An Improved Newton's Method Without Direct Function Evaluations

Transcription

1 Ge. Math. Notes, Vol., No., February, ISSN 9-784; Coyrght ICSRS Publcato, Avalable ree ole at htt:// A Imroved Newto's Method Wthout Drect Fucto Evaluatos Behzad Ghabar, ad Mehd Gholam Porshokouh Deartmet o Mathematcs, Takesta Brach, Islamc Azad Uversty, Takesta, Ira E-mal: b.ghabary@yahoo.com Deartmet o Mathematcs, Takesta Brach, Islamc Azad Uversty, Takesta, Ira E-mal: m.gh.orshokouh@tau.ac.r (Receved: 4-- /Acceted: 3--) Abstract Due to the act that systems o olear equatos arse requetly scece ad egeerg, they have recetly attracted researchers terest. I ths work, we reset a ew Newto-lke aroach whch s deedet o ucto evaluato ad has bee rovded usg a orgal dea that mroves some detos ad otos o a recetly roosed method [] or solvg systems o olear. Also, the covergece o roosed method has bee dscussed. The comutatoal advatages ad covergece rate o the roosed method are also tested va some umercal eermets. From the obtaed umercal results t seems that reset aroach aect cosderably the overall erormace relato to Newto's method ad ts aoremetoed varats. Keywords: Iteratve methods, System o olear equatos, Newto s method Itroducto Let us cosder the roblem o dg a real zero, say = (,,, ), o a system o olear equatos Corresodg author

2 A Imroved Newto's Method Wthout Drect 65 (,,, ) =, (,,, ) =, (,,, ) =. () Ths system ca be reerred by F ( ) =, where deretable o a oe eghborhood D D o F = (,,, ) : D. R R s cotuously The most wdely used teratve scheme or solvg systems o olear equatos s Newto s method, gvg by = ( ) ( ), () + F F Where ( F ) deote the Jacoba matr at the curret aromato = (,,, ) ad + s the et aromato. It s well kow that the method has quadratc covergece. Sce there ests o geeral method that yelds a eact soluto o (), recet years, a large umber o algorthms ad methods o solutos o deret orders have bee derved ad studed the lterature. These attemts ad cosderatos are maly because o ther umerous lueces ad alcatos real alcatos such as scece ad egeerg [, 3]. Some aroaches to solvg such systems were roosed usg decomosto method [4], quadrature ormulas [5, 6] ad other techques [7, 8]. I some cases, we may ecouter wth deret kds o (), whle the olear system s kow wth some recso oly, or eamle whe the ucto ad dervatve values deed o the results o umercal smulatos [9], or some cases the recso o the desred ucto s avalable at a rohbtve cost or whe the ucto value results rom the sum o a te seres (e.g. Bessel or Ary uctos [, ]). I [] a method wthout evaluato o olear ucto values s roosed whch ca be aled or olyomal systems. Also, these methods are deal or stuatos wth uavalable accurate ucto values or hgh comutatoal cost [3]. So, t s very mortat to obta methods whch are ree o ucto evaluatos,.e., the requred ucto values are ot drectly evaluated rom the corresodg comoet uctos but are aromated by usg arorate quattes, as have recetly used socalled IWFEN method []. Here, we geeralze some used detos IWFEN method ad the by usg a ew geometrcal terretato a method whch s a ew mroved Newto s method wthout drect ucto evaluatos s reseted. The rest o ths cotrbuto s structured as ollows. I Secto, we develo the ew method ad ts covergece theorem s roved. Some umercal eamles ad some comarsos betwee the results o deret aroaches ad our roosed method are gve Secto 3. Fally, coclusos are draw Secto 4. New Aroach to the Soluto o the Problem To develo a ew method, let us have the ollowg deto o vot ots whch eteds the corresodg estg oes.

