A Multistep Broyden s-type Method for Solving System of Nonlinear Equations

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1 Joural of Novel Appled Sceces Avalable ole at 7 JNAS Joural-7-6-/39-5 ISSN JNAS A Multstep royde s-ype Method for Solvg System of Nolear Equatos M.Y.Wazr, M.A. Alyu ad A.Wal 3 - Departmet of Mathematcal Sceces, aculty of Sceces, ayero Uversty, Kao, Kao State, Ngera - Departmet of Mathematcal, aculty of Sceves, Gombe State, Gombe State 3- Departmet of Mathematcal Sceces, aculty of Sceces, ederal Uversty, Looja, Koj State, Ngera Correspodg author: M.Y.Wazr ASRAC: he paper proposes a approach to mprove the performace of royde s method for solvg systems of olear equatos. I ths wor, we cosder the formato from two precedg terates rather tha a sgle precedg terate to update the royde s matr that wll provde suffcet formato appromatg of the Jacoba matr each terato. Uder some sutable assumpto, the covergece aalyss s establshed. he umercal results verfy that the proposed method has clearly ehaced the umercal performace of royde s Method. Keywords: Mult-Step royde, olear systems of equatos, Performace profle. INRODUCION Solvg systems of olear equatos s becomg more essetal the aalyss of comple problems may research areas. he problem cosdered s to fd the soluto of olear equatos (), () Where : D has cotuous frst partal dervatves. he assumpto here s there est a vector,,,, wth ad where s the Jacoba matr of at s assumed to be locally Lpschtz cotuous at. Newto s method s a well ow method for solvg (). he method geerates a teratve sequece gve tal guess va ( ) ( ) () where,,. from a However, despte the fact that the Newto s method s smple to mplemet ad has a quadratc rate of covergece, stll t requres the computato ad storage of Jacoba matr ad ts verse, ad also requres solvg systems of lear equatos at each terato. I practce, computatos of some fuctos dervatves are quet costly ad sometmes they are ot avalable or could ot be doe precsely. I ths case, Newto s method caot be used drectly. It s mperatve to meto that some efforts have already bee carred out order to reduce or elmate the wellow shortcomgs of Newto s method for solvg systems of olear equatos. Such methods clude Chord s Newto s method, eact Newto s method, quas-newto s method [3,,,7]. Oe of the wdely used quas- Newto method for solvg system of olear equatos s royde s method []. he method appromates the

2 Jacoba matr ad ts verse by a dervatve free matr (ow as royde s matr) ad therefore calculatg the dervatve every terato s avoded. royde s method uses the formato from the last terate to compute the curret pot whch maes t has lesser formato of the prevous terates. It wll be ce f we ca corporate the formato of the last two or more terates ad update the royde s matr. y dog ths, the royde matr wll have suffcet formato from the prevous terates ad hece mprove the accuracy of the Jacoba appromato. hs motvated the paper. Wazr et al [] corporated the cocept of two-step to dagoal updatg ad solve system of olear equatos ad they succeeded ehacg the method developed by Leog et al [5]. he ma reaso for developg ths approach s to mprove the accuracy royde s matr va mult-step whch we use the formato from two precedg terates to compute the curret pot. hs paper s orgazed as follows: the et secto descrbes our ew approach; secto 3 gves the covergece aalyss of our proposed method. he umercal results are preseted secto 4 ad the cocluso s gve secto 5.. MULI-SEP ROYDEN-LIKE APPROACH hs secto descrbes our ew mult-step royde-le approach whch geerates a sequece of vectors, (3) va Where s the royde appromato of the Jacoba matr updated each terato va mult-step approach. Our target s to come up wth a matr through a royde updatg scheme. o acheve ths, we mae use of a terpolatg curve the varable space to develop a modfed secat equato whch was derved tally by Des ad Wolowcz [3]. hs s made possble by cosderg some of the most successful two-step methods [5,4,9,6] for more detal. hrough tegratg ths two-step formato, we ca preset a mprove secat equato as follows: s s y y (3) s s y y y lettg ad (4), we have, (4) Sce we corporate the formato from the last two precedg teratos stead of a precedg terato (4) ad v yv, v (5), we requre to buld a terpolatg quadratc curves ad where terpolates the last two precedg terates ad, y ad v terpolates the last two precedg fucto evaluato ad. v, v ad Usg the approach troduced [5], the value of (4) ca be determe by computg the values of v. y lettg, v v v v v j j = ( v ) ( v ) = = = s s s ca be computed as follows: (5) 4

3 v v v = ( v ) ( v ) = = = s s s s s s (6) Let us defe by v v, v v (7) herefore ad hat s, ca be computed by the followg relatos, s s, y y. (4) s gve by update as [8], the Mult-Step royde s matr s obtaed as () Now, the algorthm for our method s as follows; Algorthm. (Mult-Step royde s method (MSM)) Step : Choose a tal guesss et I ad let : 4 : ( ). ( ),. Step Compute If stop where Usg the same approach as [] ad cosderg the royde s Step 3: If : defe ( ). Else f : set s ad y ad go to 5. Step 4: If compute v, v ad va (6) (8) respectvely ad fd ad usg (9) ad () respectvely If set s ad y 4.. Step 5: Let ( ) ad update as defed by (). Stepv6 : Chec f, f yes reta, that s computed by step 5. Else set, Step 7: Set : ad go to... CONVERGENCE ANALYSIS hs secto dscusses the covergece aalyss of our method. he followg theorem s stated wthout proof as t wll be useful provg the superlear covergece of the Mult-step royde update. heorem 3. [4]. Let : that for some be cotuously dfferetable a ope, cove set D, D, s cotuous at ad ( ) s osgular. Let L ad assume be a sequece of 4

4 osgular matrces ad suppose for some D the terates geerated by ( ), remas ' ( ) lm. D ad coverges to at a superlear rate f ad oly f Now, we show the superlear covergece of our method heorem3. Let : be cotuously dfferetable fucto a ope cove set C. Assume that there est postve costats ad ' ( ). such that ad he the sequece geerated by Mult-step royde s update ( ) () () s well defe ad covergece to superlearly. Proof: y theorem (3.), t s eough to show uder the codtos of the theorem that ' ( ) lm (3) rom () ad (3), ( ) ( ) ag orms o both sdes gves ( ) ( ) = ( ) ( ) = ( ) I (4) ( ) ( ) ( ) I (5) Sce I (6) ( u) ( v) ( )( u v) u v u v Ad by the fact that We have ( ) ( ) ( ) ( ) ( ) ( ). Hece (7) 4

5 E ( ). Now let rom (6), we have ( ) E E I Sce E Ad We get E E tr E E = tr 4 tr 4 4 E E E = = E E E E = E E I herefore Hece Sce tr E tr E E I, E E E I E E I E (9) for ay E EI E E Now, by usg (), (8), ad the fact that. () We ca wrte (9) as E that s,, () mples that. () E 4 (8) E 3 E E 3 E E E 4 E E E I, whch s () 43

6 ( ) ut from (3) ca be wrtte as E ad (), we have that 3 4 E E. 4 y summg both sdes, we obta E E E 3 4 E 3 4 E, E (3) E E, ad. hus, Whch hold. herefore as. Hece the proof s complete. 3. NUMERICAL RESULS I ths secto, we aalyze ad compare the performace of MSM wth that of Newto s method NM, ed Newto s method NM ad royde s method M for solvg systems of olear equatos. he algorthms are wrtte MALA7.. (Ra) ad are tested for some classcal bechmar problems. All the problems were ru o a PC wth AMD E-APU wth Radeo(tm) CPU wth.ghz speed. o descrbe the results of these epermets we gve the dmeso of problem (N), the umber of teratos performed(ni) ad the CPU tme (secods). We declare a termato of the methods wheever, ( ) 4 (4) he detty matr has bee chose as a tal appromate Jacoba. he symbol " " s used to dcate a falure due to: () he umber of terato s at least 5 but o pot of that satsfes () s obtaed; () CPU tme secod reaches 5; (3) Isuffcet memory to tate the ru. Dola ad More [] gave a ew tool of aalyzg the effcecy of Algorthms. hey troduced the oto of a performace profle as a meas to evaluate ad compare the performace of set of solvers S o a set P. Assumg t ps, there est s solvers ad p problems, for each problem p ad solvers s, they defed computg tme (the umber of fucto evaluatos or others) requred to solve problem p by solvers s. Requrg a base le for comparsos, they compared the performace o problem p but solver s wth the best performace by ay solver o ths problem; usg the performace rato r ps, tps, m t : ss ps, (5) rm Suppose that a parameter rp, s p, s r for all s chose, ad p, s rm f ad oly f s does ot solve problem p. he performace of solver s o ay gve problem mght be of terest, but we would le to obta a overall assessmet of the performace of the solver, the they defed s( t) szep P: rp, s t p (6) 44

7 hus s() t s the probablty for solver s S r, that a performace rato ps s wth a factor t R of the best possble rato. he fucto s s the (cumulatve) dstrbuto fucto for the performace rato. he s : R, performace profle for a solver s a odecreasg, pecewse costat fucto, cotuous from s () the rght at each breapot. he value of s the probablty that the solver w over the rest of the solvers. Accordg to the above rules, we ow that oe solver whose performace profle plot s o the top rght wll w over the rest of the solvers. elow are the bechmars problems used to test the proposed methods ths research. problem (Artfcal ucto) f ( ) cos,,,, ad.5,.5,,.5 problem (rgoometrc system of eyog, ) f ( ) cos,,,, ad.5,.5,,.5 problem 3(eyog et al, ) f ( ), f ( ) +,,,, ad.5,.5,,.5 problem 4(Artfcal ucto) f ( ) s cos cos,,,, ad.5,.5,,.5 problem 5(eyogetal, ) f ( ),,,, ad.5,.5,,.5 problem 6 f ( ) s 4ep, f ( ) s 4ep +cos ep,,,, ad.5,.5,,.5 problem 7(Artfcal ucto) f ( ),,,, ad.5,.5,,.5 problem 8(Darvsh ad Sh, ) f ( ) e,,,, ad.5,.5,,.5 problem 9(Artfcal ucto) f ( ), f ( ) cos +,,,, ad,,, problem (eyog, ) f ( ) 4,,,, ad.5,.5,,.5 problem (eyog, ) f ( ),,,, ad.5,.5,,.5 problem (Artfcal ucto) f ( ) 4, 3 f( ) 4,,,, ad.5,.5,,

8 problem 3(Artfcal ucto) f ( ) 3 log 3 9,,,, ad.9,.9,,.9 problem 4(Artfcal ucto) f e ( ) cos 93 8, f e ( ) cos 93 8,,,, ad.5,.5,,.5 problem 5(Artfcal ucto) f ( ).5.5,,,, ad.5,.5,,.5 problem 6(Hafzad aghat, ) f ( ) cos,,,, ad.5,.5,,.5 problem 7(Artfcal ucto) s f ( ) 3.66,,,, ad.5,.5,,.5 3 problem 8(Artfcal ucto) f ( ) ep cos,,,, ad.5,.5,,.5 problem 9(Artfcal ucto) f ( ),,,, ad.5,.5,,.5 Problem(Rooseetal.,99) f( ) +,,,, ad.5,.5,,.5 Problem(System of olear equatos) f( ) 3 cos + s,,,, ad,,, Problem(rgoometrc System) f ( ) 3 cos, f ( ) 3 cos,,,, ad.5,.5,,.5 problem 3(System of olear equatos) f ( )., f ( ).,,,, ad.5,.5,,.5 Problem4(System of olear equatos) f ( ) cos s,,,, ad.5,.5,,.5 Problem5(System of olear equatos) f( ) uh log cosh, f uh ( ) log cosh, f uh ( ) log cosh,,,,.5,.5,,.5 ad u h 46

9 elow we gve the tables ad graphs that show the performace of Mult-step royde method comparso wth Newto method (NM), royde s method (M) ad ed Newto method (NM). We deote by MSM the method defe algorthm (.). able. Numercal Results of the methods whe solvg problems -6 NM NM M MSM Problems N NI CPU me NI CPU me NI CPU me NI CPU me P P P P P P able. Numercal Results of the methods whe solvg problems 7-3 NM NM M MSM Problems N NI CPU me NI CPU me NI CPU me NI CPU me P P P P P

10 P P able 3. Numercal Results of the methods whe solvg problems 4- NM NM M MSM Problems N NI CPU me NI CPU me NI CPU me NI CPU me P P P P P P P

11 able 4. Numercal Results of the methods whe solvg problems -5 NM NM M MSM Problems N NI CPU me NI CPU me NI CPU me NI CPU me P P P P P gure. Performace profle of NM, NM, M ad MSM methods term of Number of Iterato 49

12 gure. Performace profle of NM, NM, M ad MSM methods term of CPU tme I the above fgures, the left as of the plot represets the percetage of the test problems for whch a method s the best, whle the rght sde correspods to the percetage of the test problems that were successfully solved by these methods. gure ad represet the performaces profle terms of the umber of terato ad CPU tme (secods) respectvely. ables 3 ad fgures ad have show that usg the two-step approach buldg the royde s updatg scheme has sgfcatly ehaced the performace of the classcal royde method. hs observato s glarg whe cosderg CPU tme ad umber of teratos (NI). I addto, t s worth metog that the result of MSM solvg problem7, whe the dmeso creases, shows that our ew approach becomes a better caddate. 5. CONCLUSION I ths paper, we have preseted a ew method (MSM) for solvg system of olear equatos. Ule the sgle step, the method employs a two-step to update the o-sgular royde s matr appromatg the Jacoba matr. Numercal epermets show strog dcato that our ew approach requres less computatoal cost ad umber of teratos as compared to the NM, NM ad M methods. Hece, we ca wd up that our method (MSM) s a better caddate whe compared wth NM, NM ad M methods solvg system of olear equatos. REERENCES [] C.G. royde. A class of methods for solvg olear smultaeous equatos. Math. Compute. 9: , 965. [] E.D. Dola ad J.J. More. echmarg optmzato software wth performace profles. Math. Program. Ser, 9: 3,. [3] Jr.J.E. Des ad H. Wolowcz. Szgad least-chage secat methods. SIAM Joural o Numercal Aalyss, 3(5):9 34, 993. [4] I.A. Moghrab J.A.ord. Mult-step quas-ewto methods for optmzato. J. Compute. Appl. Math., 5:35 33, 994. [5] I.A. Moghrab J.A. ord. Alteratg mult-step quas-ewto methods for ucostraed optmzato. J. Compute. Appl. Math., 8:5 6, 997. [6] S. harmlt J.A. ord. New mplcte updates mult-step quas-ewto methods for ucostraed optmzato. J.Comput. Appl. Math., 5:33 46, 3. [7] M. Mamat K. Muhammad ad M.Y. Wazr. Abroyde sle method for solvg systems of olear equatos. World Appled Aceces Joural, :68 73, 3. [8] C.. Kelly. Solvg olear equatos wth Newto s method. SIAM, 3. [9] W.J. Leog M. ardad M.A.Hassa. Aew two-step gradet-type method for large- scale ucostraed optmzato, computers ad mathematc swtch applcatos. Computers ad Mathematcs wth Applcatos, 59():33 337,. [] W.J. Leog M.Y. Wazr ad M. Mamat. A two-step matr-free secat method for solvg large-scale systems of olear equatos. Joural of Appled Mathematcs, do:.55// (Artcle ID: ):9pages,. [] K.Natasa ad L.Zora. Newto-le method wth modfcato of the rght-had vector. Joural of Computatoal Mathematcs, (7):37 5,. 5

13 [] C.S.Esestat R.S.Demboad Stehaug. Ieact ewto method. SIAM JNumer.Aal., 9():4 48,98. [3] J.E. Des R..Jr. Schabel. Numercal Methods for Ucostraed Optmzato ad Nolear Equatos. Pretce-Hall, Eglewood Clffs, NJ, 983. [4] S.WENYU ad Y.YUAN. Optmzato heory ad Methods. Sprger Optmzato ad Its Applcatos, 6. [5] M.Y. Wazr W.J.Leog ad M.A. Hassa. A matr-freequas-ewto method for solvg olear systems. Comput. Math. Appl., 6: ,. 5