On fourth order simultaneously zero-finding method for multiple roots of complex polynomial equations 1

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1 General Mathematcs Vol. 6, No. 3 (2008), 9 3 On fourth order smultaneously zero-fndng method for multple roots of complex polynomal euatons Nazr Ahmad Mr and Khald Ayub Abstract In ths paper, we present and analyse fourth order method for fndng smultaneously multple zeros of polynomal euatons. S. M. Ilč and L. Rančč modfed cubcally convergent Ehrlch Aberth method to fourth order for the smultaneous determnaton of smple zeros [5]. We generalze ths method to the case of multple zeros of complex polynomal euatons. It s proved that the method has fourth order convergence. Numercal tests show ts effcent computatonal behavour n the case of multple real/complex roots of polynomal euatons Mathematcs Subject Classfcaton: 65H05 Key words and phrases: Smultaneous teratve methods, mult-pont teratve methods, multple zeros, order of convergence, polynomal euatons. Receved 7 August, 2007 Accepted for publcaton (n revsed form) 24 January,

2 20 Nazr Ahmad Mr and Khald Ayub Introducton The methods for smultaneous fndng of all roots of polynomals are very popular as compared to the methods for ndvdual fndng of the roots.these methods have a wder regon of convergence and are more stable, (see, [2,6-7,9-2]) and references cted theren. For fourth order smple zero-fndng smultaneous methods,(see, [4,8,3,5-6]). S. M. Ilč and L. Rančč modfed cubcally convergent Ehrlch-Aberth method to fourth order for smultaneous fndng of smple complex zeros of polynomal euatons [5]. We generalse ths method to the case of multple zeros of complex polynomal euatons. It s proved that the method has fourth order convergence, f the roots have known multplctes. Recently, X. Zhang, H. Peng and G. Hu establshed a ffth order zero-fndng method for the smultaneous determnaton of smple complex zeros of polynomal euatons[5]. However, n case of multple zeros, t has lnear convergence as s also obvous from the numercal tests. Results of numercal tests show effcent computatonal behavour of our method n case of multple real/complex zeros of complex polynomal. The method and ts convergence analyss s consdered n Secton 2, where as results of numercal tests are presented n Sectn 3. Secton 4 contans concluson. 2 The method and ts convergence analyss Let us consder a monc algebrac polynomal P of degree n havng zeros w j m wth multplctes, such that = n, m () P(z) = z n + a n z n a z + a 0 = (z w j ). j= j=

3 On fourth order smultaneously zero-fndng method... 2 We propose the followng method for fndng the multple zeros of complex polynomal (): (2) z = z N m j= α z z j + m j=, αjn 2 j (z z j ) 2 where N = P(z ) s the Newton s correcton. The method (2) s the generalzaton of the method presented by S. M. Ilč and L. Rančč [5] to P (z ) the case of multple zeros of a complex polynomal.we name ths method as NMM-method. We clam that the NMM-method s of convergence order four. (3) Frst, let us ntroduce the notatons, d = mn w w j, = 2n and,, nstead of,j d j Further, suppose that the condtons, (4) ǫ < d 2n =, ( =,...,m) m m,, respectvely. hold for all, where ǫ = z w. We assume here that n 3. We prove the followng lemma: Lemma. Let z,...,z m be the dstnct approxmatons to the zeros w,...,w m respectvely. Also, let ε = z w,where z s the new approxmaton produced by the NMM-method (2). If (4) holds, then the followng neualtes also hold: () () ǫ 3 ǫ < ǫ j 2, (n ) ǫ 2 d 2n =, ( =,...,m). Proof. Consderng (3), we get j= j=

4 22 Nazr Ahmad Mr and Khald Ayub (5) z w j = (w w j ) + (z w ) w w j z w > d d 2n = 2n 2, Consderng (4) and (5), we get (6) z z j = (z w j ) + (w j z j ) z w j w j z j > 2n 2 = 2n 3. Defnng the notaton we have =, z, w j αj, = z w j. Thus, usng (5), we have α j z, w j < = 2(n ) Snce n α < n for all, we have (7) Also <, N = P (z ) P(z ) =, = α ε +, (n ) 2(n ) = 2. z w j = α 2(n ) (n α ). z w +, = α + ε, ε.

