Effective Modified Hybrid Conjugate Gradient Method for Large-Scale Symmetric Nonlinear Equations

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1 Avalable at Appl Appl Math ISSN: Vol Iue December 07 pp Applcato ad Appled Mathematc: A Iteratoal Joural AAM Eectve Moded Hbrd Cojuate Gradet Method or Lare-Scale Smmetrc Nolear Equato Jamlu Sab'u ad Mohammed Yuu Wazr Abtract Departmet o Mathematc acult o Scece Northwet Uvert Kao Kao Nera abujamlu@malcom Departmet o Mathematcal Scece acult o Scece Baero Uvert Kao Kao Nera mwazrmat@buedu Receved: December 06; Accepted: September 07 I th paper we propoed hbrd cojuate radet method u the cove combato o R ad PRP cojuate radet method or olv Lare-cale mmetrc olear equato va Adre approach wth omootoe le earch Local ormula or obta the cove parameter u Newto ad our propoed drecto wa alo propoed Uder approprate codto lobal coverece wa etablhed Reported umercal reult how that the propoed method ver prom Keword: Bactrac le earch; Secat equato; mmetrc olear equato; Cojuate radet method MSC 00: 90C0 65K05 90C5 49M7 5A8 Itroducto Let u coder the tem o olear equato 06

2 AAM: Iter J Vol Iue December where : R R a olear mapp Ote the mapp ollow aumpto: aumed to at the A here et a A A * R t * 0 a cotuoul deretable mapp a ehborhood o vertble * A4 he Jacoba ' mmetrc * he promet method or d the oluto o the clacal Newto' method whch eerate a equece o terate rom a ve tal pot va { } 0 ' where 0 he attractve eature o th method are rapd coverece ad eae o mplemetato Neverthele Newto' method requre the computato o the Jacoba matr whch requre the rt-order dervatve o the tem I practce computato o ome ucto dervatve are qute cotl ad ometme the are ot avalable or could ot be doe precel I th cae Newto' method caot be appled drectl I th wor we are tereted hadl lare-cale problem or whch the Jacoba ether ot avalable or requre a low amout o torae he bet method CG approach It vtal to meto that the cojuate radet method are amo the popular ued method or ucotraed optmzato problem he are partcularl ecet or hadl lare-cale problem due to ther coverece properte mplct to mplemet ad low torae Zhou ad She05 Not wthtad the tud o cojuate radet method or lare-cale mmetrc olear tem o equato cat ad th what motvated u to have th paper I eeral CG method or olv olear tem o equato eerate a teratve pot { } rom tal ve pot 0 u d where > 0 attaed va le earch ad drecto d are obtaed u d d 0 4 term a cojuate radet parameter h problem uder tud ma are rom a ucotraed optmzato problem a addle pot problem Karuh-Kuh-ucer KK o equalt cotraed optmzato problem the dcrtzed two-pot boudar value problem the dcrtzed ellptc boudar value problem ad

3 08 Jamlu Sab u ad Mohammed Yuu Wazr etc Equato the rt-order ecear codto or the ucotraed optmzato problem whe the radet mapp o ome ucto or the equalt cotraed problem : R R m R 5 m 6 t h z 0 h where a vector-valued ucto he KK codto ca be repreeted a the tem wth z v ad z v z h z v h z 7 where v the vector o Larae multpler Notce that the Jacoba mmetrc or all ee e Ortea ad Rheboldt 970 Problem ca be coverted to the ollow lobal optmzato problem 5 wth our ucto deed b z v z v 8 A lare umber o ecet olver or lare-cale mmetrc olear equato have bee propoed aalzed ad teted the lat decade Amo them the mot clac oe etrel due to L ad uuhma 999 whch a Gau-Newto-baed BGS method developed he lobal ad uperlear coverece are alo etablhed It perormace urther mproved b Gu et al 00 where a orm decet BGS method deed Norm decet tpe BGS method epecall coorporat wth trut reo trate are preeted the lterature whch howed ther moderate eectvee epermetall Yua et al 009 Stll the matr torae ad olv o -lear tem are requred the BGS tpe method preeted the lterature he recet deed omootoe pectral radet alorthm Che ad Che 0 all wth the rame wor o matr-ree he cojuate radet method or mmetrc olear equato ha receved a ood atteto ad tae a approprate prore However L ad Wa 0 propoed a moded lectcher-reeve cojuate radet method whch baed o the wor o Zha et al 006 ad the reult llutrate that ther propoed cojuate radet method prom I le wth th developmet urther tude o cojuate radet are pred or olv lare-cale mmetrc olear equato Zhou ad She 04 eteded the decet three-term pola-rebere-pola o Zha et al 006 or olv b comb wth the wor o L ad uuhma 999 Meawhle the clac pola-rebere-pola ucceull ued to olve mmetrc Equato b Zhou ad She 05 Subequetel Xa et al05 propoed a method baed o well-ow cojuate radet o Haer ad Zha 005 he propoed method covere loball Eteve umercal

