On the Interval Zoro Symmetric Single Step. Procedure IZSS1-5D for the Simultaneous. Bounding of Real Polynomial Zeros

Transcription

1 It. Joural of Math. Aalyss, Vol. 7, 2013, o. 59, HIKARI Ltd, O the Iterval Zoro Symmetrc Sgle Step Procedure IZSS1-5D for the Smultaeous Boudg of Real Polyomal Zeros 1 Atyah Wa Mohd Sham, 1 Masor Mos ad 2 Nasrudd Hassa * 1 Departmet of Mathematcs, Faculty of Scece, Uverst Putra Malaysa, UPM Serdag, Selagor DE, Malaysa 2 School of Mathematcal Sceces, Faculty of Scece ad Techology, Uverst Kebagsaa Malaysa UKM Bag, Selagor DE, Malaysa as@ukm.my Copyrght 2013 Atyah Wa Mohd Sham, Masor Mos ad Nasrudd Hassa. 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 A ew modfed method IZSS1-5D for the smultaeously boudg all the zeros of a polyomal s formulated ths paper. The effcecy of ths method s measured o the CPU tmes ad the umber of teratos after satsfyg the covergece crtera where the results are obtaed usg fve test polyomals. The R-order of covergece of ths method s at least fve. Keywords: CPU tmes, tal dsot tervals, terval, umber of terato. 1. Itroducto I ths paper, we refer to the methods establshed by Alefeld ad Herzberger [1, 2], Bakar et.al [3], Mos [4, 5], Mos et.al [6], Mos ad Wolfe [7], Petkovc [8], Jamaludd et.al [9, 10, 11] ad Sham et.al [12, 13] order to crease the rate of

2 2948 Atyah Wa Mohd Sham, Masor Mos ad Nasrudd Hassa covergece of the terval symmetrc sgle-step method ISS1-5D. The am of ths paper s to preset the terval zoro symmetrc sgle-step procedure, IZSS1-5D whch s the modfcato of terval zoro symmetrc sgle-step procedure, IZSS1 Rusl et. al [14]. IZSS1-5D method has bee modfed order to crease the effcecy of the method. We repeat the same er loopg ad add up aother step to test the accuracy. The effcecy of the algorthm s measured umercally by takg the CPU tme ad also the umber of teratos of the algorthm. 1 1 Cosder p : R R a polyomal of degree > 1 defed by 1 2 * = 1 p( x) = a x + a x + a x a = ( x x ) (1) where a 1 are gve. Suppose that p has dstct values R ( = 1,..., ) * (0) x R ( = 1,..., ) ad that X I( R) (set of real tervals) ( = 1,..., ) are such that ad * x X ( 0 ), ( = 1,..., ) (2) (0) (0) X X =,(, = 1,..., ; ). (3) 2. The Iterval Zoro Symmetrc Sgle-Step Procedure IZSS1-5D The terval symmetrc sgle-step procedure IZSS1-5D s a exteso of the terval sgle-step procedure IS ad ISS1 of [2] ad [7]. The terval sequece ( k X ) ( = 1,..., ) of IZSS1-5D are geerated as follows. Step 1: Set k = 0, (4a) Step 2: For ( k) ( k ) ( ) k 0, x = md X,( = 1,..., ); (4b) ( k ) ( k ) ( k ) ( k ) = = p x x x Step 3: Let δ δ ( ) Step 4: Step 5: ( ) (4c) ( k) p( x ) X = x I X 1 ( k) ( k,1) ( k) ( k,0) ( x X ) ( x X 5δ ) = 1 = + 1 ( k,1) ( k) ( k) ( = 1,..., ) ( k ) p( x ) X = x I X 1 ( k) ( k,1) ( k) ( k,2) ( x X ) ( x X ) = 1 = + 1 ( k,2) ( k) ( k,1) ( =,...,1) (4d) (4e)