5 On fourth order smultaneously zero-fndng method Now, usng (4) and (7) n the above result, we have: ε (8) N = α + ε, < ε ε,.e., (9) N < 2. From (2), we have <. 2, ǫ = z w = z α N z z j + w α 2 j N j (z z j ) 2 = ǫ α α ǫ + z w j z z j + α 2 j N j (z z j ) 2 = ǫ α + ǫ Usng Newton s correcton, we have α ǫ [ αj ], (z j w j )(z z j )+α 2 j N j(z w j ) (z w j )(z z j ) 2 ǫ α ǫ = ǫ [ ǫ j (z z j )+ α + ǫ α +ǫ P,j α (z w j )(z z j ) 2 «(z w j ) α ǫ = ǫ [ ] ǫ j{ (z z j )( +ǫ j P,j α + ǫ α α )+ z w j} j (z w j )(z z j ) 2 ( +ǫ j P,j α ) = ǫ = ǫ = ǫ + ǫ α αj ǫ j + ǫ α αj ǫ j ǫ [ [ + ǫ αj α ǫ 2 j A, j ǫ (z j w j ) (z z j )ǫ j P,j α (z w j )(z z j ) 2 ( +ǫ j P,j α ) ǫ ǫ j (z j z j )ǫ j P,j α (z w j )(z z j ) 2 ( +ǫ j P,j α ) ] ] ]

6 24 Nazr Ahmad Mr and Khald Ayub where mples or (0) From (8), we have mples A j = (z z j ),j α ( ) (z w j )(z z j ) 2 + ǫ j,j α ǫ = ǫ = ǫ + ǫ2 α α + ǫ2 ǫ 2 ja j ǫ ǫ 2 j A, j ǫ 2 α α + ǫ2 ǫ 2 ja j ǫ 2 j A j ǫ j N j = + ǫ j,j α < 2, + ǫ j,j α. < 2 ǫ j Now usng euatons (5), (6) and (7), and the above result, we have + z z j,j α A j z w j z z j 2 αj + ǫ j, j α z z j + α,j = z w j z z j + ǫ j,j α n 2n 3 + ( ) ( 2 ) = 2 2n + (2n 3) 2n 2 2n 3 2 ǫ 2(n ) (2n 3) 2 ǫ j j 2 4n 3 = 2(n ) ǫ j (2n 3) 2.

7 On fourth order smultaneously zero-fndng method Snce 4n 3,n 3 s monotoncally decreasng seuence, so that fndng (2n 3) the least upper bound for n 3 of the seuence, we have () A j < and ǫ α j ǫ 2 α ja j ǫ α 2 2(n ) ǫ j. ǫ j 2 A j < ǫ. ǫ j 2 2 α 2(n ). ǫ j = ǫ 2 ǫ j. α 2(n ) <. <. < 2(n ).. 2 2(n ) 2(n ) (n α ), snce α = α and ǫ α for all. Usng = n α < n for all, we obtan ǫ (2) α j ǫ 2 α ja j 2(n ) (n ) = 2. Also further, usng (2), mples (3) + ǫ ǫ 2 ǫ α ja j ǫ 2 A j > 2 Fnally, usng euaton (), (3) n (0), we get (4) 2 ǫ ǫ j 2 2 ǫ 2(n ) < 2 ǫ j = 2 (n ) ǫ 2 ǫ j. Ths completes, the proof of Lemma (). Now from (4), we obtan ǫ 2 < α (n ) 2 j = (n ) = (n ) (n α ).

