On the Interval Zoro Symmetric Single-step Procedure for Simultaneous Finding of Polynomial Zeros

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1 Appled Mathematcal Scences, Vol. 5, 2011, no. 75, On the Interval Zoro Symmetrc Sngle-step Procedure for Smultaneous Fndng of Polynomal Zeros S. F. M. Rusl, M. Mons, M. A. Hassan and W. J. Leong Department of Mathematcs, Faculty of Scence Unverst Putra Malaysa, Serdang, Malaysa Abstract The am of ths paper s to present the nterval zoro symmetrc snglestep procedure IZSS1 whch s the modfcaton of nterval symmetrc sngle-step procedure ISS1. Ths procedure has a faster convergence rate than does ISS1. We start wth sutably chosen ntal dsont ntervals where each nterval contans a zero of a polynomal. The IZSS1 method wll produce successvely smaller ntervals that are guaranteed to stll contan the zeros. The convergence rate of the procedure IZSS1 wll be shown n ths paper. The procedure s run on fve test polynomals and the results obtaned show that the modfed method s better n comparson wth the procedure ISS1. Keywords: Interval analyss; Interval procedure; Smultaneous ncluson; Smple zeros; R-order of convergence; R-factor of a sequence 1 Introducton The teratve procedures for the smultaneous determnaton of all zeros of a polynomal are the mportant root solvers snce they overcome deflaton (McNamee [?]).These procedures start wth some dsont ntervals whch contan the polynomal zeros. The nterval teratve procedures wll determne bounded ntervals each of whch s guaranteed to stll contan the zero. It s a very sgnfcant way of obtanng relable bounds on the zeros as the ntervals sequences generated by the procedures are always converges to the zeros. Snce then, researchers have concentrated on teratve methods such as the famous Newton s method; see for examples, Gargantn and Henrc (1971), Gargantn (1978), Petkovc (1982), Alefeld and Herzberger (1983), Mlovanovc

2 3694 S.F.M. Rusl, M. Mons, M.A. Hassan and W.J. Leong and Petkovc (1983), Petkovc and Stefanovc (1986), Mons(1988),Carstensen (1993), Sun and L (1999). In ths paper, we wll descrbe the nterval zoro symmetrc sngle-step procedure IZSS1. The sgnfcant of usng the nterval analyss for determnng the convergence rate of the procedure s that the analyss s very straght forward compared to the analyss of the pont procedures. We use the Intlab V5.5 toolbox (S.M. Rump [?]), for MATLAB R2007a n order to determne the numercal results. It s a necessary tool to determne the narrow computatonally rgorous bounds on the zeros of the polynomals. The R-order of convergence analyss of an teratve procedure used n ths paper s as a measure of the asymptotc convergence rate of the procedure. The concept of R-order of convergence s dscussed n detal n Ortega and Rhenboldt [?] and Alefeld and Herzberger [?]. The R-order of the procedure I whch converge to x s denoted by O R (I,x ) and the R-factor of a null sequence w (k) generated from the procedure I s denoted by R p (w (k) ), where p 1 and w (k) s a null sequence generated from the procedure I. The followng theorem s proved n Ortega and Rhenboldt [?]. Theorem 1.1 Let I be an teraton procedure wth the lmt x, and let Ω(I,x ) be the set of all sequences ( ) generated by I havng the propertes that lm x = x and x,k 0. If there exsts a p 1 and a constant γ such that for all ɛω(i,x ) and for a norm, t holds that h (k+1) γ h (k) p, k k( )then t follows that the R-order of I satsfes the nequalty O R (I,x ) p. Furthermore, f there exst a p 1 such that for any null sequence w (k) generated from then the R-factor of such sequence s defned to be ( R ) lm p w (k) = sup w (k) 1 k,p=1 k lm sup w (k) 1 p k,p>1, k where R p s ndependent of the norm. Suppose that R p ( w (k) ) < 1 then t follows from Ortega and Rhenboldt [?] that the R-order of I satsfes the nequalty O R (I,x ) p. We wll use ths result n order to calculate the R-order of convergence of IZSS1 n the subsequent secton.