3 66 Behzad Ghabar et al. Deto. For ay {,,, } ad =,, ad based o comoets o curret ot, we geeralze the deto o vot ots [] = (,,,, ) (3),, vot as ts ew ollowg orm = (,,,,,, ). (4),, vot j ( ) It s obvous that the oly derece betwee our deto (4) ad (3) s that (3) j ( ) has cosdered to be the value o, but our deto j ( ) {,,, } s ot kow ror ad should be obtaed such way that wll be troduced later. Moreover, we mose the vot ots (4) to be led o a arallel le to j ( ) as, whch asses through the curret ot ay terato o the algorthm. From the deto o the vot ots, t s obvous, that these ots have the same comoets wth the curret ot ad der oly at the j ( ) th comoet., Deto. Let s dee the uctos g : j ( ) R R, as g ( t ) = (,,,, t,,,, ). (5), j ( ) j ( ) j ( ) + From (5) ad the mosed roerty to vot ots (4), t s evdet that the ukow j ( ) -th, comoet o (4), j ( ), ca be oud by solvg each o the corresodg oe dmesoal equatos g t = (6), j ( ) ( ) Accordg to Imlct Fucto Theorem [4] there est uque mags φ such that = φ y y φ y = ad thereore j ( ) ( ), ( ; ( )), j ( ) = φ ( y ). Where y = j ( ) j ( ) + (,,,,,,, ). Smlar [], ths aer the sg-ucto based method [5] s used or solvg the corresodg oe-dmesoal equatos. It s clear that the soluto o (6) s deedg o the eresso o the comoets ad the curret aroach. That s, ay o the Eq. (6) has o zeros, we are ot able to aly our roosed method o a system o equatos. Here, smlar to what was dscussed [], we ca adoted some techques to guaratee the estece o vot ots. For eamle choosg a lear combato betwee the comoets lke [6] or alyg ether a reorderg techque lke [7]). For the eeds o ths work we cosder that we are always able to d the zeros o (6) s ossble. The key dea ths aer s to substtute the ucto value o F ( ) = ( ( ), ( ),, ( ), ( )), Newto s method () to the ts sutable

4 A Imroved Newto's Method Wthout Drect 67 aromato. Hece, let us use the rst orders Taylor easo o g aroud the ot j ( ) t = as j ( ) dg g t g dt t,,, j ( ) ( ) ( ( ) ( ) ( ) ) ( )( ). j j j + j ( ) j ( ), Settg t = at (7) ad vew o (5), we have j ( ) (7) g ( ) ( ) + ( )( ) (8),,, j ( ) j ( ) j ( ) j ( ) j ( ),, Due to the roerty o vot ots (.e. g ( ( ) ( ) ) j j =, the relato (8) becomes ( ) ( )( ) (9), j ( ) j ( ) j ( ) The relato (9) s so mortat or the develomet o our aroach, because usg (9) () wll trasorm t to a ew Newto's method, whch wll ot deed drectly o the ucto, values ( ), but o j ( ) ( ) ad the -th comoets o the ots ad vot., I Fg., we ca see the behavor o ucto g ( t ) or ay j ( ) {,,, }. I we brg rom j ( ), the vot ot B = ( vot, j ( ),), the arallel le to the taget o the ucto at the ot P = ( j ( ), ( )). From the smlar tragles, the ucto value ( ), deoted by the, segmet AP, ca be aromated by the quatty j ( ) ( )( j ( ) ), deoted by the j ( ) segmet AQ. Ths lot ca gve us a good dea to choose a sutable drecto o j ( ).Usg smlarty tragles, t ca be vered that wheever segmet BC has a ewer legth, the, aromato j ( ) ( )( j ( ) ) stead o ( ) j ( ) s more vald. So, we should choose that drecto j ( ) whch mmzes the eresso ( ) BC = j ( ) ( ) j ( ), j ( ) () From tragular equalty (), we have BC, j ( ) j ( ) + j ( ) ( ) ( ) () It s clear that, the eresso ( ) mmzg wheever the umerator eresso ( ) j ( ) acheves ts mamum value. Hece, ths aer we let j ( ) = J whereas or ay k {,,, }, J be the smallest de whch we have ( ) ( ),.e. that drecto whch has the steeest sloe amog the J k comoets o the gradet vector o ucto at the ot Now, usg (9) Newto method (), we have V ( ) L ( ) F ( )( ). + = ()

