A New Newton s Method with Diagonal Jacobian Approximation for Systems of Nonlinear Equations

Transcription

1 Joural of Mathematcs ad Statstcs 6 (3): 46-5, ISSN Scece Publcatos A New Newto s Method wth Dagoal Jacoba Appromato for Systems of Nolear Equatos M.Y. Wazr, W.J. Leog, M.A. Hassa ad M. Mos Departmet of Mathematcs, aculty of Scece, Uversty Putra Malaysa 434 Serdag, Malaysa Abstract: Problem statemet: The maor weaesses of Newto method for olear equatos etal computato of Jacoba matr ad solvg systems of lear equatos each of the teratos. Approach: I some etet fucto dervatves are qut costly ad Jacoba s computatoally epesve whch requres evaluato (storage) of matr every terato. Results: Ths storage requremet became urealstc whe becomes large. We proposed a ew method that appromates Jacoba to dagoal matr whch ams at reducg the storage requremet, computatoal cost ad CPU tme, as well as avodg solvg lear equatos each teratos. Cocluso/Recommedatos: The proposed method s sgfcatly cheaper tha Newto s method ad very much faster tha fed Newto s method also sutable for small, medum ad large scale olear systems wth dese or sparse Jacoba. Numercal epermets were carred out whch shows that, the proposed method s very ecouragg. Key words: Nolear equatos, large scale systems, Newto s method, dagoal updatg, Jacoba appromato INTRODUCTION Cosder the system of olear equatos: () = () where, () : R R wth the followg propertes: There est * wth (*) = s cotuously dfferetable a eghbourhood of * '(*) = J (*) The most well-ow method for solvg (), s the classcal Newto s method. However, the Newto s method for olear equatos has the followg geeral form: Gve a tal pot, we compute a sequece of correctos {s } ad terates { } as follows: Algorthm CN (Newto s method): where, =,,... ad J ( ) s the Jacoba matr of, the: Stage : Solve J ( ) s = -( ) Stage : Update = s Stage 3: Repeat - utl coverges. Correspodg Author: M.Y. Wazr, Departmet of Mathematcs, aculty of Scece, Uversty Putra Malaysa 434 Serdag, Malaysa 46 The covergece of Algorthm CN s attractve. However, the method depeds o a good startg pot (Des, 983). Newto s method wll coverges to * provded the tal guess s suffcetly close to the * ad J (*) wth J () Lpchtz cotuous ad the rate s quadratc (Des, 983),.e.: * h * () or some h. Eve though t has good qualtes, CN method has some maor shortfalls as the dmeso of the systems creases whch cludes (Des, 983) for detals): Computato ad storage of Jacoba each terato Solvg system of lear equatos each terato More CPU tme cosumpto as the equatos dmeso creases There are several strateges to overcome the above drawbacs. The frst s fed Newto method,.e., by settg J ( ) J ( ) for >. ed Newto s the easest ad smplest strategy to overcome the shortfalls

