An Efficient without Direct Function Evaluations. Newton s Method for Solving Systems of. Nonlinear Equations

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1 Aled Mathematcal Sceces, Vol. 3, 9, o., - 3 HIKARI Ltd, htts://do.org/.988/ams.9.88 A Effcet wthout Drect Fucto Evaluatos Newto s Method for Solvg Systems of Nolear Equatos Elefthera N. Malhoutsak ad Theodoula N. Grasa Dvso of Comutatoal Mathematcs ad Iformatcs Deartmet of Mathematcs, Uversty of Patras GR-654 Patras, Greece Coyrght 9 Elefthera N. Malhoutsak ad Theodoula N. Grasa. Ths artcle s dstrbuted uder the Creatve Commos Attrbuto Lcese, whch ermts urestrcted use, dstrbuto, ad reroducto ay medum, rovded the orgal work s roerly cted. Abstract The toc of ths work s the roblem of solvg systems of olear equatos. A ew method Newto s form s roosed whch retas the mortat advatage of Newto's method to coverge quadratcally ad moreover t s deal for mrecse fucto roblems. It may be cosdered as a mrovemet of IWFEN method, reseted revous work. A elargemet of radus covergece area s roved based o a Des modfed Newto-Katorovch theorem makg the method to be workg better tha Newto s method ots far away from the soluto. The umercal results are romsg eve cases wth sgular or llcodtoed Jacoba matrces. Mathematcs Subject Classfcato: 65H, 65H5 Keywords: Imrecse Fucto Values, Newto's Method, Nolear Systems, Pvot Pots, Quadratc Covergece Itroducto The roblem of solvg systems of olear equatos s of great sgfcace the felds of comutatoal scece, egeerg ad mechacs ad may realworld roblems fall to ths research sectrum []. It s defed as:

2 Elefthera N. Malhoutsak ad Theodoula N. Grasa where F x (),,, : F f f f D s cotuously dfferetable o a oe eghborhood D D of a soluto,,, x x x x D of the system (). Newto's method costtutes the most used ad famous local method for solvg such roblems, gve by the followg teratve scheme: where,,, x x F x F x,,,, () x x x x s a aroxmato of the soluto x a -terato ad F s the Jacoba matrx of F for all x x, x,, x. It s a owerful techque due to ts quadratc covergece, whch s esured wheever a suffcetly good tal aroxmato ad a osgular Jacoba matrx are avalable. If these codtos are ot vald, the alcato of Newto's method s restrcted, sce a slow covergece or dvergece may be occurred [, 3, 4, ]. A class of modfcatos of Newto's method to cotrbute reducg the above drawbacks s the DR methods [7, 8], whch are also of quadratc covergece. Ther key ot s the usage of vot ots, ots roerly selected from the soluto surfaces of the fuctos volved the system () [4, 5, ]. For cases of accurate fucto values or fucto values wth hgh comutatoal cost, mrecse fucto methods have bee develoed, such as [6], [] ad [9]. DR methods are also deal for mrecse fucto roblems. I ths work, the roosed method costtutes a ew modfed Newto s scheme for solvg systems of olear equatos, whch retas the quadratc covergece of Newto s method ad smultaeously t s deal for mrecse fucto roblems. The dervato of the method s smlar to Newto's method, by costructg a ew lear system. Ths costructo deeds o how the vot ots wll be utlzed Taylor's exaso to determe the aroxmatos of the volved fuctos system (). The roosed method costtutes a mroved modfcato of IWFEN method reseted revous work [] whch s also deal for mrecse fucto roblems. The dfferetato from IWFEN method s that the Jacoba matrx s aroxmated by aother oe evaluated at vot ots. A tal verso of the roosed method was troduced []. Furthermore, a study o the radus of covergece area takes lace. The theorem reseted cocers the elargemet of the radus of covergece area of the ew method comared wth Newto's method ad ts roof s based o a Des modfed Newto-Katorovch theorem [], [5]. Numercal results show the suerorty of the roosed method comared wth Newto s oe. Certaly, ts cost er terato s hgher tha the oe of Newto's method due to the comutato of vot ots. Nevertheless, the accelerato of the covergece whch haes may cases makes the overall erformace of the roosed method better, cludg roblems wth sgular or ll-codtoed Jacoba matrx. Addtoally, the roosed method s comared wth IWFEN method ad from the umercal results we ca say that the ew method costtutes