5 68 Behzad Ghabar et al. Where V ( ) ( ) j () j () ( ) = ad j ( ) ( ), j () φ ( y ) j () j (), () ( ) j φ y j () j () L ( ) = =., j ( ) φ ( y ) j ( ) j ( ) Uder the assumtos o Imlct Fucto Theorem the dagoal matr V ( ) s vertble ad () becomes V F L ( ) ( )( ) = ( ) Now, we cosder the ucto L ( ) = ( φ ( y ), φ ( y ),, φ ( y )) T (3) j () j () j ( ) Utlzg aga the Imlct Fucto Theorem to derve φ ( ), we have j ( ) ( ) ( ) j () ( ) j () ( ) j ( r ) ( ) ( ) s ( ) ( ) L ( ) = j () ( ) j () ( ) j () ( ) = V ( ) F ( ), ( ) ( ) ( ) j ( ) ( ) j ( ) ( ) j ( ) ( ) (4) Where the etres o occur at j ( ) -th colum or -th row. Fally, Eqs. (3) ad (4) troduce teratve method gve by = ( ) ( ), (5) + L L A smlar covergece theorem to what stated [], ca be reseted ad roved, as ollows. Theorem. Let F = (,,, ) : D R R be sucetly deretable at each ot o a oe eghborhood D o R, that s a soluto o the system F ( ) =. Let us suose that F ( ) s cotuous ad osgular. The the sequece { } k teratve scheme (4) or sucetly close tal guess, coverges to order o two. Proo. Usg the mag l ( ) = φ ( y ), L = ( l, l,, l ) : D R R, where T k obtaed usg the wth covergece

6 A Imroved Newto's Method Wthout Drect 69 + The terato o our method s gve by = L ( ) L ( ) For the above mag the well-kow codtos o Newto's theorem (see [4]) are obvously ullled because o the orm o L ( ), stated (4) ad the roerty o vot ots ϕ ( y ), or =,,,. The covergece theorem or mag L s roved, ad the teratos o (5) coverge to quadratc ally. Usg ths roosed dea, we may eect that our method wll have better covergece tha the method o IWFEN method. The results o eermets Secto 3 wll corm ths cojecture. Also, t should be oted out that the roosed ew rocedure remas the cost o IWFEN method. Ths s because o t does ot eed more comutatoal cost, sce the used ew artal dervatves have already bee evaluated the corresodg Jacoba matr. 3 Numercal Eamles I ths secto, we erorm some umercal eermets or testg the covergece o the teratve method roosed the revous secto. I order to comare the results, we take the same eamles whch were used []. I Tables -, we reset the results obtaed, or varous tal ots by Newto s method, IWFEN ad our ew roosed method. Eamle. The rst system has two roots = (.,.,.) ad = (.,.,.). It s gve by: 3 (,, ) = =, 3 3 (,, ) = =, 3 3 (,, ) = +. = Eamle. The secod eamle s (,, ) = e + =, (,, ) = ( + ) + ( ) =, 3 3 (,, ) = + = Wth the soluto 4 (.9999, =, ). Results were obtaed by usg Male sotware va 3 dgt loatg ot arthmetc (Dgts:=3). The teratve rocess wll sto <. k + k 4 From Tables -, we see that the results o comutato or the roosed teratve method admt the theoretcal order o covergece Theorem. Furthermore the roosed teratve method coverges much aster tha the other comared methods.

