a system of three algebrac equatons n threevarables: F(, y, z) = + y+ z 3=, F y z y y z F y z y z (,, ) = + + 4 7 =, 8 4 9 3(,, ) = + + 3=. (49) Obvously = y = z = s the soluton. In ths test we tae m =, h =., ν =. As shown n Fg. 4 the resdual norm converges very fast wth 4 steps for satsfyng the convergent crteron of ε = 3. We get the solutons (, y, z) = (.9996,.999,.7) wth the resdual errors (F, F, F 3 ) = ( 6.46 4, 7.574 4, 3.5 6 ). Resdual Norm 6 4-5 4 6 8 4 Fg. 4. The resdual norm of Eample 4. 5. Eample 5 The followng eample s gven by Roose, Kulla, Lomb and Meressoo [44]: F 4 =, =. = 3 ( + + ) + ( + ), n+ (5) Intal values are fed to be =, =,, n. For ths case we use m = and a large ν = to speed up the rate of convergence, whch needs 749 steps wth a tme stepsze h = 4 used n the GPS ntegratng method. When the convergent crteron s gven by 6, the resdual error ( ) / F = of numercal solutons s about 9.8 7. In Fg. 5 we plot the resdual error wth respect to, and the numercal solutons of, =,, are recorded n Table 3. Resdual Norm 5 5-5 4 6 8 Fg. 5. The resdual norm of Eample 5.

S. N. Atlur et al.: A Modfed Newton Method for Solvng Non-Lnear Algebrac Equatons 45 Table 3. The numercal solutons of Eample 5 wth n =..5 3 4 5 3.83 5.383 7.395 9.4.969 6 7 8 9.6 4.86 5.75 7.76 8.66 As compared wth those reported by Spedcato and Huang [46] for the Newton-le methods, the present modfed Newton method s more accurate and tme savng, where the computatonal tme s lesser than. sec by usng a PC586. Resdual Norm..5. 6. Eample 6 Then, we consder a smlar test eample gven by Krzyworzca []: F = (3 5 ) +, F = (3 5 ) +, =,, 9, (5) F = (3 5 ) +. 9 For ths case we use m = and a large ν = to speed up the rate of convergence, whch needs only 55 steps wth a tme stepsze h =. used n the GPS ntegratng method. The ntal values are fed to be =., =,,. When the convergent crteron s gven by 6, the resdual error ( F ) / = of numercal solutons s about 9.6 7. In Fg. 6 we plot the resdual error wth respect to, and the numercal solutons of, =,,. are recorded n Table 4. As reported by Mo, Lu and Wang [4] the Newton method cannot be appled for ths eample, and ther solutons obtaned by the conjugate drecton partcle swarm optmzaton method are dfferent from the present solutons. For ths eample t may have multple solutons, but Krzyworzca [] ddn t gve soluton for ths eample. Obvously, our method converges faster than that n the above cted paper by Mo, Lu and Wang. 7. Eample 7 In ths eample we apply the MNM to solve the followng boundary value problem [6]: The eact soluton s 3 u = u, u() = 4, u() =. 4 u ( ) =. ( + ) (5) (53) -.5 4 6 Fg. 6. The resdual norm of Eample 6. By ntroducng a fnte dfference dscretzaton of u at the grd ponts we can obtan 3 F u u u u ( ) u = 4, u =, = ( + + ), n+ (54) where = /(n + ) s the grd length. Usng the followng parameters m = 3, n =, h = 3, ν =.4 and ε = 4 we compute the roots of the above system. In Fg. 7(a) we plot the resdual error wth respect to, and compare the numercal soluton wth eact soluton n Fg. 7(b), whch can be seen that the error as shown n Fg. 7(c) s very small n the order of 3. V. CONCLUSIONS Snce the wor of Newton, teratve algorthms were developed by many researchers, etendng to contnuous type of systems by ntroducng an artfcal tme. The present paper transformed the contnuous form j() t + Bj F( j) = of the Newton s algorthm nto another contnuous form ( s) B j ' j (s) + F ( j ) = through a new tme varable of s = e t. A dscretzaton of the above equaton by a bacward dfference s performed, and an ODEs system s derved by ntroducng a fcttous tme. The teratve algorthm, whch was obtaned by applyng the GPS to the resultant ODEs, does not need the nverse of B j, and s computatonally far more effcent than the Newton s algorthm. In dong so, we found that the modfed Newton method not only can remove the drawbacs of the Newton s method, but also can preserve the quadratcally convergent speed, as shown n the plots of the