3 A effcet wthout drect fucto evaluatos Newto s method 3 a mrovemet of IWFEN method. The rest of the aer s orgazed as follows: I secto the aalyss ad the dervato of the roosed method s reseted. The secto 3 covers the covergece theorem of the ew method whle at secto 4 ts algorthm s llustrated. I secto 5 uder secfc codtos the elargemet of the radus of covergece area of the roosed method comared wth the oe of Newto's method takes lace. I secto 6, umercal alcatos show the effcecy of the ew roosed schema. The last secto summarzes the cotrbuto of ths aer. The Proosed Method As t s kow, Newto's method s obtaed by lear aroxmatos of the volved fuctos f x through Taylor's exaso, whch -terato are gve by: j j j j f x f x f x x x,,,,,,,, (3) wth x to be the curret terato. As a result of the above, the Newto's terato () s roduced. Remag to the basc dea of Newto's method to aroach learly each olear equato of the system (), our dea s the dervato of ew lear u, u,, u,,, u,, aroxmatos. I order to acheve ths, dfferet ots,,,,,, for each fucto comoet f x stead of x are utlzed. The roduced ew lear aroxmatos accordg to Newto's methodology are the followg: j j j j,,, f x f u f u x u. (4) Havg the exerece from revous work ad research, a good choce for the dervato of the roosed lear aroxmatos seems to be the usage of vot,, ;, y u,, u,,, u, x, x,, x ots u xvot y x, wth ad the -th comoet equatos: Thus: x results from the soluto of the oe-dmesoal, f x, x,, x, x. (5),, f x f x, x,, x, x. vot (6) If Eq. (5) has soluto, t s obvous that the vot ots soluto surfaces of f x are lyg o the, vot x ad moreover they le o a le arallel to x -axs. If the ukow comoet x does ot aear to ay f the these f are substtuted

4 4 Elefthera N. Malhoutsak ad Theodoula N. Grasa by a lear combato wth other fucto comoets whch cota the x. For locatg the vot ots, stead of the ukow comoet x ay comoet x t, t,,, ca be selected for all fuctos f [9, 4, 5, ]. Ay method ca be used for the soluto of Eq. (5) but ths work we have selected to use the oe-dmesoal secto method, whch s based o the algebrac sgs of fucto values. I ths method, order to solve a oedmesoal equato :, s cotuous, the t, where t t sg t sg t h /, k,, k k k k teratve scheme whch s used s, f wth t ad h ad for ay real umber : sg, f., f For more detals regardg ths method see [7, 8, 8]. Moreover, case of accurate fucto values, there s o ssue for obtag the vot ots from (5) as the oe-dmesoal secto method demads oly the algebrac sgs of fucto values ad ot the exact fucto values [8, ].. Modfed Imroved Wthout drect Fucto Evaluatos Newto Method The roosed method s referred as MIWFEN (Modfed Imroved Wthout drect Fucto Evaluatos Newto) method. It retas the mortat advatage of Newto's method to coverge quadratcally ad moreover t s deal for mrecse fucto roblems. For the dervato of MIWFEN method, exadg the fuctos, f x,,,,,, about the corresodg vot ots xvot,,,, holdg oly frst order of formato from the Taylor seres, we have:,,,, vot j vot j j vot j f x f x f x x x f x x x,,, j vot j j vot j f x x x f x x x (7) due to (6). Takg x x the relato (7), we have a aroxmato of f x,, f x f x x x (8) vot ad from ths relato, a aroxmato of the -th comoet ots s gve by: Substtutg (9) to (7), we have: x f x, x, f xvot x of vot,. (9)

5 A effcet wthout drect fucto evaluatos Newto s method 5 j vot j j j, f x f x x x f x. () Usg the relatos (3) ad (6) for x x we have the followg, vot aroxmatos of f x :, f x f x x x. () Substtutg () to (), we have at last the ew aroxmatos of fucto comoets: j vot j j j,, f x f x x x f x x x. () Thus, the ew lear aroxmated system of the roosed method s gve by:,, j vot j j j whch matrx form s as follows: f x x x f x x x (3), vot T L l, l,, l wth, where F x x x L x, (4) l x f x x x. So, the teratve scheme of MIWFEN method s the followg: vot, x x F x L x,,,,,,,. (5) 3 The Covergece of the Proosed Method I order to study easer the covergece of the roosed method, we wll gve system (3) aother form. To do ths, we cosder the mag T Q q, q,, q : D, wth: j vot j j j, q x f x x x l x,,,,,,, (6) ad thus, our system (3) s gve by For q x. x x, the exresso (6) becomes q x l x ad so: Q x L x. (7)

