A family of root-finding methods with accelerated convergence

Keywords--Determinat ion of polynomial zeros, Simultaneous iterative methods, Convergence analysis, Accelerated convergence, R-order of convergence. 1. I N T R O D U C T I O N The problem of determining polynomial zeros has a great impor tance in theory and practice (for instance, in the theory of control systems, digital signal processing, s tabil i ty of systems, analysis of transfer functions, various mathemat ical models, differential and difference equations, eigenvalue problems). Iterative methods for the s imultaneous finding all polynomial zeros belong to the most efficient approaches and became practically applicable with the rapid development of digital computers, see, e.g., [1-9] and references cited therein. Since the corresponding iterative formulas run in identical versions, s imultaneous methods are very sui table for the implementa t ion on parallel computers, which addit ionally increases their importance, see [10-13]. Quant i ta t ive (initial) conditions for predicting the immediate appearance of a safe and fast convergence of the simultaneous methods, in the spirit of Smale's point es t imat ion theory [14], were studied in details in [8,15]. *Author to whom all correspondence should be addressed. The authors would like to thank the anonymous referees who made valuable comments and suggestions that improved the presentation. 0898-1221/06/$ see front matter ~) 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2005.10.013 Typeset by .AA~-TEX I000 h4. S. PETKOVIC AND L. Z. RAN(~Id In this paper, we use a fixed-point relation involving a complex parameter to construct a new family of simultaneous methods of the fourth order for solving the polynomial equation P(z) = 0 in ordinary complex arithmetic. Interval versions of this family for the inclusion of simple and multiple zeros were studied in [16,17]. The motivation and reasons for developing higher-order methods were discussed in [13]. In order to decrease the computational cost of the basic method, we state another modification of this family which possesses very fast convergence (Section 2). The proposed methods have a high computational efficiency since the acceleration of convergence is attained with only few additional computations. Actually, the increase of the convergence rate is attained by means of Newton's and Halley's corrections which use the already calculated values of P, P~, pH at the points Z l , . . . , z~-the current approximations to the wanted zeros. The main convergence theorems for the total-step as well as single-step methods are established in Section 3. The results of numerical experiments are presented in Section 4. 2. S I M U L T A N E O U S M E T H O D S F O R F I N D I N G S I M P L E Z E R O S Let P be a monic polynomial with simple zeros 41, . . . , ~, and let Z l , . . . , Zn be their mutually distinct approximations. For the point = z~ (i e I~ := { 1 , . . . , ~ } ) , and a complex parameter c~ ¢ -1 , let us introduce the notations, Ex,i = Sx,i = (A = 1, 2) si = zi ~i, J=~ (z~ G ) ~ ' ~=, ( ~ z j ) ~ ' j # i j # i P'(z~) P'(z~) e P(z,)P"(z~) ~1,i = p ( z i ) , ~2,i-~p ( z i ) 2 , F~ = (a + 1)E2,i (~(a + 1)E~,i, fi = (a + 1)$2# ~(c~ + 1)S~,,, (2.1) LEMMA 2.1. For i E In the following identity is valid, (~ + 1)~,~ ~ , ~ F~ = ( ~ ~ + 1 -0~51,i) 2 . (2.2) PROOF. Starting from the identities, P'(zi) ~ 1 1 j=l and 5~i P'(zi)~-P(z~)P"(zi) (P ' (z , )~ ' ~ 1 "-~1 . . . . + E2,i, ' P (zd ~ \ ~ ( z d 7 ( z ~ G ) ~ E~ j = l we obtain )(1 (~ + 1)&,~ ~ , ~ , ~ / * = ( ~ + 1) 1 + 22,~ ~ + Z~,~ 2c~ E 1 2~E + a 2 E ~ i l i E i £i 1,~ ' £i ' ( 1 251# ~1, i + a 2 3 1 # 2 £ i Ei _ (o~+ 1 zi O~(~1'i) 2 (a + 1)E2,i + a(o~ + 1)E2,~ Root -F ind ing Methods 1001 From identity (2.2), we obtain the following fixed-point relation, c ~ + l ~i = Z i r ~ 1/2 ( i E I n ) , (2 .3) OZ(~l,/ + [(a + 1 ) 5 2 # a521 , , F i J . assuming tha t two values of the square root have to be taken in (2.3). The symbol • points to the choice of the proper value of the square root appearing in (2.3) as well as later iterative formulas and some expressions used in the convergence analysis. Let us introduce some additional notation. (m) . z (m) of the zeros at the m TM iterative step are denoted by 1 ° The approximations z I , . . , z l , . . . , Zn, and the new approximations z~ re+ l ) , . . . , Z(m+l)n , obtained by some simultaneous iterative method, by ~1, • • •, in, respectively. 2 ° o 1 P (z,) N i = N (z , ) = ~1,, P ' ( z i ) = (z,)= [P_,(z,) P,,(z,) n,-' LP(zd 2P' (zdJ 2513 52# + ~2,~ (Newton's correction), (2.4) (Halley's correction). (2.5) , 1 S k i ( a , b ) = ~ 1 1 ' j.'= ( z i ~ a J ) k ~-j=i-t-1 ( z i 2 b j ) k ' f / ( a , b) = (c~ + 1) S2,i (a , b) o~ (o~ + 1) S12,i (a, b) , where a = ( a l , . . . , an) and b = ( b i , . . . , bn) are vectors of distinct complex numbers. If a = b = z = (zi . . . . . z,~), then we will write S k , i ( z , z ) = Sk, i and f i ( z , z ) = f , . 4 ° z = (Z l , . . . , zn) (the current approximation), £ = (~71,..., in) (the new approximation), ZN = ( Z N . 1 , . . . , ZN, n ) , ZN# = Zi -N (z i ) (the Newton approximation), Z H = ( Z H , 1 , . . . , ZH,n) , ZH, i = z, -H (z i ) (the Halley approximation). We recall tha t the correction terms (2.4) and (2.5) appear in the iterative formulas, = z N (z) (Newton's method)