8 26 Nazr Ahmad Mr and Khald Ayub Snce = n α < n for all, we have namely ǫ < d 2n = = ǫ < ǫ < (n ) (n ) =, and hence Lemma () s also vald. Let z (0),...,z m (0) be reasonably good ntal approxmatons to the zeros w,...,w m of the polynomal P, and let ǫ (k) = z (k) w, where z (k),...,z m (k) be the approxmtons obtaned n the kth teratve step by the smultaneous method (NMM-method). Usng Lemma, we now state the man convergence theorem concerned wth NMM-method. Theorem. Under the condtons (5) ǫ (0) = z (0) d w < 2n =, ( =,...,m), the NMM-method s convergent wth the convergence order four. Proof. In Lemma () we establshed result (4) under the condtons (4). Usng the same argument under condton (5) of Theorem, we have from (4) ǫ () 3 (n ) ǫ (0) So by Lemma (), we have: ǫ (0) < 2 d 2n = ǫ () < ǫ (0) 2 <, ( =,...,m). d 2n =, ( =,...,m). Usng the mathematcal nducton, we can prove that the condton (5) mples (6) ǫ (k+) 3 (n ) ǫ (k) 2 ǫ (k) j 2 <,

9 On fourth order smultaneously zero-fndng method for each k = 0,,... and =,...,m. Puttng ǫ (k) = t(k), (6) becomes (7) Let t (k) = max t (0) t (k+) m t(k) ( ) 2 t (k) ( (n ) t (k) j ) 2, ( =,...,m).. Then from condton (5), t follows that t (0) < for =,...,m, and from (7), we have t (k) k = 0,,... and =,...,m. Thus, from (7), we get (8) t (k+) < ( t (k) ) 2 (n ) (n ) ( t (k)) 2 ( ) 2 t (k) (n ) ( ) j t (k) 2 ( ) t (k) 4. (n ) ǫ (0) = < for each { } Ths shows that the seuences t (k) ; =,...,m converge to 0. Conseuently, the seuences also converge to 0,.e., z (k) w for all { } ǫ (k) as k ncreases. Fnally, from (8) t can be concluded that the method (2) (NMM-method) has convergence order four. 3 Numercal Tests We consder here some numercal examples of algebrac polynomals wth repeated real and complex zeros to demonstrate the performance of fourth order method (2) (NMM-method). We use the abbrevatons as GHN(0), GHN() and GHN(2) to refer to the formulae of convergence order two, three and three for multple zeros n [8] and ZPH to refer to the formula of convergence order fve for dstnct zeros n [5].

10 28 Nazr Ahmad Mr and Khald Ayub All the computatons are performed usng Mapple 7.0. We take ǫ = 0 8 as tolerance and use the followng stoppng crtera for estmatng the zeros: x n+ x n < ǫ. Frstly, takng the multplctes eual to one n our method (2), we reestmated the examples from [5]. We got the same estmates as by the fourth order convergent method n [5] for dstnct zeros. Secondly, numercal tests for the algebrac polynomals wth real and complex repeated zeros from [8] are provded n tables 3.(a) to 3.(b). The roots obtaned by the methods GHN(0), GHN() and GHN(2) are accurate to 8 dgts n tables 3.(a) to 3.(b), where as the roots obtaned by the NMM-method are accurate to 20 to 30 dgts n table 3.(a) and accurate to 2 to 32 dgts n table 3.(b). Thrdly, the NMM-method s also compared wth ZPH-method[5]. We got the roots accurate to 64 dgts at the frst teraton, where as the ZPHmethod obtaned roots accurate to 2 dgts at ffth teraton. Table 3.(a) : Example : z 3 + 0z 2 90z 000z z z z z z z z z z = (z 5) 3 (z 2) 4 (z + 3) (z + 6) 5 Exact Root: α = (5, 2, 3, 6) Intal Pont Number of teratons for dfferent methods x 0 GHN(0) GHN() GHN(2) NMM (5.9, 2.7, 3.9, 6.7) (5.6,.5, 2.7, 6.5) (5.5,.4, 2.6, 6.3) (5.4, 2.2, 2.8, 5.9) Absolute Errors are eual to0 8 Absolute Errors le between0 8 to0 30