6 6 Elefthera N. Malhoutsak ad Theodoula N. Grasa Obvously, the Jacoba matrx of fucto Q s the followg:,,, vot vot vot,,, f x f x f x f x f x f x Q x Fx,,, f xvot f xvot f xvot x,,,. vot vot vot, vot, (8) For x x : vot Q x F x,. (9) The teratve scheme (5) ca be equvaletly wrtte the form: x x Q x Q x,,,, () whch s a Newto s form ad thus the covergece theorem of the roosed method that follows, s based o the equvalet theorem of Newto s method. 3. The covergece theorem of the roosed method Theorem 3. Let,,, : F f f f D s twce cotuously dfferetable o a oe eghborhood D D of a soluto,,, x x x x D. The the teratos x,,, of the MIWFEN method gve by () wll coverge to x rovded the Jacoba matrx of F at x s osgular ad also rovded the tal guess x s suffcetly close to x. Moreover the order of covergece wll be two. Proof: Usg the mag Q q, q,, q T : D wth q x gve by (6), the teratos of MIWFEN method are gve by (). As roved above,, Q x s osgular matrx, sce by hyothess Q x F x vot, where F x s osgular. Thus, by the well kow Newto's covergece theorem [3, 5, 6] for our fucto Q x for a tal guess the teratos x,,, coverge to x suffcetly close to x, x ad the order of covergece s two.

7 A effcet wthout drect fucto evaluatos Newto s method 7 4 The Algorthm of the Proosed Method Algorthm : Algorthm of MIWFEN Method Etry: {, x, y,max ter,max ter, tolx, tolf, tolx, tolf, F, F, Q, Q } {where: s the umber of equatos ad ukows, x s the tal terato, y x, x,, x, max ter s the maxmum umber of teratos for secto method, max ter s the maxmum umber of teratos for the roosed method, tolx ad tolf are the desred accuracy values for the termato crtera of the secto method used to comute the -th comoet of vot ots, tolx ad tolf are the desred accuracy values for the termato crtera of the roosed method, F s gve by (), F s the Jacoba matrx of F, Q s the fucto wth comoets matrx of Q } for,,max ter do for,, do for m,,max ter do edfor edf edfor edf edfor q x as defed by (6) ad Q s the Jacoba, call secto algorthm f y x y tolx tolf x,, f ( x m m ; m x tolx f y x tolf ) the retur, Comute Q x ad Solve the lear system Q x v Qx v x x {correcto} f retur, ( ;,,,, ) x { -th comoet of the vot ots} Q x {From relatos (7) ad (9)} x x tolx Q x tolf the x {Number of Iteratos, Aroxmate Soluto} 5 The Elargemet of Radus of Covergece Area The followg theorem cocers the elargemet of the radus of covergece Q x as gve above, comared wth area of MIWFEN method wth Newto's method. The roof of the theorem s based o a Des modfed Newto-

8 8 Elefthera N. Malhoutsak ad Theodoula N. Grasa Katorovch theorem [, 5]. I what follows, K, ad r are referred to the resectve arameters of the Newto's method as they are used ths theorem. Also, ote that Newto's method s aled o the system F x whle the roosed method o the system Q x as gve to the revous secto. Theorem 5. Suose that:,, : Q q q has a cotuous Fréchet secod dervatve o a oe covex subset D of some oe subset D of. Let the followg codtos be satsfed:.,, xd F x vot exsts ad vot. x D, Q x K, wth K K Q x, wth for some F x wth ˆ. x D., ˆ Let addto that, I Qx Fxvot x, wth Let l, ad assume, as log as, vot l defed, that x x. ˆ K h. 5. N x, r D, for Uder these hyotheses the followg hold: x. x x F x Q x s ˆ l r ad a) the sequece x s defed ad each b) x coverges to x N x, r, for whch c) l x ˆ x, ll x x r, Q x, d) the sequece of error bouds of the teratve method (5) s of order l ad e) the radus of covergece r satsfes the equalty r r. Proof: Smlar to the roof of the Theorem.9 [5]. y modfyg the codtos of Theorem 5. ad more secfc the bouds of ˆ, K ad, we coclude to the results lsted Table. Each set of codtos reresets a searate case ths table. Case corresods to the oe that s gve to Theorem 5.. Workg a smlar way, we have the rest of the cases. h.