4 AAM: Iter J Vol Iue December epermet howed that each over-metoed method perorm qute well Some related paper o mmetrc olear tem are Romero-Cadava et al 0 Sab'u 07 Sab'u ad Sau 06 ad Wazr ad Sab'u 06 I th wor we propoe to preet a hbrd CG method u R ad PRP CG parameter Our atcpato to uet a ood CG parameter that wll lead to a oluto wth le computatoal cot We orazed the paper a ollow: I the et ecto we preet the detal o the propoed method Coverece reult are preeted Secto Some umercal reult are reported Secto 4 all cocluo are made Secto 5 Eectve Moded Hbrd CG Method MHCG h ecto preet eectve moded hbrd cojuate radet method R ad PRP u ome udametal approach o Adre 008 b corporat the oeatve retrcto o the CG parameter ueted b Powell 984 We are motvated b the wor o L ad uuhma 999 e loball ad uperlearl Gau-Newto-baed BGS method or mmetrc olear tem I ther wor a appromate radet obtaed wthout ta the dervatve e 9 ad the earch drecto d produced b olv the lear equato the tepze to be obtaed b ome le earch ad the matr ormula B B d where updated b the BGS B B B B B I vew o the above act we urther preet the cove combato o R ad PRP cojuate radet method to obta: where R 0 H* PRP ad R PRP a calar at 0 B ubttut to 0 we have

5 040 Jamlu Sab u ad Mohammed Yuu Wazr * H U 4 ad our ew drecto become: d d d or equvaletl d 4 However order to uaratee a ood electo o we equate the Newto drecto wth our propoed drecto due to the act that "It remarable that the pot cloe eouh to a local mmzer * the a ood drecto to ollow the Newto drecto" J 5 to et J 6 It well-ow that hereore we obta J 7 B the deto o we arrve at J 8 Multpl 8 b J we have J J J 9

6 AAM: Iter J Vol Iue December ad hece ater ome alebrac mapulato we have J J J 0 Due to the eetal propert o low memor requremet or the CG method we appl the moded ecat equato propoed b Babae-Kaa ad Ghabar 04 J or equvaletl z J Subttut to 0 we obtaed the ollow hbrdzato parameter z z z Replac the term ad b ad 9 repectvel eld z z z 4 Hav derved the CG parameter H* 0 we the preet our drecto a * 0 0 d d d H 5 where * PRP R H 6 ad ve b 4 wth PRP R ad 7

7 04 Jamlu Sab u ad Mohammed Yuu Wazr It vtal to ote that the hbrdzato parameter ve b 4 ma be outde the terval [0] However order to have cove combato 6 we adopt the coderato o Adre 008 the ee that the we let ad the we let all we preet our cheme a < 0 0 > 0 d 8 Moreover the drecto ve b 5 ma ot be a decet drecto o 8 whch cae the tadard wole ad Armjo le earche caot be ued to compute the tepze drectl hereore we ue the omootoe le earch propoed Zhou ad She 04 to compute our tepze Let be cotat ad be a ve potve equece uch that d > 0 > 0 r 0 0 < 9 Let ma r at d d 0 Now we ca decrbe the alorthm or our propoed method a ollow: Alorthm MHCG Step : Gve 0 > 0 0 r 0 ad a potve equece at 9 ad et 0 Step : et a topp crtero I e the top; otherwe cotue wth Step Step : Compute d b 5 Step : Compute b the le earch 0 Step : Compute d Step 6 : Coder ad o to tep 4 5 Coverece Reult h ecto preet lobal coverece reult o hbrd CG method o be wth let u dee the level et a: e 0 o aalze the coverece o our method we wll mae the ollow aumpto o olear tem