7 7 Behzad Ghabar et al., Fg. The behavor o ucto g ( ) ( t ) j 3 Table Comarso betwee deret methods or eamle. Newto IWFEN Preset Method IT FE IT FE IT FE Table Comarso betwee deret methods or eamle. Newto IWFEN Preset Method IT FE IT FE IT FE

8 A Imroved Newto's Method Wthout Drect Cocluso I ths aer we reset a geeralzato o a recetly roosed varat o Newto's method or solvg olear systems. Ths ew method has the order o covergece two ad s deedet o ucto evaluato. Also, the roosed method ca be used some systems where the ucto calculatos are qute costly or caot be doe recsely. As see Tables [-], the umercal results o the roosed method are qute satsactory ad admt the geometrcal elaatos. I some cases the results o ourselves are very accetable ad there s a sucet reducto o the umber o teratos ad hece the roosed method seems to be a relable reemet or Newto s method ad some ts recet modcatos. Reereces [] E.N. Malhoutsak, I.A. Nkas ad T.N. Grasa, Imroved Newto s method wthout drect ucto evaluatos, Joural o Comutatoal ad Aled Mathematcs, 7 (9), 6-. [] R.L. Burde ad J.D. Fares, Numercal Aalyss, (7th ed.), PWS Publshg Comay, Bosto, (). [3] J.M. Ortega, W.C. Rheboldt, Iteratve Soluto o Nolear Equatos Several Varables, Academc Press, (97). [4] E. Babola, J. Bazar ad A.R. Vahd, Soluto o a system o olear equatos by Adma decomosto method, Al. Math. Comut., 5 (4), [5] M. Frot ad E. Sorma, Thrd-order methods rom quadrature ormulae or solvg systems o olear equatos, Al. Math. Comut., 49 (4), [6] M.T. Darvsh ad A. Barat, Suer cubc teratve methods to solve systems o olear equatos, Al. Math. Comut., 88 (7), [7] D.K.R. Babajee, M.Z. Dauhooa, M.T. Darvsh, A. Karam ad A. Barat, Aalyss o two Chebyshev-lke thrd order methods ree rom secod dervatves or solvg systems o olear equatos, Joural o Comutatoal ad Aled Mathematcs, 33 (), -.

9 7 Behzad Ghabar et al. [8] J. Bazar ad B. Ghabary, A ew aroach or solvg systems o olear equatos, Iteratoal Mathematcal Forum, 3(38) 8, [9] M. Kuerschmd ad J.G. Ecker, A ote o soluto o olear rogrammg roblems wth mrecse ucto ad gradet values, Math. Program. Study, 3 (987), [] M.N. Vrahats, T.N. Grasa, O. Ragos ad F.A. Zarooulos, O the localzato ad comutato o zeros o Bessel uctos, Z. Agew. Math. Mech., 77(6) (997), [] M.N. Vrahats, O. Ragos, F.A. Zarooulos ad T.N. Grasa, Locatg ad comutg zeros o Ary uctos, Z. Agew. Math. Mech., 76 (7) (996), [] W. Che, A Newto method wthout evaluato o olear ucto values, CoRR, cs.ce/996, (999). [3] T.N. Grasa ad E.N. Malhoutsak, Newto's method wthout drect ucto evaluatos, : E. Ltaks (Ed.), Proceedgs o 8th Hellec Euroea Research o Comuter Mathematcs & ts Alcatos-Coerece, HERCMA 7, (7). [4] J.M. Ortega ad W.C. Rheboldt, Iteratve Soluto o Nolear Equatos Several Varables, Academc Press, New York, (97). [5] T.N. Grasa ad M.N. Vrahats, A dmeso-reducg method or solvg systems o olear equatos, It. J. Comut. Math.,3 (99), 5-6. [6] T.N. Grasa ad M.N. Vrahats, A dmeso-reducg method or ucostraed otmzato, J. Comut. Al. Math., 66(-) (996), [7] D.G. Sotrooulos, J.A. Nkas ad T.N. Grasa, Imrovg the ececy o a olyomal system solver va a reorderg techque, : D.T. Tsa-hals (Ed.), Proceedgs o 4th GRACM Cogress o Comutatoal Mechacs, III ().