9 A effcet wthout drect fucto evaluatos Newto s method 9 Secfcally, Cases ad gve r r, Cases 3, 4, 5, ad 6 gve r r ad Cases 7 ad 8 gve r r. Table : Cases of Theorem 5. where r (MIWFEN) s greater tha r (Newto) Case Case Case 3 Set of codtos ˆ K K ˆ K K ˆ K K Cocluso r r Case 4 Case 5 ˆ ˆ K K K K r r Case 6 ˆ K K Case 7 ˆ K K Case 8 ˆ K K r r 6 Results The ew method for solvg systems of olear equatos has bee aled o three test roblems wth dfferet characterstcs as gve [] ad mlemeted usg Fortra rograms o a PC Itel Core 5. Our exerece from these roblems, as summarzed uder the umercal results of Table, shows that the ew method rovdes qute satsfactory results comared wth Newto's oe eve cases of sgular or ll-codtoed Jacoba matrces. For stace, for Examle wth sgular Jacoba matrx, we have dcatvely gve the tal value (3,3,5), for whch Newto s method has a very slow covergece, (6 teratos) whle MIWFEN method eeds oly teratos. Addtoally, MIWFEN method s comared wth the also mrecse fucto method IWFEN [] as t s a modfcato of ths method. From the umercal results we ca say that MIWFEN method costtutes a mrovemet of IWFEN method.

10 Elefthera N. Malhoutsak ad Theodoula N. Grasa Sce the usage of vot ots s crucal the roosed method, order to rema o the soluto surfaces we demad for ther calculato a hgh accuracy ad thus we use the secto method accuracy tolx ad tolf to be 5 for the soluto of the oe-dmesoal equatos whle for the root calculatos 4 the accuracy tolx ad tolf to be. The ext crtera have bee gve uder Algorthm. As otato to Table we troduce: x the tal ot, IT the total umber of teratos requred to obta root, FE the total umber of fucto evaluatos (cludg dervatves), AS the total umber of algebrac, sgs for the comutato of -th comoet x of vot ots ad r the root to whch each method coverges. Examle : The frst system has two roots, r.,.,. ad r.,.,. ad ts Jacoba matrx s osgular. Its dffculty s that at some ots the fucto values caot be acheved accurately. 3 f x, x, x x x x x f x, x, x x x x f x, x, x x x x x () Examle : The secod examle has oe root r , , ad ts Jacoba s sgular. x 4 3 x x3 x3e ,, 3 3 f x, x, x f x, x, x x ( x x ) x ( x x ) f x x x x x Examle 3: The thrd examle s the row's almost lear system for 5. It has the three roots: r,,,,, r r ,, , , ,, , ad ts Jacoba s osgular. The dffculty of ths system s that ts Jacoba s ll-codtoed. f x, x, x, x, x x + x x x x f x, x, x, x, x x + x x x x f x, x, x, x, x x + x x x x f x, x, x, x, x x + x x x x f x, x, x, x, x x x x x x () (3)

11 A effcet wthout drect fucto evaluatos Newto s method Table : Comarso betwee Newto, IWFEN ad MIWFEN Newto IWFEN MIWFEN Ex. x IT FE r IT FE AS r IT FE AS r (.4,.5,.5) r 8 6 r 7 84 r (-4,-,) r r r (-,-,.6) 5 6 r r r (,-,-) r r r (,,) 4 54 r r 4 6 r (3,3,3) 464 r r 4 6 r (3,3,5) 9 4 r r 4 6 r (3,3,5) r r 4 6 r (4,4,4) r r 4 6 r (-8,-3,4,,.5) r r r 3 (,3,4,,.5) r r r (-.,-.,-.,-.,-.) 36 8 r r r 3 (-.,-.,-.,-.,-.) r r r 3 7 Cocluso I ths work, a ew method for solvg systems of olear equatos s troduced whch costtutes a mroved modfcato of IWFEN method. Its key feature s the dervato of ew lear aroxmatos of the fucto comoets f x, whch are used to costruct ew lear system havg as dfferetato from Newto's method the usage of other ots, the vot ots. The roosed method, o oe had t retas the quadratc covergece of Newto s method due to ts Newto s form ad o the other had t s arorate for mrecse fucto roblems. Deste the extra comutatoal cost due to the usage of vot ots, the umercal results show a fast covergece ad combato wth the elargemet of radus of covergece area whch s roved, the ew method seems romsg.