46 Journal of Marne Scence and Technology, Vol. 7, No. 3 (9) Table 4. The numercal solutons of Eample 6. 3 4 5.8445.7754.69879739.584467655.6648975 6 7 8 9.754694.949854.6677.794687.69376395 Numercal Error E+ E+ E+ E- E- E-3 E-4 Resdual Norm E+3 u 5 4 3...4.6.8..5E-3 (c).e-3 5.E-4.E+ (a) 4 6 8 (b) Numercal Eact...4.6.8. Fg. 7. Applyng the MNM to a boundary value problem: (a) resdual norm, (b) comparng numercal and eact solutons, and (b) dsplayng the numercal error. resdual norm vs. teraton number for many eamples eamned n ths paper. Numercal eamples confrmed that the modfed Newton method s hghly effcent to fnd the true solutons wth the resdual errors beng very small. The modfed homotopy method s more comple than the modfed Newton method; however, the accuracy and effcency of the modfed Newton method are much better than that of the modfed homotopy method. REFERENCES. Alber, Y. I., Contnuous processes of the Newton type, Dfferental Equatons, Vol. 7, pp. 46-47 (97).. Allgower, E. L. and Georg, K., Numercal Contnuaton Methods: An Introducton, Sprnger, New Yor (99). 3. Ascher, U., Huang, H., and van den Doel, K., Artfcal tme ntegraton, BIT, Vol. 47, pp. 3-5 (7). 4. Ascher, U., Matthej, R., and Russell, R., Numercal Soluton of Boundary Value Problems for Ordnary Dfferental Equatons, SIAM, Phladelpha (995). 5. Atlur, S. N., Methods of Computer Modelng n Engneerng and Scences, Tech. Scence Press, 4 pages (). 6. Atlur, S. N., Lu, H. T., and Han, Z. D., Meshless local Petrov-Galern (MLPG) med collocaton method for elastcty problems, CMES: Computer Modelng n Engneerng & Scences, Vol. 4, pp. 4-5 (6). 7. Atlur, S. N. and Shen, S., The meshless local Petrov-Galern (MLPG) method: a smple & less-costly alternatve to the fnte and boundary element methods, CMES: Computer Modelng n Engneerng & Scences, Vol. 3, pp. -5 (). 8. Atlur, S. N. and Zhu, T. L., A new meshless local Petrov-Galern (MLPG) approach n computatonal mechancs, Computatonal Mechancs, Vol., pp. 7-7 (998). 9. Atlur, S. N. and Zhu, T. L., A new meshless local Petrov-Galern (MLPG) approach to nonlnear problems n computer modelng and smulaton, Computer Modelng and Smulaton n Engneerng: CMSE, Vol. 3, pp. 87-96 (998).. Boggs, P. and Denns, J. E., A stablty analyss for perturbed nonlnear analyss methods, Mathematcs of Computaton, Vol. 3, pp. 99-5 (976).. Broyden, C. G., A class of methods for solvng nonlnear smultaneous equatons, Mathematcs of Computaton, Vol. 9, pp. 577-593 (965).. Chow, S. N., Mallet-Paret, J., and Yore, J. A., Fngng zeroes of maps: homotopy methods that are constructve wth probablty one, Mathematcs of Computaton, Vol. 3, pp. 887-889 (978). 3. Chu, M. T., On the contnuous realzaton of teratve processes, SIAM Revew, Vol. 3, pp. 375-387 (988). 4. Davdeno, D., On a new method of numercally ntegratng a system of nonlnear equatons, Dolady Aadem Nau SSSR, Vol. 88, pp. 6-64 (953). 5. Denns, J. E., On the convergence of Broyden s method for nonlnear systems of equatons, Mathematcs of Computaton, Vol. 5, pp. 559-567 (97). 6. Denns, J. E. and More, J. J., A characterzaton of superlnear convergence and ts applcaton to quas-newton method, Mathematcs of Computaton, Vol. 8, pp. 549-56 (974). 7. Denns, J. E. and More, J. J., Quas-Newton methods, motvaton and theory, SIAM Revew, Vol. 9, pp. 46-89 (977). 8. Deuflhard, P., Newton Methods for Nonlnear Problems: Affne Invarance and Adaptve Algorthms, Sprnger, New Yor (4). 9. Hrsch, M. and Smale, S., On algorthms for solvng f () =, Communcatons on Pure and Appled Mathematcs, Vol. 3, pp. 8-3 (979).. Hsu, S. B., The Numercal Methods for Nonlnear Smultaneous Equatons, Central Boo Publsher, Tape, Tawan (988).. Kellogg, R. B. T., L, T. Y., and Yore, J. A., A constructve proof of the Brouwer fed-pont theorem and computatonal results, SIAM Journal

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