8 AAM: Iter J Vol Iue December Aumpto he level et deed b bouded here et uch that cotuou or all I ome ehborhood o the Jacoba Lpchtz cotou e there et a potve cotat uch that * L > 0 N * 0 L or all N Properte ad mpl that there et potve cotat M M ad L uch that Lemma Zhou ad She 04 M J M N L J M N 4 Let the equece be eerated b the alorthm above he the equece covere ad N or all 0 Lemma Let the properte o above hold he we have lm d lm 0 5 lm 0 6 Proo: B 9 ad 0 we have or all > 0 d d 7 B umm the above equalt we obta d 8 0 0

9 044 Jamlu Sab u ad Mohammed Yuu Wazr rom ad the act that ate 9 the ere mple 5 B a mlar wa we ca prove that 6 hold 0 d coveret h he ollow reult how that Moded hbrd CG method alorthm loball coveret heorem Let the properte o above hold he the equece eerated b Moded hbrd CG method alorthm covere loball; that lm 0 9 Proo: We prove th theorem b cotradcto Suppoe that 9 ot true the there et a potve cotat uch that 0 40 Sce J 40 mple that there et a potve cotat at 0 4 Cae : lmup > 0 he b 6 we have lm 0 h ad Lemma how that lm 0 whch cotradct 40 Cae : lmup 0 Sce 0 th cae mple that lm 0 4 B deto o 9 ad the mmetr o the Jacoba we have J

10 AAM: Iter J Vol Iue December J t 0 J dt LM 4 where we ue ad 4 the lat equalt Equato ad/or equalte 9 0 ad 40 how that there et a cotat uch that > B 9 ad we et J t dt MM rom 45 ad 4 we obta LM L 46 h toether wth 4 ad 6 how that lm 0 Clearl thereore rom 46 ad 44 we have z bouded ad z z 0 47 z mea there et a cotat 0 uch that or ucetl lare 48 Aa rom the deto o our * we obta H* 0 M M 49 whch mple there et a cotat 0 uch that or ucetl lare H* 50

11 046 Jamlu Sab u ad Mohammed Yuu Wazr Wthout lo o eeralt we aume that the above equalte hold or all 0 he we et H* d d MM d 5 whch how that the equece at 0 amel d bouded Sce lm 0 the ' r doe ot ' ' ' d > d 5 whch mple that ' d ' ' > ' d 5 B the Mea Value heorem there et 0 uch that ' d ' ' d d 54 Sce have * bouded wthout lo o eeralt we aume B 9 ad 5 we lm d H* * lm lm d 55 where we ue 49 0 ad the act that the equece d bouded O the other had we have lm ' * d 56 * * * Hece rom 5-56 we obta 0 whch mea 0 h cotradct 40 he proo the completed 4 Numercal reult I th ecto we compare the perormace o our method or olv olear Equato wth orm decet cojuate radet method or mmetrc olear equato Xa et al 06 Moded hbrd CG method MHCG: We et ω ω 0 4 α0 00 r 0 ad

12 AAM: Iter J Vol Iue December or the orm decet NDCG cojuate radet method or mmetrc olear equato we et ad θ 0 he code or both MHCGM ad NDCGM method were wrtte Matlab 74 R00a ad ru o a peroal computer 8 GHz CPU proceor ad 4 GB RAM memor We topped the terato the total umber o terato eceed 000 or 0 4 We ue - to repreet alure due oe o the ollow: Memor requremet Number o terato eceed 000 I ot a umber NaN We teted the method o te tet problem wth deret tal pot ad value Problem -7 are rom Zhou ad She 05 whle problem 8 ad 0 are rom La Cruz 006 Problem he trctl cove ucto: e ; Problem ; Problem multple o or / e e Problem 4 he varable bad ucto: 05 5 or 05 Problem 5 he Epoetal ucto:

13 048 Jamlu Sab u ad Mohammed Yuu Wazr Problem 6 roometrc ucto: Problem 7 0 e 0 e ; co co co or j j Problem 8 e e he dcretzed Chadraehar' H-equato: c or j j j 05 wth c [0 ad or I our epermet we tae c 09 Problem 9 he Haboo ucto: 005 Problem 0 he Sular ucto: j j j j j j or

14 AAM: Iter J Vol Iue December able Numercal comparo o MHCG ad NDCG method where e oe Problem P 0 Iter me MHCG Iter NDCG me p 0e E E E E E E E E E E E E E E E E E E E E E E-05 e E E E E E E E p e E E E E E E E E-05

15 050 Jamlu Sab u ad Mohammed Yuu Wazr able cotued MHCG NDCG Problem P 0 Iter me Iter me e E E-05 0e E E E E E E E E E E E E-05 p e E E E E E E E E E E E E E E E E E E E E-05 p4 00e E E E E E E E E-05

16 AAM: Iter J Vol Iue December able cotued MHCG NDCG Problem P 0 Iter me Iter me 000e E E E E E E E E E E E E E E E E-05 p5 e E E E E E E E E E E E E E e E E E E

17 05 Jamlu Sab u ad Mohammed Yuu Wazr able cotued MHCG NDCG Problem P 0 Iter me Iter me E E E E E p6 E E E E E E E E e E E E E E E E E E p7 E E E E E E E E E E E E e E E E E E E E E E E E E-05

18 AAM: Iter J Vol Iue December able cotued MHCG NDCG Problem P 0 N Iter me Iter me P8 E E E E E E E E E E E E E E E E E-05 0e E E E E E E E E E E E E E E E E E E-06 p9 0e E E E e E E E E p0 00e E E E E E E-05

19 054 Jamlu Sab u ad Mohammed Yuu Wazr E E-05 τ ure Comparo o the perormace o MHCG ad NDCG method term o CPU tme τ ure Comparo o the perormace o MHCG ad NDCG method term o umber o terato able lt the umercal reult where Iter ad me tad or the total umber o all terato ad the CPU tme ecod repectvel the orm o the redual at the topp pot Oe ca ee that MHCG olve mot o the problem ucceull whle NDCG aled to olve more tha tet problem ad th a clear dcato that MHCG more ecet tha NDCG compared to the umber o terato ad CPU tme repectvel urthermore o the averae our ver mall whch e that the oluto obtaed a better appromato o the eact oluto compared to the NDCG However rom ure ad oe ca eal ee that our clam juted e le umber o terato ad CPU tme to covere to appromate oluto It mportat to meto that th paper β H obtaed u the cove combato o β R ad β PRP whch qute deret rom our method wazr ad Sab u 05 where β wa obtaed b comb Br ad Mart ez drecto wth clacal Newto drecto However th reearch we propoed a hbrdzato parameter σ [0 ] 4 whch wll uaratee a ood cove combato a ueted badre Cocluo I th paper we developed eectve hbrd cojuate radet method baed o Adre

20 AAM: Iter J Vol Iue December approach o hbrdz CG parameter u well-ow cove combato a Adre 008 Adre 008 A ew cove parameter wa propoed u the propoed drecto th paper ad the amou Newto drecto A moded ecat equato wa ued obta the hbrdzato parameter toether wth the oeatve retrcto o the cojuate radet parameter a ueted b Kaa ad Ghabar 0 he propoed method ha le umber o terato ad CPU tme compared to the et alorthm I addto the teret apect o method that the method a ull dervatve-ree teratve procedure wth lobal coverece propert uder ome reaoable codto Numercal comparo u a et o lare-cale tet problem how that the propoed method ver prom However to eted the method to eeral mooth ad o-mooth olear equato wll be our urther reearch REERENCES Adre N 008 Aother hbrd cojuate radet alorthm or ucotraed optmzato Numercal Alorthm Vol 47 No pp 4-56 Babae-Kaa S ad Ghabar R 04 wo hbrd olear cojuate radet method baed o a moded ecat equato Optmzato Vol 6 No 7 pp Che W ad Che Z 0 Nomootoe pectral method or lare-cale mmetrc olear equato Numercal Alorthm Vol 6 No pp 49-6 Gu G Z L D H Q L ad Zhou S Z 00 Decet drecto o qua-newto method or mmetrc olear equato SIAM Joural o Numercal Aal Vol40 No 5 pp Haer W W ad Zha H 005 A ew cojuate radet method wth uarateed decet ad a ecet le earch SIAM Joural o optmzato Vol 6 No pp 70-9 La Cruz W Martíez J ad Rada M 006 Spectral redual method wthout radet ormato or olv lare-cale olear tem o equato Mathematc o Computato Vol 75 No 55 pp L D ad uuhma M 999 A Globall ad Super learl Coveret Gau--Newto-Baed BGS Method or Smmetrc Nolear Equato SIAM Joural o Numercal Aal Vol 7 No pp 5-7 L D H wa Wa X L 0 A moded letcher-reeve-tpe dervatve-ree method or mmetrc olear equato Numercal Alebra Cotrol Optmzato Vol No pp 7-8 Lu H Yao Y Qa X ad Wa H 06 Some olear cojuate radet method baed o pectral cal ecat equato Computatoal ad Appled Mathematc Vol 5 No pp Ortea J M ad Rheboldt W C 000 Iteratve oluto o olear equato everal varable Socet or Idutral ad Appled Mathematc Powell M J 984 Nocove mmzato calculato ad the cojuate radet method Numercal aal Vol 066 No pp -4 Sprer Berl Hedelber Romero-Cadaval E Spauolo G Garca L Ramo PCA Suto ad Xao WM 0 Grp-coected phovoltac eerato plat compoet operato IEEE Idutral electroc maaze Vol 7 No pp 6-0 Sab'u J 07 Eectve Alorthm or Solv Smmetrc Nolear Equato Joural o

21 056 Jamlu Sab u ad Mohammed Yuu Wazr Cotemporar Appled Mathematc Vol 7 No pp Sab u J ad Sau U 06 A ecet ew cojuate radet approach or olv mmetrc olear equato Aa Joural o Mathematc ad Computer Reearch Vol No pp4-4 Wazr M Y ad Sab'u J 06 A alteratve cojuate radet approach or lare-cale mmetrc olear equato Joural o Mathematcal ad Computatoal Scece Vol 6 No 5 p 855 Wazr M Y ad Sab u J 05 A dervatve-ree cojuate radet method ad t lobal coverece or olv mmetrc olear equato Iteratoal Joural o Mathematc ad Mathematcal Scece Vol 05 Xao Y Wu C ad Wu S Y 05 Norm decet cojuate radet method or olv mmetrc olear equato Joural o Global Optmzato Vol 6 No 4 pp Yua G Lu X ad We Z 009 BGS trut-reo method or mmetrc olear equato Joural o Computatoal ad Appled Mathematc Vol 0 No pp Zha L Zhou W ad L D 006 Global coverece o a moded letcher Reeve cojuate radet method wth Armjo-tpe le earch Numerche Mathemat Vol 04 No 4 pp Zha L Zhou W ad L D H 006 A decet moded Pola Rbère Pola cojuate radet method ad t lobal coverece IMA Joural o Numercal Aal Vol 6 No 4 pp Zhou W ad She D 05 Coverece properte o a teratve method or olv mmetrc o-lear equato Joural o Optmzato heor ad Applcato Vol 64 No pp Zhou W ad She D 04 A eact PRP cojuate radet method or mmetrc olear equato Numercal uctoal Aal ad Optmzato