12 Elefthera N. Malhoutsak ad Theodoula N. Grasa Refereces [] J.E. Des Jr., O Newto-Lke Methods, Numersche Mathematk, (968), o. 4, htts://do.org/.7/bf66685 [] J.E. Des Jr ad R.. Schabel, Numercal Methods for Ucostraed Otmzato ad Nolear Equatos, Pretce-Hall, Eglewood Clffs, N.J., 983. [3] P. Deuflhard, A Modfed Newto Method for the Soluto of Ill- Codtoed Systems of Nolear Equatos wth Alcato to Multle Shootg, Numersche Mathematk, (974), o. 4, htts://do.org/.7/bf46969 [4] T.N. Grasa, Imlemetg the Italzato-Deedece ad the Sgularty Dffcultes Newto's Method, Tech. Re. 7-3 (7), Deartmet of Mathematcs, Uversty of Patras. [5] T.N. Grasa, A modfed Newto drecto for ucostraed otmzato, Otmzato, 63 (4), o. 7, htts://do.org/.8/ [6] T.N. Grasa ad E.N. Malhoutsak, Newto's method wthout drect fucto evaluatos, Proceedgs of the 8th Hellec Euroea Coferece o Comuter Mathematcs ad ts Alcatos (HERCMA'7), E.A. Ltaks ed., Athes, Hellas, (7). T.N. Grasa ad M.N. Vrahats, The mlct fucto theorem for solvg systems of olear equatos, Iteratoal Joural of Comuter Mathematcs, 8 (989), 7-8. htts://do.org/.8/ [8] T.N. Grasa ad M.N. Vrahats, A dmeso-reducg method for solvg systems of olear equatos, Iteratoal Joural of Comuter Mathematcs, 3 (99), 5-6. htts://do.org/.8/ [9] T.N. Grasa ad M.N. Vrahats, A dmeso-reducg method for ucostraed otmzato, J. Comut. Al. Math., 66 (996), o. -, htts://do.org/.6/377-47(95)74-3 [] C. Lu, New Evolutoary Algorthm for Solvg Nolear System of Equatos, Joural of Iformato & Comutatoal Scece, (3), o. 3, htts://do.org/.733/jcs85

13 A effcet wthout drect fucto evaluatos Newto s method 3 [] E.N. Malhoutsak, T.N. Grasa ad I.A. Nkas, Imroved Newto's method wthout drect fucto evaluatos, Joural of Comutatoal ad Aled Mathematcs, 7 (9), o., 6 -. htts://do.org/.6/j.cam [] E.N. Malhoutsak ad T.N. Grasa, A Effcet wthout Drect Fucto Evaluatos Newto s Method for Solvg Systems of Nolear Equatos, I: ook of Abstracts of XIII alka Coferece o Oeratoal Research (ALCOR 8), M. Martć, G. Savć, ad M. Kuzmaovć, Eds, elgrade, Serba, 8. [3] J.M. Ortega, Numercal Aalyss, Academc Press, New York, 97. [4] J.M. Ortega ad W.C. Rheboldt, Iteratve Soluto of Nolear Equatos Several Varables, Academc Press, New York, 97. [5] A.M. Ostrowsk, Soluto of Equatos Eucldea ad aach Saces, Academc Press, 3rd ed., New York ad Lodo, 973. [6] J.F. Traub, Iteratve Methods for the Soluto of Equatos, Pretce-Hall, Eglewood Clffs, N.J., 964. [7] M.N. Vrahats, Algorthm 666: Chabs: amathematcal software ackage for locatg ad evaluatg roots of systems of olear equatos, ACM Trasactos o Mathematcal Software (TOMS), 4 (988), o. 4, htts://do.org/.45/ [8] M.N. Vrahats ad K.I. Iordads, A Rad Geeralzed Method of secto for Solvg Systems of No-lear Equatos, Numersche Mathematk, 49 (986), o., htts://do.org/.7/bf3896 [9] M.N. Vrahats, D.G. Sotrooulos ad E.C. Tratafyllou, Global Otmzato for Imrecse Problems, Develomets Global Otmzato, I. omze, T. Csedes, R. Horst ad P.Pardalos, (Ed.), Srger Kluwer, The Netherlads, 37-54, 997. htts://do.org/.7/ _3 [] M.Y. Wazr, W.J. Leog, M.A. Hassa ad M. Mos, A Effcet Solver for Systems of Nolear Equatos wth Sgular Jacoba va Dagoal Udatg, Aled Mathematcal Sceces, 4 (), o. 69, Receved: December 5, 8; Publshed: Jauary 8, 9