It. Joural of Math. Aalyss, Vol. 7, 2013, o. 20, 983-988 HIKARI Ltd, www.m-hkar.com O Modfed Iterval Symmetrc Sgle-Step Procedure ISS2-5D for the Smultaeous Icluso of Polyomal Zeros 1 Nora Jamalud, 1 Masor Mos, 2 Nasrudd Hassa * ad 3 Syarfah Kart 1 Departmet of Mathematcs, Faculty of Scece, Uverst Putra Malaysa, 43400 UPM Serdag, Selagor DE, Malaysa 2 School of Mathematcal Sceces, Faculty of Scece ad Techology, Uverst Kebagsaa Malaysa 43600 UKM Bag, Selagor DE, Malaysa as@ukm.my 3 Isttute for Mathematcal Research, Uverst Putra Malaysa, 43400 UPM Serdag, Selagor DE, Malaysa Copyrght 2013 Nora Jamalud et al. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. Abstract I ths paper, we preset a ew modfed terval symmetrc sgle-step procedure ISS2-5D whch s the exteso from the prevous procedure ISS2. The algorthm of ISS2-5D cludes the troducto of reusable correctors δ ( 1,..., ) for k 0. The procedure s tested o fve test polyomals ad the results are obtaed usg MATLAB 2007 software assocato wth ItLab V5.5 toolbox to record the CPU tmes ad the umber of teratos. Keywords: terval procedure, polyomal zeros, symmetrc sgle-step, smultaeous cluso
984 N. Jamalud, M. Mos, N. Hassa ad S. Kart 1 Itroducto Iterval teratve procedure for smultaeous cluso of smple polyomal zeros were dscussed [1,3,5,8,11] I ths paper, we cosder the procedures developed by [4,6,7,9,13] order to descrbe the algorthm of the terval symmetrc sgle-step procedure ISS2-5D. Ths procedure eeds some pre-codtos for tal tervals X (0) ( 1,..., ) to coverge to the zeros x * ( 1,..., ) respectvely, startg wth some dsjot tervals X (0) ( 1,..., ) each of whch cotas a polyomal zero. It wll produce bouded closed tervals whch wll trap the requred zero. The forward step [13] s modfed by addg a δ δ ( 1,..., ) ( k 0) (1(c)) o the secod part of the summato of the deomator (see (1(d)). The backward step of ths procedure comes from [8]. The terval aalyss s very straght forward compared to the aalyss of the pot procedures [6,9]. The programmg laguage used s Matlab 2007a wth the Itlab V5.5 toolbox [12]. The effectveess of our procedure s measured umercally usg CPU tme ad the umber of teratos. 2 The Iterval Symmetrc Sgle-Step Procedure ISS2-5D The terval symmetrc sgle-step procedure ISS2-5D s a exteso of the terval sgle-step procedure ISS2 [13] based o [2,3,6,8,9,10]. The sequeces ( k X ) ( 1,..., ) are geerated as follows. (1,0) (0) Step 1: X X (Ital tervals) (1a) Step 2: For Step 3: Let Step 4: 0, ( k) ( k) ( ), ( 1,..., ) k x md X (1b) px δ ( 1,..., ) (1c) p x ( ) '( ) δ X x X ( k,1) ( k) + 1 ( ) 1 1 k 1+ δ + ( k) ( k,1) ( k) ( k) ( k) j 1 x X j j + 1x X j 5δ j ( 1,..., ) (1d) Step 5:
Modfed terval symmetrc sgle-step procedure ISS2-5D 985 δ X x X ( k,2) ( k) ( k,1) + 1 ( ) 1 1 k 1+ δ + ( k) ( k,1) ( k) ( k,2) j 1 x X j j + 1x X j ( 1,..., ) (1e) ( k+ 1) ( k,2) Step 6: X X ( 1,..., ) (1f) Step 7: If k+ 1 ( ) wx < ε, the stop. Else set k k+ 1 ad go to Step 2. (1g) Step 4 s from [6] ad poted out wthout δ by [9], whle Step 5 s from [8]. The procedure ISS2-5D has the followg attractve features: (k ) (a) The use of 5δ j stead of δ as [6]. (b) The values δ computed for use Step 4 are reused Step 5. (k ) 1 1 ( k) ( k,1) j 1 x xj ( 1,..., ) used Step 4 are reused Step 5. ( k,1) ( k,2) x x ( k 0) so that eed ot be computed. (c) The summatos (d) ( k,2) x 3 Numercal Results ad Dscusso We used the Itlab V5.5 toolbox [12] for MATLAB R2007 to get the followg results below. The algorthms ISS ad ISS2-5D are ru o fve test polyomals where the stoppg crtero used s w 10 10. Cosder the followg polyomal [3] p( λ) det( λi A), (2) where ad a1 b1 b1 a2 O 0 A O O O 0 O a 1 b 1 b 1 a (3)
