Sains Malaysiana 43(7)(2014): 1101–1104

Interval Symmetric Single-step Procedure ISS2-5D for Polynomial Zeros

(Prosedur Selang Bersimetri Langkah-tunggal ISS2-5D untuk Punca Polinomial)

NORAINI JAMALUDIN¹, MANSOR MONSI¹ & NASRUDDIN HASSAN²*

¹Mathematics Department, Faculty of Science, Universiti Putra Malaysia

43400 Serdang, Selangor, D.E. Malaysia

²School of Mathematical Sciences, Faculty of Science and Technology

Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received: 24 June 2013/Accepted: 14 October 2013

ABSTRACT

We analyzed the rate of convergence of a new modified interval symmetric single-step procedure ISS2-5D which is an extension from the previous procedure ISS2. The algorithm of ISS2-5D includes the introduction of reusable correctors δi(k) (i = 1, …, n) for k ≥ 0. Furthermore, this procedure was tested on five test polynomials and the results were obtained using MATLAB 2007 software in association with IntLab V5.5 toolbox to record the CPU times and the number of iterations.

Keywords: Interval procedure; polynomial zeros; rate of convergence; simultaneous inclusion; symmetric single-step

ABSTRAK

Satu analisis dilakukan terhadap kadar penumpuan bagi prosedur terubahsuai selang bersimetri langkah-tunggal ISS2-5D baru yang merupakan lanjutan daripada prosedur ISS2 sebelumnya. Algoritma ISS2-5D termasuk pengenalan pembetulan yang boleh diguna semula δi(k) (i = 1, …, n) untuk k ≥ 0. Prosedur ini diuji ke atas lima jenis polinomial dan keputusan diperoleh menggunakan perisian MATLAB 2007 dan peralatan IntLab V5.5 untuk merekod masa CPU dan bilangan lelaran.

Kata kunci: Kadar penumpuan; kemasukan serentak; prosedur selang; punca polinomial; selang langkah tunggal bersimetri

REFERENCES

Aberth, O. 1973. Iteration methods for finding all zeros of a polynomial simultaneously. Mathematics of Computation 27: 339-334.

Aitken, A.C. 1950. On the iterative solution of linear equation. Proceedings of the Royal Society of Edinburgh Section A 63: 52-60.

Alefeld, G. & Herzberger, J. 1983. Introduction to Interval Computations. New York: Computer Science Academic Press.

Bakar, N.A., Monsi, M. & Hassan, N. 2012. An improved parameter regula falsi method for enclosing a zero of a function. Applied Mathematical Sciences 6(28): 1347-1361.

Gargantini, I. & Henrici, P. 1972. Circular arithmetics and the determination of polynomial zeros. Numerische Mathematik18(4): 305-320.

Iliev, A. & Kyurkchiev, N. 2010. Nontrivial Methods in Numerical Analysis: Selected Topics in Numerical Analysis. Saarbrucken: Lambert Academic Publishing.

Jamaludin, N., Monsi, M., Hassan, N. & Kartini, S. 2013a. On modified interval symmetric single step procedure ISS2- 5D for the simultaneous inclusion of polynomial zeros. International Journal of Mathematical Analysis 7(20): 983-988.

Jamaludin, N., Monsi, M., Hassan, N. & Suleiman, M. 2013b. Modification on interval symmetric single-step procedure ISS-5δ for bounding polynomial zeros simultaneously. AIP Conf. Proc. 1522: 750-756.

Kyurkchiev, N. 1998. Initial Approximations and Root Finding Methods. Mathematical Research, Volume 104. Berlin: Wiley-VCH.

Kyurkchiev, N. & Markov, S. 1983a. Two interval methods for algebraic equations with real roots. Pliska Stud. Math. Bulgar. 5: 118-131.

Kyurkchiev, N. & Markov, S. 1983b. A two-sided analogue of a method of A.W. Nourein for solving an algebraic equation with practically guaranteed accuracy. Ann. Univ. Sofia, Fac. Math. Mec. 77: 3-10.

Markov, S. & Kyurkchiev, N. 1989. A method for solving algebraic equations. Z. Angew. Math. Mech. 69: 106-107.

Milovanovic, G.V. & Petkovic, M.S. 1983. A note on some improvements of the simultaneous methods for determination of polynomial zeros. Journal of Computational and Applied Mathematics 9: 65-69.

Monsi, M., Hassan, N. & Rusli, S.F. 2012. The point zoro symmetric single-step procedure for simultaneous estimation of polynomial zeros. Journal of Applied Mathematics Article ID: 709832.

Monsi, M. & Wolfe, M.A. 1988. An algorithm for the simultaneous inclusion of real polynomial zeros. Applied Mathematics and Computation 25: 333-346.

Nourein, A.W. 1977. An improvement on two iteration methods for simultaneous determination of the zeros of a polynomial. International Journal of Computer Mathematics 6(3): 241- 252.

Ortega, J.M. & Rheinboldt, W.C. 1970. Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press.

Petkovic, M.S. 1989. Iterative methods for simultaneous inclusion of polynomial zeros. Lecture Notes in Mathematics. Volume 1387. Berlin: Springer Verlag.

Petkovic, M.S. & Stefanovic, L.V. 1986. On a second order method for the simultaneous inclusion of polynomial complex zeros in rectangular arithmetic. Archives for Scientific Computing 36(33): 249-261.

Rump, S.M. 1999. INTLAB-INTerval LABoratory. In Tibor Csendes, Developments in Reliable Computing. Dordrecht: Kluwer Academic Publishers.

Salim, N.R., Monsi, M., Hassan, M.A & Leong, W.J. 2011. On the convergence rate of symmetric single-step method ISS for simultaneous bounding polynomial zeros. Applied Mathematical Sciences 5(75): 3731-3741.

Sham, A.W.M., Monsi, M. & Hassan, N. 2013a. An efficient interval symmetric single step procedure ISS1-5D for simultaneous bounding of real polynomial zeros. International Journal of Mathematical Analysis 7(20): 977-981.

Sham, A.W.M., Monsi, M., Hassan, N. & Suleiman, M. 2013b. A modified interval symmetric single step procedure ISS-5D. AIP Conf. Proc. 1522: 61-67.

*Corresponding author; email: nas@ukm.edu.my