3 Interval Zoro symmetrc sngle-step procedure The Smultaneous Incluson Procedures of Real Zeros of Polynomals. Let p : R 1 R 1 be a polynomal of degree n defned by p(x) =a n x n + a n1 x n a 1 x + a 0 n = a x (1) =0 where a R 1 (1,..., n) are gven. Suppose that p has n dstnct zeros x R (1,..., n) and that I(R) (the set of real ntervals) (1,..., n) are such that and x X(0) (1,..., n) (2) = φ (, =1,..., n; ), (3) It assumed henceforth that a n = 1, so that p(x) = n (x x ). (4) =1 By (4), for 1,..., n ( x x ( =1,..., n)) If x x p(x) 1 (x (5) x ). x (0) m( (1,..., n), (6) are the mdponts of the nterval ( 1,..., n) respectvely. Then by (2) and (3), So by (5) we have, x x(0) x (0) x (1,..., n; ). (7) p(x (0) 1 (x(0) x ) (1,..., n). (8)

4 3696 S.F.M. Rusl, M. Mons, M.A. Hassan and W.J. Leong Furthermore, by (3), (6), x (0) / x X (1) x (0) (, =1,..., n; ), whence 0 / (x (0) ) (1,..., n). (9) So, by (2), (8), and the ncluson monotoncty (Alefeld and Herzberger [?]) of real nterval arthmetc, p(x (0) (x(0) ) (1,..., n). (10) Ths gves rse to the nterval total step procedure IT of Alefeld and Herzberger defned by = m(x (k) (1,..., n), (11a) p( X (k+1) (x(k) X (k) ) X (k) (1,..., n) (k 0), (11b) The rate of convergence of IT procedure s at least two or O R (IT,x 2. (Alefeld and Herzberger [?]) A modfcaton of IT s known as sngle-step procedure IS (Alefeld and Herzberger [?]) and t s defned by (for k 0) = m(x (k) (1,..., n), (12a) X (k+1) (1,..., n), 1 =1 (x(k) X (k+1) p( ) n =+1 (x(k) X (k) ) X (k) (12b) It have been proved n Alefeld and Herzberger [?] that for 1,...n, O R (IS,x 1 ± σ. where σ (1, 2). s the greatest postve zero of t n t 1. The natural extenson of the nterval sngle-step procedure IS s the Interval symmetrc sngle-step procedure ISS1 of Mons [?] and t s defned by (for k 0) = m(x (k) (1,..., n), (13a) X (k,1) X (k+1) (1,..., n), ( n,..., 1), 1 =1 (x(k) 1 =1 (x(k) X (k,1) X (k,1) p( ) n =+1 (x(k) p( ) n =+1 (x(k) X (k,0) ) X (k+1) ) X (k,0) X (k,1) (13b) (13c)

5 Interval Zoro symmetrc sngle-step procedure 3697 The rate of convergence of procedure ISS1 s at least three or O R (ISS1,x 3 (Alefeld and Herzberger [?]). 3 The Interval Zoro Symmetrc Sngle-step Procedure An extenson of the dea of Atken (1950), Alefeld (1977) and Mons (1988) are used to establsh the new modfed method so-called the nterval zoro symmetrc sngle-step procedure IZSS1. The zoro s referred to the pattern of the z steps n the procedure. The procedure IZSS1 conssts of generatng the sequences (X (k) (1,..., n) from X (1,0) (ntal ntervals) (14a) for k 1, (14b) = m(x (k,0) (1,..., n), (14c) X (k,1) X (k,2) X (k,3) X (k+1) X (k+1,0) (1,..., n), ( n,..., 1), (1,..., n), 1 =1 (x(k) 1 =1 (x(k) 1 =1 (x(k) X (k,1) X (k,1) X (k,3) p( ) n p( ) n p( ) n =+1 (x(k) =+1 (x(k) =+1 (x(k) X (k,0) ) X (k,2) ) X (k,2) ) X (k,0) X (k,1) X (k,2) (14d) (14e) (14f) = X (k,3) ( n,..., 1), (14g) = X (k+1) (1,..., n), (14h) Theorem 3.1 If () (2) and (3) holds; () the sequences X (k) (1,..., n), are generated from (14), then ( k 0) x X (k+1) X (k) (1,..., n), If also () 0 / D I(R) s such that p (x) D ( x (1,..., n), then X (k) x (k )(1,..., n) and w(x (k+1) 1 (1 d I 2 d S )w(x (k) holds. Then for (1,..., n), O R (IZSS1,x 4. Proof The proof that x X (k+1) X (k) (1,..., n) ( k 0) and that (12) holds s almost dentcal wth the correspondng proofs n Alefeld and Herzberger [?], and s therefore omtted. It remans to prove that for

6 3698 S.F.M. Rusl, M. Mons, M.A. Hassan and W.J. Leong (1,..., n), O R (IZSS1,x 4. From (Alefeld and Herzberger [?]) t may be shown that α >0 such that ( k 0), and where and Let w (k,1) w (k,2) w (k,3) βw (k,0) βw (k,0) βw (k,0) w (k,s) 1 =1 1 =1 1 =1 w (k,1) + w (k,1) + w (k,3) + n =+1 n =+1 n =+1 w (k,0) w (k,2) w (k,2) (1,..., n), (15) ( n,..., 1), (16) (1,..., n), (17) =(n 1)αw(X (k,s) (s =0, 1, 2, 3), (18) β = 1 n 1. (19) u (1,1) 2 (1,..., n 1) 3 ( n) u (1,2) 3 ( n,..., 2) 4 (1), (20), (21) and u (1,3) 4 (1,..., n 1), (22) 5 ( n) and for (r =1, 2, 3), let u (k+1,r) 4u (k,r) (1,..., n 1) 4u (k,r), (23) +1 ( n) Then by (20)-(23), for ( k 0) u (k,1) u (k,2) 2(4 (k1) ) (1,..., n 1) 10 3 (4(k1) ) 1, (24) ( n) (4(k1) ) 1 ( n) 3 3(4 (k1) ) ( n 1,..., 2), (25) 4(4 (k1) ) (1)

7 Interval Zoro symmetrc sngle-step procedure 3699 and u (k,3) 4(4 (k1) ) (1,..., n 1) 16 3 (4(k1) ) 1. (26) ( n) 3 Suppose, wthout loss of generalty, that w (0,0) h<1 (1,..., n). (27) Then by nductve argument t follows from (15)-(27) that for (1,..., n) (k 0) and w (k,1) w (k,2) w (k,3) whence by (27) and (14g), for ( k 0) So, by (17)-(27), for ( k 0) Let Then, by (28), So, h u(k+1,1), h u(k+1,2), h u(k+1,3), w (k+1) h 4(k1) (1,..., n). w(x (k) ( ) β h 4(k) (1,..., n). (28) α w (k) = max w(x (k). 1 n w (k) ( ) β h 4k ( k 0). α R 4 (w (k) )= lm sup k (β = lm k = h < 1. (w (k) ) 1 (4 k ) α ) 1 (4 k ) h Therefore, the R-order of convergence of IZSS1 defned by (14) s at least 4 or O R (IZSS1,x 4(1,..., n).

8 3700 S.F.M. Rusl, M. Mons, M.A. Hassan and W.J. Leong 4 Numercal Results In order to show the sgnfcant of the IZSS1 method, we use the MATLAB R2007a n co-operated wth the Intlab V5.5 toolbox for nterval arthmetc developed by S.M. Rump [?] and t s tested on fve test polynomals. The stoppng crteron used s w (k) Test Polynomals Test Polynomal 1 :[?] The polynomal s gven by wth and the ntal ntervals: p(x) =(x 2)(x 3.4)(x 5.2)(x 7.1) n =4, x 1 = 2, x 2 =3.4, x 3 =5.2, x 4 =7.1, 1 =[0.9, 2.1] ; 2 =[2.9, 3.9] ; 3 =[4.9, 6.3] ; 4 =[6.6, 8.1]. Test Polynomal 2 :[?] The polynomal s gven by wth p(x) =(x 3)(x 11)(x 30)(x + 3)(x + 11)(x + 30) n =6, x 1 = 3, x 2 = 11, x 3 = 30, x 4 = 3, x 5 = 11, x 6 = 30, and the ntal ntervals: 1 =[1, 2] ; 2 =[3, 4] ; 3 =[5, 6] ; 4 =[2, 1] ; 5 =[4, 3] ; 6 =[6, 5]. Test Polynomal 3 :[?] The characterstc polynomal p(λ) =det(λi A), (29a)

9 Interval Zoro symmetrc sngle-step procedure 3701 where a 1 b 1. b 1 a A = an1 b n1 b n1 a n and f (0) (λ) =1, f (1) (λ) =(λ a 1 ), f (k) (λ) =(λ a k )f (k1) (λ) (b k1 ) 2 f (k2) (λ) p(λ) =f (n) (λ). For ths polynomal (Alefeld and Herzberger [?]): (2 k n), (29b) n =9, b 1 (1,..., n 1), a 1 = 15, a 2 = 10, a 3 = 7, a 4 = 4 a 5 =0, a 6 =4, a 7 =7, a 8 =10, a 9 =15. Intal ntervals: 1 =[17.2, 13.8] ; 2 =[12.1, 8.9] ; 3 =[8.7, 6.1] ; 4 =[6.0, 2.1] ; 5 =[2.0, 2.3] ; 6 =[2.4, 6.1] 7 =[6.3, 8.9] ; 8 =[9.1, 12.9] ; 9 = [13.1, 17.2]. Test Polynomal 4 :[?] The polynomal s gven by (29) wth Intal ntervals: n =5, b 1 (1,..., 4), a 1 =0, a 2 =3, a 3 =6, a 4 =9, a 5 =12. 1 =[2.5, 2.1] ; 2 =[2.2, 4.5] ; 3 =[4.6, 7.9] ; 4 =[8.0, 10.8] ; 5 = [10.9, 13.1].

10 3702 S.F.M. Rusl, M. Mons, M.A. Hassan and W.J. Leong Test Polynomal 5 :[?] The polynomal s gven by (29) wth Intal ntervals: n =6, b 1 (1,..., 5), a 1 =35, a 2 =27, a 3 =21, a 4 =16, a 5 =9, a 6 =5. 1 = [30, 40] ; 2 = [25, 29] ; 3 = [20, 24] ; 4 = [13, 19] ; 5 =[7, 12] ; 6 =[3, 6]. 4.2 Results and Dscussons The followng tables summarze the results of all test polynomals Table 1: Number of teratons and CPU tmes Test Polynomal n ISS1 IZSS1 No. of teratons k CPU tmes No. of teratons k CPU tmes Fgure 1: Fgure 2:

11 Interval Zoro symmetrc sngle-step procedure 3703 Table 2: Fnal ntervals of each component,. (Test polynomal 5) ISS1 IZSS1 n Iteraton Interval Iteraton Interval Fnal Intervals Fnal Intervals k =3 wdths k =2 wdths 1 X (3) 1 [ , ] e-12 X (2) 1 [ , ] e-14 2 X (3) 2 [ , ] e-15 X (2) 2 [ , ] e-13 3 X (3) 3 [ , ] e-15 X (2) 3 [ , ] e-14 4 X (3) 4 [ , ] e-15 X (2) 4 [ , ] e-15 5 X (3) 5 [ , ] e-15 X (2) 5 [ , ] e-15 6 X (3) 6 [ , ] e-15 X (2) 6 [ , ] e-12 Table 3: The wdth of the fnal ntervals (w (k) polynomal 3) 1 Step 1 (15d) Interval Wdth w (k) = w(x (k) ) at teraton k.(test of X (k) Iteraton Steps k Method ISS1 Method IZSS Step 2 (15e) Step 3 (15f ) (Not Applcable)

12 3704 S.F.M. Rusl, M. Mons, M.A. Hassan and W.J. Leong Table 3:(contnue). 2 Step 1 (15d) Table 4: * Interval Wdth w (k) of X (k) Iteraton Steps k Method ISS1 Method IZSS e e-04 3 Step 2 (15e) Step 3 (15f ) Step 1 (15d) Step 2 (15e) Step 3 (15f ) 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-08 (Not Applcable) 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 e e e e e e e e e e e-14 (Not Applcable) (Converged)

13 Interval Zoro symmetrc sngle-step procedure 3705 Table 5: The wdth of fnal ntervals at the teraton k.(test polynomal 4) Iteraton Interval Wdth Methods k w (k) ISS1 IZSS e e e e e e e e e e e e-15 (Converged) e e e-15 Table 1 shows the comparson of the number of teratons and the CPU tmes n seconds, between the procedures ISS1 and IZSS1 whle Fgure 1 and Fgure 2 are the graphs of the number of teratons and the CPU tmes respectvely. For Table 2, we take the results of test polynomal 5 to show the fnal ntervals of the procedure IZSS1 compare to the procedure ISS1 ncludng ther wdths before the algorthm stop at the stoppng crteron,w (k) Table 3 s a result of the fnal nterval wdth of test polynomal 3. It shows that even though the numbers of teratons of both procedures are the same, the algorthm IZSS1 converge earler by one step. We observe n Table 4 that for test polynomal 4, the method IZSS1 s better than the method ISS1 n term of number of teratons. 5 Concluson We have shown analytcally that the nterval symmetrc sngle-step procedure IZSS1 gves hgher rate of convergence, where the R-order of convergence of IZSS1 s at least 4 or O R (IZSS1,x 4. Whle the R-order of convergence of ISS1 (Mons [?]) s at least 3, that s O R (ISS1,x 3. It s clear from Table 1, Table 2, Table 3 and Table 4 that the procedure IZSS1 numercally requres less CPU tmes and number of teratons and furthermore the fnal ntervals of IZSS1 have a better accuracy than does ISS1 where the stoppng crteron used s w (k)

14 3706 S.F.M. Rusl, M. Mons, M.A. Hassan and W.J. Leong References [1] G. Alefeld and J. Herzberger,Introducton To Interval Computatons, Translated By Jon Rokne, Academc Press, New York, [2] R. Butt, Introducton to Numercal Analyss Usng Matlab, Infnty Scence Press, [3] O.Capran, K. Madsen and H. B. Nelsen,Introducton to Interval Analyss, 2002, DTU. [4] G.I. Hargreaves, Interval Analyss n Matlab, Manchester Insttute for Mathematcal Scences, [5] L. Jauln, M.Keffer, O.Ddrt,E.Walter, Appled Interval Analyss, Sprnger, [6] J.M. McNamee, C.K. Chu, L. Wuytack, Numercal Methods For Zeros of Polynomals Part 1, Unted Kngdom : Elsever Publshng Company, Frst Edton, [7] G.V. Mlovanovc, Petkovc, M.S., On the Convergence of a modfed Method for Smultaneous Fndng of Polynomal Zeros, Computng (30),1983. [8] M. Mons, Prvate Communcaton, Department of Mathematcs, Unverst Putra Malaysa, [9] M. Mons, Some Applcatons of Computer Algebra and Interval Mathematcs, Unversty of St. Andrews,1988. [10] J.M. Ortega, W.C. Rhenboldt, Iteratve Soluton of Nonlnear Equatons n Several Varables, Academc Press, New York,1970. [11] M.S. Petkovc, Iteratve Methods for Smultaneous Incluson of Polynomal Zeros, Sprnger, [12] S.M. Rump,INTLAB-Interval Laboratory, Receved: Aprl, 2011

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