986 N. Jamalud, M. Mos, N. Hassa ad S. Kart (0) f ( λ) 1, (1) f ( λ) ( λ ak ), ( k) ( k 1) 2 ( k 2) f ( λ) ( λ ak) f ( λ)( bk 1) f ( λ) (2 k ), ( ) p( λ) f ( λ). (4) Test Polyomal 1: Salm [13] For ths example from [6] 5, a1 1; a2 1; a3 2, a4 0.5, a5 0.5 b 1( 1,..., 1), Ital Itervals: X [ 1.55, 0.35], X [0.89,1.59], X [1.60, 2.59], X [ 0.99, 0.25], X [0.39, 0.80] (0) 1 2 3 4 5 Test Polyomal 2: Salm [13] The polyomal s gve by (3) wth 9, a1 15; a2 15; a3 10; a4 7; a5 4; a6 10a7 7; a8 4 b 1( 1,..., ) Ital tervals: X [12.0,17.0]; X [8.6,11.2]; X [5.2,8.4]; X [2.4, 5.0]; X [ 2.0, 2.2]; X [6.4, 2.9]; X [ 8.2, 6.5]; X [ 11.8, 8.0]; X [ 17.2, 13.5] (0) 1 2 3 4 5 6 7 8 9 Test Polyomal 3: Mos [8] The polyomal s gve by (3) wth 9, a1 2.4519; a2 1.9021; a3 1.9275; a4 1.5765; a5 1.1867; a6 0.3210; a7 0.2674; a8 0.1254; a9 0.0435 b 1( 1,..., 1) Ital tervals: X [ 2.5, 2.0]; X [ 2.0, 1.89]; X [ 0.13, 0.10]; X [0.03,0.05]; X [0.25,0.28]; X [0.30,0.39]; X [1.09,1.31]; X [1.49,1.70]; X 1 2 3 4 5 6 7 8 (0) 9 [1.80,2.10] Test Polyomal 4: Mos [8] The polyomal s gve by (2) wth
Modfed terval symmetrc sgle-step procedure ISS2-5D 987 6, a1 2; a2 0.3; a3 0.4; a4 1; a5 2; a6 4 b 1( 1,..., ) Ital tervals: X [ 3.9, 1.5]; X [ 1.2,0.39]; X [0.4,0.99]; X [1.01,1.99]; X 1 2 3 4 (0) (0) 5 [2.09, 3.1]; X6 [3.19, 5.3] Test Polyomal 5: Mos [8] The polyomal s gve by (3) wth 10, a1 33; a2 21; a3 13; a4 7; a5 1; a6 1; a7 7; a8 13; a9 21; a10 33 b 1( 1,..., ) Ital tervals: X [ 35, 30]; X [ 25, 19]; X [ 15, 11]; X [ 9, 5]; X [ 2, 0.5]; X [0.4,2]; X X X X (0) (0) 1 2 3 4 5 6 7 [5,9]; 8 [11,15]; 9 [19, 25]; 10 [30,35]; Table 1: Number of Iteratos ad CPU Tmes Polyomal Degree ISS2 ISS2-5D No. of CPU No. of CPU teratos tme teratos tme 1 5 2 0.086719 1 0.040234 2 9 2 0.124219 2 0.123438 3 9 1 0.089844 1 0.084766 4 6 2 0.099609 1 0.055859 5 10 2 0.134375 2 0.130078 Table 1 shows that the procedure ISS2-5D requred less CPU tmes tha the procedure ISS2 for all 5 test polyomals, ad requred less umber of teratos meag ISS2-5D coverges faster tha ISS2. However, for test polyomals 2, 3 ad 5, the umber of teratos for both procedures s the same, but the tme cosumed for procedure ISS2-5D s stll less tha the ISS2 procedure. 4 Cocluso The above results have show umercally that the procedure ISS2-5D performs better tha ISS2 terms of CPU tmes ad umber of teratos. The attractve features of our procedure metoed Secto 2 cotrbute to these performaces.
988 N. Jamalud, M. Mos, N. Hassa ad S. Kart Ackowledgemet. We are debted to Uverst Kebagsaa Malaysa for fudg ths research uder the grat UKM-GUP-2011-159. Refereces [1] O.Aberth, Iterato methods for fdg all zeros of a polyomal smultaeously, Maths. of Computato, 27(1973), 339 334. [2] A.C.Atke, O the teratve soluto of lear equato. Proc. Roy. Soc. Edburg Sec. A, 63(1950), 52 60. [3] G. Alefeld ad J. Herzberger, Itroducto to Iterval Computatos, Academc Press, New York, 1983. [4] N.A. Bakar, M. Mos ad N. Hassa. A mproved parameter regula fals method for eclosg a zero of a fucto, Appled Mathematcal Sceces, 6(28) (2012), 1347 1361. [5] I. Gargat ad P. Herc, Crcular arthmetcs ad the determato of polyomal zeros, Numer. Math., 18(1972), 305 320. [6] G.V. Mlovaovc ad M.S. Petkovc, A ote o some mprovemets of the smultaeous methods for determato of polyomal zeros, Joural of Computatoal ad Appled Mathematcs, 9(1983) 65 69. [7] M. Mos, N. Hassa ad S.F. Rusl, The pot zoro symmetrc sgle-step procedure for smultaeous estmato of polyomal zeros. Joural of Appled Mathematcs, Volume 2012, Artcle ID 709832, 11 pages, do:10.1155/2012/709832. [8] M.Mos ad M.A.Wolfe, A algorthm for the smultaeous cluso of real polyomal zeros, Appled Mathematcs ad Computato, 25(1988), 333 346. [9] A.W.Noure, A mprovemet o Noure s method for the smultaeous determato of the zeroes of a polyomal(a algorthm), It.Comput. Math., 3(1977), 109 112. [10] J. M. Ortega ad W. C. Rheboldt, Iteratve Soluto of Nolear Equatos Several Varables, Academc Press, New York, 1970. [11] M.S.Petkovc ad L.V.Stefaovc, O secod order method for the smultaeous cluso of polyomal complex zeros rectagular arthmetc, Computg, 36(1986), 249 261. [12] S.M.Rump, INTLAB-INTerval LABoratory: I Tbor Csedes, Developmet Relable Computg, Kluwer Academc Publsher, Dordrecht, 1999. [13] N.R.Salm, M.Mos, M.A.Hassa ad W.J.Leog, O the covergece rate of symmetrc sgle-step method ss for smultaeous boudg polyomal zeros, Appled Mathematcs Sceces, 5(2011), 3731 3741. Receved: Jauary, 2013