2 J. Math. & Stat., 6 (3): 46-5, for systems of olear equatos ad t follows the followg steps: Algorthm N (fed Newto): Let be gve: Step : Solve J ( )s = -( ) Step : Set = s for =,,... N method dmshes both the computato of the Jacoba (ecept for the frst terato) as well as avodg solvg lear system each terato but s sgfcatly slower (Natasa ad Zora, ). The secod strategy s eact Newto method. Ths method avods solvg Newto equato (Stage of Algorthm CN) by tag the correcto {s } satsfyg (Dembo et al., 98; Esestat ad Waler, 985): r = J ( )s ( ) Ieact Newto method s gve by the followg algorthm: Algorthm INM (Ieact Newto): Let be gve: Step : d some s whch satsfes: Where: Step : Set: J ( ) s = -( ) γ γ η ( ) = s CPU tme, that s why Newto method caot hadle large-scale system of olear equatos. I ths study we propose a method that reduces computatoal cost, storage requremet, CPU tme ad also elmates the eed for solvg lear system each terato. Ths s made possble by appromatg the Jacoba to dagoal matr. The proposed method s sgfcatly cheaper tha Newto method, so much faster tha ed Newto s method ad s sutable for both small, medum ad large scale systems of equatos. MATERIALS AND METHODS A ew Newto method wth dagoal Jacoba: Cosder the Taylor epaso of () about : () = ( ) '( )( ) o( ) (3) The the complete Taylor seres epaso of () s gve by: ˆ() ( ) '( )( ) o( ) = (4) where, '( ) s the Jacoba of at. I order to corporate correct formato o the Jacoba matr to the updatg matr, from (4) we mpose the followg codto (Albert ad Syma, 7): ˆ( ) = ( ) (5) where, ˆ( ) s a appromated evaluates at. The (4) turs to: where, {η } s a forcg sequece. Lettg η t ( ) ( ) '( )( ) (6) gves Newto method. Aother modfcato s quas-newto s method, the Hece we have: method s the famous method that replaces dervatves computato wth drect fucto computato ad also '()( ) ( ) ( ) (7) replaces Jacoba or ts verse wth a appromato whch ca be updated at each teratos (Lam, 978; Des, 97). There are qute may modfcatos We propose the appromato of '( ) by a troduced to coquer some of the shortfalls dagoal matr..e.: (Dragoslav ad Natasa, 996; Hao ad Q, 8; Natasa ad Zora, ), but most of the modfcatos '( ) D (8) requres to computes ad store a matr (Jacoba) each teratos (Natasa ad Zora, ). where D s a gve dagoal matr, updated at each I some cases whe the umber of equatos s terato. The (7) turs to: suffcetly large t becomes computatoally epesve ad requres evaluato (ad storage) of geerally fully D populated J ( ) of dmeso whch requres more ( ) ( ) ( ) (9) 47

3 Sce we requre D to be a dagoal matr, says D = dag(d, d,..., d ), we cosder to let compoets of ( ) ( ) the vector as the dagoal elemets of D, from (9) t follows that: d ( ) ( ) = () () () Hece: () J. Math. & Stat., 6 (3): 46-5, () D = dag(d ) () for =,,..., ad =,,,...,. Where: ( ) = The th compoet of the vector ( ) ( ) = The th compoet of the vector ( ) () = The th compoet of the vector () = The th compoet of the vector () d = The th dagoal elemet of D respectvely () () We use (4) (to safeguard very small f oly deomator s ot equal to zero () () 8 > for =,,..., else set () d d. I order to demostrate the performace method NDJ, four promet methods are compared ad the comparso was based upo the followg crtero: 48 = () We propose the update for our proposed method (NDJ) as below: = D ( ) () where, D s defed by (), provded () () 8 >. Else set () d () = d for =,,.... Algorthm NDJ: Cosder (): R R wth the same property as (): Step : Gve ad D = I, set = Step : Compute ( ) Step 3: Compute = = D ( ) where D Step 4: If s defed by (), provded () else set d = d () for =,,..., ( ) > () () 8 8 stop else set = ad go to Step RESULTS Number of teratos, CPU tme secods, storage requremet ad robustess de. The methods are amely: NDJ stads for method proposed ths study The Newto method (CN) The ed Newto method (N) The Icomplete Jacoba Newto method (IJN) MRV deotes Newto-le method wth the modfcato of rght-had sde vector The MRV was proposed (Natasa ad Zora, ) ad IJN proposed by (Hao ad Q, 8). The stoppg crtero used s - ( ) 8. We mplemeted the fve methods (CN, N, MRV, IJN ad NDJ) usg MATLAB 7.. All the calculatos were carred out double precso computer. We troduced the followg otatos: N: umber of teratos ad CPU: CPU tme secods. Problem -3 s to show the ftess of our method (NDJ) to small scale ad Problem 4- are for large scale systems wth dese or sparse Jacoba. Problem : Cosder the system of two olear equatos (Des, 983): 3 () = 9 = (,5) Problem : Cosder the system of three olear equatos (Hao ad Q, 8): ( 3 )( ) ( 3) ( 3 )( ) ( 3) () = ( 3 )( 3 ) 3( ) = (3, 3,3) Problem 3: Cosder the system of fve olear equatos (Hao ad Q, 8): ( 5 )( ) ( 3 4) 4 ( 5 )( ) ( 3 4) 4 () = ( 5 )( 3 ) 3( 4) 4 ( 5 )( 4 ) 4( 3) 4 ( 5 )( 5 ) = (.5,3.5,.5,3.5,.5)

4 J. Math. & Stat., 6 (3): 46-5, Problem 4: Sgular Broyde (Gomes-Ruggero et al., 98; Broyde, 965): f () = ((3 h ) ) f () = ((3 h ) ) f () = ((3 h ) ) = ad h = Problem 5: Geeralzed fucto of Rosebroc (Lusa, 994): f () = 4c( ) ( ) f () = c( ) 4c( ) ( ), =,..., f = c( ), =.. ad c = Problem 6: Eted Rosebroc (Hao ad Q, 8): = f () = 4 ( ) ( ) ( ) f () = ( ) 4 ( ) ( ) ( ), =,..., f = ( ) ( ) = (.,,.,,.,...) T Problem 7: Trgoometrc-Epoetal system (Lusa, 994): f () = f () = f () = =, =,..., Problem : Broyde trdagoal (More et al., 98): f () = (3 ) f () = (3 ) f () = (3 ) = Problem : Spedcato4 (More et al., 98): () = { ( ) f eve f odd = (.,...,.,) T DISCUSSION I Table, the robustess de s gve by (Natasa ad Zora, ): t V = f () = 3 5 s( )s( ) f () = 3 5 s( )s( ) 4 ep( ) 3 f () = 4 ep( ) 3 =, =,..., Problem 8: System of lear equato (Hao ad Q, 8): = () = ( )( ), = () = ( )( ) =,,..., = (.53.5,.5,3.5,...) Problem 9: Structured Jacoba problem (Lusa, 994): T 49 where, t umber of success by method ad s the umber of problem attempted by method ad the large value of de shows better result ad the best possble result s. rom Table, t ca be see that our proposed method (NDJ) s Cheaper tha Newto method (CN), ed Newto method (N) ad better tha (IJN) ad (MRV). However our method s slower tha Newto method (CN) but much faster tha ed Newto method (N). I addto CN, MRV ad IJN requred to computes ad stores the Jacoba each terato where as NDJ oly vector storage. Table : Results of problems -3 (umber of terato/cpu tme) Problems CN N MRV IJN NDJ Problem 4/. 6/. 5(α =.5) * 6/. Problem 7/.3 /. *(α =.8) * /. Problem 3 6/.8 59/.4 *(α =.8) * 8/. *: Meas that partcular method fals to coverge