11 On fourth order smultaneously zero-fndng method Table 3.(b) : Example2 : z 7 3z 6 +5z 5 7z 4 +7z 3 5z 2 +3z =(z ) 2 (z+) 2 (z ) 3 =`z (z ) 3 Exact Root: α = (,,) Intal Pont Number of teratons for dfferent methods x 0 GHN(0) GHN() GHN(2) NMM (0. 0.8, , ) ( , , ) ( , , ) (0. 0.9, , ) Absolute Errors are eual to0 8 Absolute Errors le between0 3 to0 32 Table 3.2 Example3 : z 4 4z 3 + 6z 2 4z + = (z ) 4 Absolute error by NMM= E + 00.Accuracy upto 64 dgts n frst teraton Absolute error by ZPH Iteraton e (k) e (k) 2 e (k) 3 e (k) 4 k = E E E E + 00 k = E E E E + 00 k = E E E E 0 k = E E E E 0 k = E E E E 02 4 Concluson From the numercal comparson, we observe that the NMM-method s compareable wth fourth order methods n case of dstnct zeros. It has got better performance n case of multple zeros over the thrd order methods for multple zeros and hgher order methods for dstnct zeros. References [] O. Aberth, Iteraton methods for fndng all zeros of a polynomal smultaneously, Math. Comput. 27 (993), [2] M. Cosnard and P. Fragnaud, Fndng the roots of a polynomal on an MIMD multcomputer, Parallel Computng 5 (990),

12 30 Nazr Ahmad Mr and Khald Ayub [3] L. W. Ehrlch, A modfed Newton method for polynomals, Comm. ACM 0 (967), [4] G. H. Ells and L. T. Watson, A parallel algorthm for smple roots of polynomals, Comput. Math. Appl. 2 (984), [5] S. M. Ilč and L.Rančč, On the fourth order zero fndng methods for polynomals, Flomat 7 (2003), [6] S. Kanno, N. Kjurkchev and T. Yamamoto, On some methods for the smultaneous determnaton of polynomal zeros, Japan J. Industr. Appl. Math. 3 (995), [7] J.M. McNamee, A bblography on roots of polynomals, J. Comput. Appl. Math. 47 (993), [8] G.H. Nedzhbov, An acceleraton of teratve procesess for solvng nonlnear euatons, Appl.Math. Comput 68 (2005) [9] A. W. M. Nouren, An mprovement on two teraton methods for smultaneous determnaton of the zeros of a polynomal, Intern. J. Comput. Math. 6 (977), [0] M. S. Petkovć, Iteratve Methods for Smultaneous Incluson of Polynomal Zeros, Sprnger-Verlag, Berln-Hedelberg-New York, 989. [] M. S. Petkov c, -D. Herceg and S. Ilć, Pont Estmaton Theory and ts Applcatons, Insttute of Mathematcs, Unversty of Nov Sad, Nov Sad, 997. [2] Bl. Sendov, A. Andreev and N. Kjurkchev, Numercal Soluton of Polynomal Euatons (Handbook of Numercal Analyss, Vol. III), Elsever Scence, New York, 994.

13 On fourth order smultaneously zero-fndng method... 3 [3] J. Stoer and R. Bulrsch, Enführung n de Numersche Mathematk II, Sprnger-Verlag, Berln, 973. [4] X. Wang and S. Zheng, A famly of parallel and nterval teratons for fndng all roots of a polynomal smultaneously wth rapd convergence (I), J. Comput. Math. (984), [5] X. Zhang, H.Peng and G.Hu, A hgh order teratve formula for the smultaneous ncluson of polynomal zeros, Appl. Maths. Comput. (2006, n press). [6] S. Zheng and F. Sun, Some smultaneous teratons for fndng all zeros of a polynomal wth hgh order of convergence, Appl. Math. Comput. 99 (999), Nazr Ahmad Mr COMSATS Insttute of Informaton Technology Department of Mathematcs Islamabad, Pakstan E-mal: namr@comsats.edu.pk Khald Ayub Bahauddn Zakarya Unversty Centre for Advanced Studes n Pure and Appled Mathematcs Multan E-mal: khaldmaths@hotmal.com