3 Iterval zoro symmetrc sgle step procedure IZSS1-5D 2949 Step 6: ( k ) p( x ) X = x I X 1 ( k) ( k,1) ( k) ( k,2) ( x X ) ( x X ) = 1 = + 1 ( k,2) ( k) ( k,1) ( =,...,1) (4f) Step 7: Step 8: If X = X (4g) ( k+ 1) ( k,3) ( k + 1 ) ε w X <, the stop, else set k = k + 1 ad go to Step 2. (4h) 3. Numercal Results ad Dscusso Table 1 shows the comparso of the umber of terato ad CPU tme secods, betwee procedures IZSS1 ad IZSS1-5D obtaed usg MATLAB 2007 software assocato wth ItLab V5.5 toolbox [15]. Test Polyomal 1: [13] where 6, , , , , , , 1 1,,5, wth tal tervals, 1,2, 3,4, 5,6, 2, 1, 4, 3, 6, 5. Test Polyomal 2: [2] where = 5, 11.5, 9.1, 7.3, 5.2, 2.5, 1 1,,4, wth tal tervals, 2.5,2.1, 2.2,4.5, 4.6,7.9, 8.0,10.8, 10.9,13.1. Test Polyomal 3: [13] where 4, 1, 2, 3, 4, 1 1,2,3, wth tal tervals, 0.6,1.3, 1.6,2.3, 2.6,3.3, 3.6,4.3. Test Polyomal 4: [2] where 9, 15, 10, 7, 4, 0, 4, 7, 10, ,,8, wth tal tervals, 12.0,17.0, 8.6,11.2, 5.2,8.4, 2.4,5.0, 2.0,2.2, 6.4, 2,9,

4 2950 Atyah Wa Mohd Sham, Masor Mos ad Nasrudd Hassa 8.2, 6.5, 11.8, 8.0, 17.2, Test Polyomal 5: [2] where 5, 0, 3, 6, 9, 12, 1 1,,4, wth tal tervals, 1.9,3.4, 4.8,5.9, 6.5,8.1, 8.3,9.8, 10.7,11.9. Table 1: Number of Iteratos ad CPU Tmes Polyomal Degree IZSS1 IZSS1-5D No. of CPU No. of CPU teratos tme teratos tme Cocluso We have developed a ew modfed method IZSS1-5D whch s better tha IZSS1 terms of the umber of teratos ad CPU tmes. From the results, we coclude that the Iterval Zoro Symmetrc Sgle-Step Procedure IZSS1-5D (wth the corrector 5δ) must have a hgher rate of covergece compared to the ( k ) 12 procedure IZSS1 usg w 10 as the stoppg crtero. Ackowledgemet. We are debted to Uverst Kebagsaa Malaysa for fudg ths research uder the grat BKBP-FST-K Refereces [1] G. Alefeld ad J. Herzberger, O the covergece speed of some algorthms for the smultaeous approxmato of polyomal roots, SIAM Joural of Numercal Aalyss, 11(1974), [2] G. Alefeld ad J. Herzberger, Itroducto to Iterval Computatos, Academc Press, New York, [3] 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), [4] M. Mos, The terval symmetrc sgle-step ISS1 procedure for smultaeously boudg smple polyomal zeros, Malaysa Joural of Mathematcal Sceces, 5(2) (2011),

5 Iterval zoro symmetrc sgle step procedure IZSS1-5D 2951 [5] M. Mos, The pot symmetrc sgle-step PSS1 procedure for smultaeous approxmato of polyomal zeros, Malaysa Joural of Mathematcal Sceces, 6(1) (2012), [6] 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 , 11 pages, do: /2012/ [7] M. Mos ad M.A. Wolfe, A algorthm for the smultaeous cluso of real polyomal zeros, Appled Mathematcs ad Computato, 25(1988), [8] M.S. Petkovc, O a teratve method for smultaeous cluso of polyomal zeros complex, Computatoal ad App. Math. 8(1982), [9] N. Jamalud, M. Mos, N. Hassa ad S. Kart, O modfed terval symmetrc sgle step procedure ISS2-5D for the smultaeous cluso of polyomal zeros, Iteratoal Joural of Mathematcal Aalyss, 7(20) (2013), [10] N. Jamalud, M. Mos, N. Hassa ad M. Sulema. Modfcato o terval symmetrc sgle-step procedure ISS-5δ for boudg polyomal zeros smultaeously, AIP Cof. Proc (2013), , do: / [11] N. Jamalud, M. Mos ad N. Hassa, The terval symmetrc sgle-step procedure ISS2-5D for polyomal zeros. To appear Sas Malaysaa. [12] A.W.M. Sham, M. Mos, ad N. Hassa, A effcet terval symmetrc sgle step procedure ISS1-5D for smultaeous boudg of real polyomal zeros, Iteratoal Joural of Mathematcal Aalyss, 7(20) (2013), [13] A.W.M. Sham, M. Mos, N. Hassa ad M. Sulema, A modfed terval symmetrc sgle step procedure ISS-5D, AIP Cof. Proc (2013), 61 67, do: / [14] S.F.M. Rusl, M. Mos, M.A. Hassa ad W.J. Leog, O the terval zoro symmetrc sgle-step procedure for smultaeous fdg of polyomal zeros, Appled Mathematcal Sceces, 5(75) (2011), [15] S.M.Rump, INTLAB-INTerval LABoratory: I Tbor Csedes, Developmet Relable Computg, Kluwer Academc Publsher, Dordrecht, Receved: October 28, 2013