5 J. Math. & Stat., 6 (3): 46-5, Table : Computatoal results for solvg problem 4 (umber of terato/cpu tme) CN N MRV(α =.8) IJN NDJ 5 /.45 69/ /3.65 /.4 /.3 5 / / /6.986 /.46 /.4 8 /.6 968/ /6.45 /.83 3/.5 3/.9 * 789/.974 3/.93 4/.7 3/.3 * 9/5.37 3/. 6/ /.456 * 978/ /.69 /.8 3/.98 * * 35/.38 4/.4 5 * * * 35/.56 4/.66 * * * 35/.99 5/ * * * 64/ /.385 *: Meas that partcular method fals to coverge Table 3: Computatoal results for solvg problem 5 (umber of terato/cpu tme) CN N MRV (α =.8) IJN NDJ 5 4/.37 4/.6 7/.36 /.3 /.9 5 4/.4 4/.3 7/.4 5/.8 3/. 8 4/.48 4/.35 7/.43 7/.3 3/.3 4/.67 4/.59 7/.54 7/.36 6/.5 4/.8 4/.45 7/. /.59 7/.3 5 4/ /.9 7/.83 /.86 7/.59 4/5.49 8/.46 9/4.39 /.989 9/ * * * 4/.46 /.98 * * * 5/.98 /.84 5 * * * 37/.569 7/.956 *: Meas that partcular method fals to coverge Table 4: Computatoal results for solvg problem 6 (Number of terato/cpu tme) CN N MRV (α =.8) IJN NDJ 5 6/.4 45/.3 4/.4 7/.3 4/.3 5 6/.3 69/.8 47/.3 9/. 6/.9 8 6/ /.3 5/.354 8/.4 /. 6/.43 73/.98 5/.9 /.4 7/.34 6/ /.87 5/34.7 6/.64 34/ / / / /.76 54/.99 6/9.64 9/ / / /.4 5 * * * 9/ /.943 * * * 9/.96 98/ * * * 8/.38 4/5.6 *: Meas that partcular method fals to coverge Table 5: Computatoal results for solvg problem 7 (umber of terato/cpu tme) CN N MRV (α =.8) IJN NDJ 5 4/.434 * /.34 4/.5 /.9 5 4/.48 * /.4 7/.36 / /.973 * /.83 7/.749 /.6 4/.56 * /.967 /.84 /.63 5/.7898 * /.47 3/.654 5/ /.39 * 4/.783 5/.534 8/.7 5/6.698 * 5/.354 8/.835 3/.56 5 * * * 3/.94 35/.494 * * * 33/ /.56 5 * * * 5/8.6 58/.86 *: Meas that partcular method fals to coverge Moreover from Table -9, our method (NDJ) also uses test problem, our method (NDJ) shows a promsg less computatoal cost ad less CPU tme tha other four result. We ca observe that as the dmeso of the methods (N, MRV, CN ad IJN), that s why our systems creases, global cost for N, MRV, CN ad method (NDJ) s sgfcatly cheaper tha CN, MRV IJN methods creases epoetally where as our ad IJN ad also much faster tha N ad MRV, ths s method (NDJ) growths learly. Ths s because they all more otceable whe the dmeso creases. I all the requre to compute Jacoba matr each teratos. 5

6 J. Math. & Stat., 6 (3): 46-5, Table 6: Computatoal results for solvg problem 8 (umber of terato/cpu tme) CN N MRV (α =.) IJN NDJ 5 8/.34 43/.9 58/.3 6/.4 8/. 5 9/.54 49/.5 6/.5 8/.8 3/.4 8 9/.67 53/ /.6 3/.4 3/. 9/ /5.8 86/5.47 3/.99 33/.73 9/ /7.64 * 3/.57 34/ /9.55 6/6.863 * 3/.63 36/.37 9/96. 69/ * 34/.48 38/. 5 * * * 34/.39 38/. * * * 34/ / * * * 44/5.7 5/.96 *: Meas that partcular method fals to coverge Table 7: Computatoal results for solvg problem 9 (umber of terato/cpu tme) CN N MRV (α =.3) IJN NDJ 5 5/.4 /. 6/.3 8/.7 /. 5 5/.5 4/. 6/.4 8/.8 6/. 8 5/.3 5/.6 7/.7 /. 8/.8 5/.57 6/.48 8/.53 /.6 8/. 5/.5 6/. 8/.9 3/.59 /.3 5 5/.99 7/.6 8/.35 5/.495 3/.75 5/6.54 7/3.73 9/ /.74 3/ * * * 7/. 4/.863 * * * 3/.59 5/.45 5 * * * 39/9.97 8/4.74 *: Meas that partcular method fals to coverge Table 8: Computatoal results for solvg problem (umber of terato/cpu tme) CN N MRV (α =.8) IJN NDJ 5 9/.4 43/.7 /.35 /.34 4/.8 5 9/.63 54/.753 4/.5 4/.4 4/.36 8 /.4 68/.64 5/.99 4/.67 6/.43 /.97 79/ /.3 6/.84 8/.69 /.64 8/ /.7 7/.6 35/.83 5 /.39 * 8/.97 7/.9 35/.94 /.79 * /.94 3/.8 35/. 5 * * * 3/ /.85 * * * 33/.88 38/.37 5 * * * 5/ /.84 *: Meas that partcular method fals to coverge Table 9: Computatoal results for solvg problem (umber of terato/cpu tme) CN N MRV (α =.5) IJN NDJ 5 8/.37 8/.7 38/.54 4/.3 34/.6 5 8/.4 35/.3 49/.96 43/.35 36/.3 8 8/. 