Some Families of Higher Order Three-Step Iterative Techniques. where is a real number and y (5)
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1 Lif Scic Jural 03;0s Sm Familis f Highr Orr Thr-Stp Itrativ Tchiqus Nair Ahma Mir Sahr Akmal Kha Naila Rafiq Nusrut Yasmi. Dpartmt f Basic Scics Riphah Itratial Uivrsit Islamaba Pakista Ctr fr Avac Stuis i Pur a Appli Mathmatics Bahaui Zakaria Uivrsit Multa Pakista. airahma.mir@gmail.cm Abstract: W suggst a aal fur w familis f highr rr thr-stp itrativ tchiqus fr slvig liar quatis. Each f ths familis has svth rr cvrgc. Pr itrati ths familis f mths rquir thr fucti a f its rivativ valuatis. Thus ach f ths familis has.67 cmputatial fficic. Svral ampls ar giv t illustrat th prfrmac a fficic f ths mths. Cmparis with thr similar mths is als giv [Nair Ahma Mir Sahr Akmal Kha Naila Rafiq Nusrut Yasmi. Sm Familis f Highr Orr Thr-Stp Itrativ Tchiqus. Lif Sci J 03;0s:43-3] ISSN: Kwrs: Itrativ mths; Thr-stp mths; Prictr-Crrctr tp mths; Cvrgc rr. Itructi Slvig -liar quati is f th mst imprtat prblms i umrical aalsis. Nwts mth is th mst sigificat itrativ mth fr slvig -liar quati. A -liar quati cat b slv i gral aalticall. Thrfr a umbr f apprimat tchiqus hav b vlp t cmput th rs f -liar quatis. Ths tchiqus hav b vlp usig rrr aalsis Amia Dcmpsiti mths Quaratur frmulas tc. I rct ars mathmaticias hav vlp ma tw-stp a thr-stp itrativ mths s [-8]. I this papr fur w familis f highr rr thrstp mths ar vlp usig variats f Chagbum Chu [] a Li. Tai fag [7] tchiqus b applig bimial apprimati thm a usig th stratg f Wihg Bi t al [6]. Cvrgc aalsis f ths familis f mths is iscuss. Each f ths familis is f svth rr cvrgc. Pr itrati. ths familis f mths rquir thr fucti a f its rivativ valuatis. Thus ach f ths familis has.67 cmputatial fficic. Cstructi f familis f thr-stp itrativ mths Csir Nwts Mth: A famil f itrativ mth b Li-Fag t.al [7] is giv b: whr is a ral umbr a is fi b. Csir th apprimati b Chu []: 0 [ ] 4 3 Usig 4 i 3 w btai a w famil f Itrativ Mths whr R a is fi b. Sc famil f mths is btai b applig bimial pasi upt first rr aml [ ] [ ] [ ] [ ] 6 whr R a is giv b. Similarl thir famil f mths is btai usig sc apprimati that is 43 lifscicj@gmail.cm
2 Lif Scic Jural 03;0s [ ] [ ] [ ] [ ] 7 [ ] [ ] whr R a is giv b. Furth famil f mths is giv b usig famil f C.Chu a Y.Ham [8] a applig first bimial apprimati it that is [ ] [ ] 8 4 Furthr csirig th stratg f Wihg t.al [6] w prps thr- stp famil f mths cmbiig quatis a : 9 a G whr f a G t rprsts a ralvalu fucti. Usig th Talr pasi a ca b apprimat as: f f 0 ; this implis f [ ] I rr t avi th cmputati f th sc rivativ w apprimat as fllws: f [ ] f [ ] 3 whr a ar sufficitl cls t wh is sufficitl big itgr. Substitutig 3 it a rplacig with apprimati i w prps th fllwig thr-stp famil f mths frm9: 4 a G f [ ] f [ ] whr f a G t rprsts a ral-valu fucti. Similarl frm 6 a th thir quati f 4 w hav sc famil f thr-stp mths: [ ] [ ] [ ] [ ] a 44 lifscicj@gmail.cm
3 Lif Scic Jural 03;0s G f [ ] f [ ] whr f a G t rprsts a ralvalu fucti. Als frm 7 a thir quati f 4 thir famil f mths is giv b: [ ] [ ] [ ] [ ] [ ] [ ] a 6 G f [ ] f [ ] whr f a G t rprsts a ral-valu fucti. Furth famil is btai b cmbiig 8 a thir quati f 4 : [ ][ ] 7 a 4 G f [ ] f [ ] whr f a G t rprsts a ralvalu fucti. W w iscuss cvrgc fr familis f thr stp mths 4 7. Cvrgc Aalsis Thrm Assum that f : D R R. is a scalar fucti sm p itrval D. Supps D 0. If th iitial pit 0 is sufficitl cls t a G is a fucti with G 0 G 0 G 0 < th th famil f thr-stp mths 4 is f cvrgc rr sv fr a R. Prf. Lt a k. Dt c k k! k 3... Usig th Talr pasi a takig it accut 0 w hav [ c c3 c4 c O ] [ c 3c 3 4c 4 c O ] 9 Frm8 a 9 w gt: c [ c c3 ] [7c c 4c 3 c ] [8c 0c c 6c 0c c 4 c ] O Thus frm w hav: c [c c ] [4c 7c c 3 c ] : [ 8c 0c c 6c 0c c 4 c ] O Agai paig abut w hav: 3 [ : c : O : ] lifscicj@gmail.cm
4 Lif Scic Jural 03;0s [ c c c 3 3 3c c 7 c c c 4c c 0c c 6c 4 c O ]. Frm 89 a w hav [ ] ] c c c 3c 6 c c c c c c 0c 4c 4c 4c 8 c c O 3 Frm w hav: : [ ] [ ] [ ] Usig a 3 i th abv rsult w btai: 3 c c c c 0c c 4c c 8c c c c c O Epaig abut w hav: 3 4 [ c c3 c 4 O ] Thus b first quati f a w hav: f [ ] [ c c O c c O ] 3 4 : : : 3 : : This implis 6 f [ ] [ c c : c : O ] 3 Usig a 4 w hav: f [ ] [c c c c c 7c c3 3 c c 4 c O ] 6 Mrvr b 8 a w hav: f [ ] f [ c c O c c c O ] 4 4 This implis f [ ] [ c c 3 c O 7 Nw ] f [ ] f [ ] f [ ] Th b 9 a 7 w btai: f [ ] [ c c 3 c O ] 3 4 Usig 4 w hav: 3 f [ ] [ c c 3 3 c 4 O ] 8 Equatis 4 furthr givs: f [ ] [ c c 3 3 c 4 ] Usig 4 w btai: f [ ] [ c 4c c c c 3 c 6c c 4 c 0 c c3 O ] 9 Equati 6 a 9 impl: f [ ] f [ ] [ cc 3 6c 3c c3 3c c 4 c 4 c3 O ] 30 Nw frm 8 a w hav: c O lifscicj@gmail.cm
5 Lif Scic Jural 03;0s Usig th Talr pasi a csirig G 0 < w gt G G G O 0 0 Usig 3 w hav: G G G c O Frm 4 w hav c cc 3 c 0c c 4c c 8c 4c c c c O 33 Thus th rrr quati fr famil f mths 4 is giv b: G f [ ] f [ ] G f [ ] f [ ] [ G 0 G 0 c O ][ c O ] c c 6c 3c c 3c c c 4 c O 34 [ G0 G 0 G 0 c G 0 c G 0 c G 0 c...] [ c c 6c 3c c 3c c c 4 c O ] O simplificati w hav: [ G 0] [ G 03 c c c3 c cc 3G 03 c c c3 c ] O 3 Thus th cvrgc rr f th famil 4 is svth rr if G 0 a th rrr quati is giv b [ G 03 c cc 3 c cc 33 c cc 3 c ] O 36 8 With G 0 rrr quati is giv b c c3 9c c 6 c 37 W hav th sam rrr quati fr famil f mths a 6 with svth rr cvrgc. 3 THE CONCRETE ITERATIVE METHODS Th thr-stp famil f mths 4 suggsts sm w mths which ar stat as ur: 3. METHOD Abbrviat as MK Fr th giv fucti G fi b t G t 38 t whr R it ca asil b s that th fucti G t f 38 satisfis th citis f Thrm. Hc w gt a w tw-paramtr svth-rr famil f mths frm 4: 39 f [ ] f [ ] 3. METHOD Abbrviat as MK Fr th fucti G fi b G t t whr 0 it ca asil b s that th fucti G t f 40 satisfis th citis f Thrm. Hc w gt a w tw-paramtr svth-rr famil f mths frm4: lifscicj@gmail.cm
6 Lif Scic Jural 03;0s f [ ] f [ ] 3.3 METHOD 3 Abbrviat as MK3 Fr th fucti G fi b 38 t G t t whr R it ca asil b s that th fucti G t f 38 satisfis th citis f Thrm. Hc w gt a w tw-paramtr svth-rr famil f mths frm : [ ] [ ] [ ] [ ] f [ ] f [ ] 3.4 METHOD 4 Abbrviat as MK4 Fr th fucti G fi b 40 G t 4 t whr 0 it ca asil b s that th fucti G t f 40 satisfis th citis f Thrm. Hc w gt a w tw-paramtr svth-rr famil f mths frm: [ ] [ ] [ ] [ ] 43 f [ ] f [ ] 3. METHOD Abbrviat as MK Fr th fucti G fi b38 t G t t whr R it ca asil b s that th fucti G t f 38 satisfis th citis f Thrm. Hc w gt a w tw-paramtr svth-rr famil f mths frm6: [ ] [ ] [ ] [ ] [ ] [ ] f [ ] f [ ] 3.6 METHOD 6 Abbrviat as MK6 Fr th fucti G fi b 40 G t 44 t whr 0 it ca asil b s that th fucti G t f 40 satisfis th citis f Thrm. Hc w gt a w tw-paramtr svth-rr famil f mths frm6: 48 lifscicj@gmail.cm
7 Lif Scic Jural 03;0s [ ] [ ] [ ] [ ] [ ] [ ] f [ ] f [ ] 4 Thrm Assum that f : D R R is a scalar fucti sm p itrval D. Supps D 0. If th iitial pit 0 is sufficitl cls t a G is a fucti with G 0 G 0 G 0 < th th famil f thr-stp mths 7 is f cvrgc rr sv fr. Prf. Lt a k. Dt c k k! k 3... Usig th Talr pasi a takig it accut 0 w hav: Frm 9a w hav: c c3 4 c 3 c 4 cc3 3 c c 64c c3 c 3 0c c4 4 c O 46 Thus usig 9 a 46 w gt ][ ] 4 c c 7 c 6 c c c 8 6 cc3 3 c c 6 94 c c 0 96 c c 38 c c O Nw frm quati7 w gt: [ ][ ] 4 48 Usig quati46 a 48 a fr w hav: c c c 3c c 36c c c c O 6 49 B Talr Sris abut w hav: 3 4 [ c c3 c 4 O ] Nw f [ ] [ c c O c c O ] 3 4 : : : 3 : : f Usig a 49 w gt: f [ ] [ c c c c c c3 3c c 4 9 c O ] 0 Nw Usig 8 a 49 w hav: f [ ] [ c c 3 c O Als b fiiti ] f [ ] f [ ] f [ ] Usig 9 a w gt: f [ ] [ c c 3 3 c 4 O ] Usig 49 w btai: 3 f [ ] [ c c 3 3 c4 O ] This implis 49 lifscicj@gmail.cm
8 Lif Scic Jural 03;0s f [ ] f c c c [ ] Usig 49 w hav: f [ ] [ c 4 cc c c c c 6cc 4 c O ] Equatis 0 a 3 impl f [ ] f [ ] [ c c 4c 3c c 0 c 3 c c O ] 4 Usig th Talr pasi a csirig G 0 < w gt: G G G O 0 0 Nw frm 8 a 49 w hav: c O Usig this rsult w hav: G G 0 G 0 c O Frm 49 w hav c cc 3 3 cc3 36c c 3 cc 4 O Thus th rrr quati fr famil f mths 7 is giv b [ G0 G 0 c O ][ c O ] c c 0c 3c c 3c c 4 c O 6 3 [ G 0] [ G 0 G 0 c c 3 ] [ G 0 G 0] c [0c 3c c 3c c 4 c ] G 0 O 7 Thus th cvrgc rr f famil f mths 7 is svth-rr if G 0 a th rrr quati is giv b [0 G 0 c c G 0 c c G 0 c 0 c c ] O 8. 8 With G 0 rrr quati is giv b c c3 c O 9 4 Th CONCRETE FAMILY OF THREE-STEP ITERATIVE METHODS Th thr-stp famil f mths 7 suggst sm w mths which ar stat as ur: 4. Mth 7 Abbrviat as MK7 Fr th fucti G fi b t G t 60 t whr R it ca b asil s that th fucti G t f 60 satisfis th citis f Thrm. Hc w gt a w -paramtr svth rr famil f mths frm7: [ ] 6 f [ ] f [ ] 4. Mth 8 Abbrviat as MK8 Fr th fucti G fi b G t 6 t whr 0 it ca asil b s that th fucti G t f 6 satisfis th citis f Thrm. Hc w gt a w -paramtr svth-rr famil f mths [ ] lifscicj@gmail.cm
9 Lif Scic Jural 03;0s f [ ] f [ ] Thus w hav cstruct fur familis f thr-stp mths MK t MK 8. Pr itrati ach f ths familis f mths rquir thr valuatis f th fucti a valuati f its first rivativ. W csir th fiiti f fficic i as p w whr p is th rr f th mth a w is th umbr f fucti valuatis pr itrati rquir b th mth. W bsrv that ach famil f mths has th fficic i 4 7 ;.67. Numrical Eampls Nw th Mths MK t MK 8 ar implmt t slv sm -liar quatis a cmpar with Ku s mth f svth rr []. Tabl shws th iffrc f tw cscutiv itrats fucti valu ttal umbr f fucti valuats TNFE a ttal umbr f itratis. Th cuti is stpp at th thir itrati fr th sak f cmparis f th mths. W us th fllwig fuctis frm []: f si cs f 4 si cs si 3cs Tabl. Cmparis f varius itrativ mths G 7 MK MK MK MK 4 f TNFE f TNFE f 3 TNFE lifscicj@gmail.cm
10 Lif Scic Jural 03;0s f 4.6 TNFE f TNFE f 6 TNFE ITERATIONS f 7.8 TNFE Tabl. Cmparis f varius itrativ mths G 7 MK MK 6 MK 7 MK 8 f TNFE f TNFE f 3 TNFE lifscicj@gmail.cm
11 Lif Scic Jural 03;0s f 4.6 TNFE f TNFE f 6 TNFE f 7.8 TNFE Cclusi I this wrk w prst ight thr-stp familis f svth rr cvrgt mths. W bsrv that ths itrativ mths ar cmparabl with svth rr mth G 7 f Ku t.al [] as cit i th Tabl a Tabl a i almst all th cass ths familis f mths as cmpar t Ku t.al mth f svth rr [] giv bttr rsults i trms f abslut rrr a fucti valu. Crrspig Authr: Nair Ahma Mir Dpartmt f Basic Scics Riphah Itratial Uivrsit Islamaba Pakista airahma.mir@gmail.cm. Rfrcs. C. Chu Y. Li X Wag Sm sith-rr variats f Ostrwski rt-fiig mths Appl. Math. Cmput J. Ku Y. Li X.Wag Sm variats f Ostrwskis mth with svth-rr cvrgc J. Cmput. Appl. Math M.A. Nr Nw classs f itrativ mths fr liar quatis Appl. Math. Cmput. 007 i:0.06/j.amc M.A. Nr Nw classs f itrativ mths fr liar quatis Appl. Math. Cmput. 007 i:006/j.amc C. Chu A simpl cstruct thir-rr mificatis f Nwts mth J. Cmput. Appl. Math. 007 i: 0.06/j.cam Wihg Bi Hgmi R Qigbia Wu Thr-stp itrativ mths with ighth-rr cvrgc fr slvig -lar quatis J. cmput. Appl. Math Li Tai-fag ^{ } Li D-shg Xu Zha-i Fag Yalig w itrativ mths fr -liar quaits Appl. Math. Cmput C. Chu Y. Ham Sm furth-rr mificatis f Nwts Mth. Appl. Math. Cmput. 007 i: 0.06/j.amc // lifscicj@gmail.cm
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