39/.94 63/ /.3 38/.89 8/.79 4/.4 * 45/.67 38/.6 8/.54 47/.463 * 48/.68 39/ /.86 54/.4835 * 5/.68 43/.449 8/.95 6/.865 * 54/.97 45/.65 5 * * * 6/.54 48/.33 * * * 68/.36 48/.59 5 * * * 87/.39 54/.79 *: Meas that partcular method fals to coverge Table : Robustess de table CN N MRV IJN NDJ V Aother advatage of our method over N, MRV, CN ad IJN methods s the storage requremet for the 5 Jacoba matr to oly a vector storage. or stace whe =, CN, MRV ad IJN stores fully populated matr whch s equvalet to 6 memory locatos but NDJ stores vectors equvalets to memory locatos oly, geeral our method has reduced the matr storage to some

7 J. Math. & Stat., 6 (3): 46-5, vector storage oly. These results dcated that NDJ s a mprovemet; wth emphass o elmatg matr storage, reducg computatoal cost ad CPU tme, as well as avodg solvg system of equatos each terato oly. CONCLUSION I ths study, a modfcato of classcal Newto s method for solvg system of olear equatos s preseted. Ths method (NDJ) appromates the Jacoba to a dagoal matr, by replacg the dervatve computato by a drect fucto computato. Hece t reduces the computatoal cost, storage requremets of a matr, CPU tme ad avods solvg lear system each terato. The umercal results of the proposed method (NDJ) are very ecouragg ad t shows that ts global cost creases learly as the dmeso of the olear systems creases whereas CN, N, MRV ad IJN methods t creases epoetally; furthermore t shows that NDJ ca acheve good performace for large olear systems. Therefore to ths ed we ca say that our method (NDJ) s sgfcatly cheaper tha the four methods ad much faster tha N ad MRV. ally we ca cocludes that our method (NDJ) s sutable for small, medum ad large scale systems especally whe the fucto dervatves are costly or ca t be doe precsely. REERENCES Hao, L. ad N. Q, 8. Icomplete Jacoba Newto method for olear equato. Comput. Math. Appl., 56: 8-7. Albert, A. ad J.E. Syma, 7. Icomplete seres epaso for fucto appromato. J. Struct. Multdsc. Optm., 34: -4. Broyde, C.G., 965. A class of methods for solvg olear smultaeous equatos. Math. Comp., 9: Dembo, R.S., S.C. Esestat ad T. Stehaug, 98. Ieact Newto methods. SIAM J. Num. Aal., 9: Des, J.E., 97. O the covergece of Broydes method for olear systems of equatos. J. Math. Comput., 5: Des, J.E., 983. Numercal Methods for Ucostraed Optmzato ad Nolear Equatos. 3rd Ed., Prce-Hall, Ic., Eglewood Clffs, New Jersey, pp: 378. Dragoslav, H. ad K. Natasa, 996. Quas-Newto s method wth correctos. Nov Sad J. Math., 6: 5-7. Esestat, S.C. ad Waler, 985. Choosg the forcg terms eact ewto methos. SIAM J. Sc. Comput., 7: 6-3. Gomes-Ruggero, M.A., J.M. Martez ad A.C. Morett, 98. Comparg algorthms for solvg sparse olear systems of equatos. SIAM J. Sc. Comput., 3: Lam, B., 978. O the covergece of a Quas-Newtos method for sparse olear. Syst. Math. Comp., 3: Lusa, 994. Ieact trust rego method for large sparse systems of olear equatos. J. Optmz. Theory Appl., 8: More, J.J., B.S. Grabow ad K.E. Hllstrom, 98. Testg ucostraed optmzato software. ACM Tras. Math. Software, 7: 7-4. Natasa, K. ad L. Zora,. Newto-le method wth modfcato of the rghthad vector. J. Math. Comp., 7: