MATEMATIKA, 2013, Volume 29, Number 2, 159 171 c Department of Mathematcal Scences, UTM Quadratc Bezer Homotopy Functon for Solvng System of Polynomal Equatons 1 Hafzudn Mohamad Nor, 2 Ahmad Izan Md. Ismal and 3 Ahmad Abdul Majd 1,2,3 School of Mathematcal Scences, Unverst Sans Malaysa, 11800 Gelugor, Penang, Malaysa e-mal: 1 hafz97my2001@yahoo.com, 2 zan@cs.usm.my, 3 majd@cs.usm.my Abstract We compare standard homotopy functon wth a proposed quadratc Bezer homotopy functon to see whch method has greater applcablty and greater accuracy. We test the methods on system of polynomal equatons by usng Newton-Homotopy Contnuaton method. The results obtaned ndcate the superor accuracy of our proposed quadratc Bezer homotopy functon. Keywords Numercal method; Polynomal equatons; Homotopy functon. 2010 Mathematcs Subject Classfcaton 65H05; 65H10; 65H20. 1 Introducton In ths paper we consder the soluton of the system of n polynomal equatons f 1 (x 1, x 2,, x n ) f 2 (x 1, x 2,, x n ) F(x) =. = 0 (1) f n (x 1, x 2,, x n ) where x = {x 1, x 2,..., x n } usng homotopy contnuaton method (HCM). A popular method for solvng (1) s the Newton method whch s derved from the Taylor expanson seres [1]. In recent years, the study of the soluton of polynomal equatons especally on system of polynomal equatons usng methods based on the concept of homotopy from Topology has attracted consderable nterest. Rafq and Awas [2] stated that the homotopy contnuaton method (HCM) has been known snce the 1930s. However, more recent work n the development of homotopy concepts for nonlnear algebrac equaton was developed n the 1970s [3]. Ths was followed by works of Garca and Zangwll [4], Melhem and Rhenboldt [5], Morgan [6] and Watson [7,8]. Alexander and Yorke [3] stated that HCM nvolves numercally fndng the soluton of a problem by startng the soluton from the soluton of a known problem and contnung the soluton as the known problem s homotoped to the gven problem. Alexander and Yorke [3] also descrbed the connecton between algebrac topology and the contnuaton method. Garca and Zangwll [4] suggested a procedure to obtan all solutons to certan systems of n equatons n the complex doman. Let z q + P (z) = 0, = 1, 2,...,n. (2) where z s the complex vector, q s the large nteger and P(z) = F(z) 1. By rewrtng (2), we obtan (z q 1) + (P (z) + 1) = 0 (3)
160 Hafzudn Mohamad Nor, Ahmad Izan Md. Ismal and Ahmad Abdul Majd To start wth, the authors defned homotopy functon as (z q 1) + t(p (z) + 1) = 0 (4) where z s the complex vector, q s the large nteger and t [0, 1] s a load parameter. Snce 1 0, thus we wll get a trval soluton when t = 0. Then, the authors rewrte (4) as z q By assgnng z q From (6), we have As t 1, we have (1 t)(z q 1) + t(p (z) + 1) = 0 (5) 1 = G(z) and P (z) + 1 = F(z), then we have (1 t)g(z) + tf(z) = 0 (6) F(z) G(z) = 1 1 t (7) F(z) lm t 1 G(z) = 0 (8) Eq. (8) showed the relatonshp between the auxlary homotopy functon and the gven functon n the homotopy functon (6). Garca and Zangwll [4] also dscussed the meanng of the terms - homotopy paths, monotoncty of the paths, smplcal pvotng and pecewse lnear paths. Melhem and Rhenboldt [5] compared several methods from local teratve and contnuaton path methods for determnng turnng ponts. For nstance, Abbott s method [9], Moore and Spence s method [10], Seydel s method [11], Smpson s method [12], Rhennboldt s method [13] and so on. The results showed the Rhenboldt s method s most approprate because t has the hghest degree of relablty among all methods [5]. Snce HCM s a method of solvng dvergence problem, therefore a body of knowledge about how to determne the turnng ponts are become mportant. Morgan [6] used SYMPOL (systems of polynomals) to conduct the mplementaton. SYMPOL was desgned to fnd all solutons to a system of n polynomal equatons wth complex coeffcents n n unknowns. The author also nvestgated three cases of choosng auxlary homotopy functon G(x) havng a theorem developed n [6]. The auxlary functons selected to form the homotopy functons were () H(x, t) = (1 t)(x x 0 ) + t( x 2 + 1) (9) () H(x, t) = (1 t)(x 2 x 2 0) + t( x 2 + 1) (10) () H(x, t) = (1 t)(x 3 x 3 0 ) + t( x2 + 1) (11) where the target functon was F(x) = x 2 + 1 = 0. Morgan [6] also descrbed the basc concepts of HCM such as homotopy functon and the auxlary homotopy functon. Homotopy functon, denoted as H(x, t), s a connecton between the start and target functons, whle the auxlary homotopy functon s the startng functon. Accordng to Watson [7], homotopy methods are theoretcally powerful, and f constructed and mplemented properly, are robust, numercally stable, accurate, and practcal.
Quadratc Bezer Homotopy Functon for Solvng System of Polynomal Equatons 161 In [8], the homotopy functon was defned n the complex doman. The choce of auxlary homotopy functon s defned by G(x) = b x d a, = 1, 2,...,n (12) where a and b are nonzero complex numbers. There are two types of total degree, denoted by d and d. The total degree of polynomal F(x), d s d = max k n d jk (13) j=1 and the total degree of the entre system (1), d s d = d 1 d 2... d n. (14) Accordng to Kotsreas [14], the total number of geometrcally solated solutons and solutons at nfnty s no more than that gven by equaton (14). By usng (12) and (13), the nvestgated homotopy functons by Morgan [6] can be smplfed as () H(x, t) = (1 t)(x x 0 ) + tf(x) (15) () H(x, t) = (1 t)(x d x d 0 ) + tf(x) (16) () H(x, t) = (1 t)(x d+1 x d+1 0 ) + tf(x). (17) Jalal and Seader [15] analyzed the stablty of multphase and reactng systems by usng HCM. The accuracy of the ntal guess was not mportant. Jalal and Seader focused on the use of the Newton, fxed-pont, and affne homotopes whch are as follows () Newton homotopy () Fxed-pont homotopy H(x, t) = tf(x) + (1 t)[f(x) F(x 0 )] (18) () Affne homotopy H(x, t) = tf(x) + (1 t)(x x 0 ) (19) H(x, t) = tf(x) + (1 t)f (x 0 )(x x 0 ) (20) where F(x) s the polynomal equatons and t [0, 1]. Grtton et al. [16] classfed HCM as a global method whch the user can use to fnd the soluton from an arbtrary ntal guess. For local methods such as the Newton method, the user should have suffcent knowledge regardng the locaton of a root to determne the ntal guess. The closer the ntal guess s to the soluton, the more effcent s the local method. Otherwse, the teratve scheme of the local method wll dverge away from the
162 Hafzudn Mohamad Nor, Ahmad Izan Md. Ismal and Ahmad Abdul Majd actual soluton. Global methods overcome the problem of choosng the approprate ntal guess. Grtton et al. [16] used HCM to study 16 chemcal engneerng problems nvolvng sothermal flash, knetcs n a strred reactor, azeotropc-pont, flow n a smooth ppe, chemcal equlbrum and others. One problem consdered nvolved the equaton, F(x) = 8(4 x)2 x 2 0.186 = 0 (21) (6 3x) 2 (2 x) where x s the fractonal converson of ntrogen. The solutons were tracked by usng HCM and adjustng auxlary functons (.e. the ntal guess). Allgower and Georg [17] defned the polynomal systems (1) as n 1 F(z) = z n + a j z j (22) and equaton (22) s a monc polynomal n whch the coeffcent of z n s equal to 1. Lettng G(z) = z n + b 0 where b 0 0, the authors defned homotopy functon as Hence, we have j=0 H(z, t) = (1 t)g(z) + tf(z). (23) n 1 H(z, t) = z n + t a j z j + (1 t)b 0 + ta 0 (24) j=0 where H(z, t) s measured n complex doman H : C [0, 1] C. Wu [18-22] has conducted extensve research on HCM. Wu [18] ntroduced Ancent Chnese Homotopy method (ACHM) and Wu [19] studed the convergence of Newton-Homotopy contnuaton method (NHCM). Wu [20] compared the tradtonal Adoman Decomposton method wth the Adoman-homotopy contnuaton method (AHCM). Further, Wu [21] nvestgated the crtera of adjustable auxlary homotopy functon and ts gudelnes and Wu [22] developed secant-homotopy contnuaton method (SHCM) from tradtonal secant method. Palancz et al. [23] descrbed HCM n a smpler way. HCM was defned as a method that deforms contnuously from the known roots of the start system nto the roots of the target system. Contnuaton graphs were used to llustrate the mathematcal problems. A smple equaton was used to demonstrate the homotopy contnuaton concepts. Then, Palancz et al. [23] extended the concepts by dscussng nonlnear geodetc problems such as resecton, GPS postonng, as well as affne transformaton. Rahman et al. [24] concerned wth the use of homotopy functons H(x, t) = tf(x) + (1 t)[(x x 0 ) + (F(x) F(x 0 )] (25) to track the approxmate solutons. The authors chose G(x) as a lnear combnaton of fxed pont and Newton functons. In most of above research, the focus was on the auxlary homotopy functon as well as homotopy contnuaton method and not the homotopy functon. The authors [1-2,4,6,14-24] were more comfortable to use standard homotopy functon rather than others. Ths paper wll ntroduce a new homotopy functon whch wll be called the Quadratc Bezer Homotopy Functon (QBHF).
Quadratc Bezer Homotopy Functon for Solvng System of Polynomal Equatons 163 2 Standard Homotopy Functon All homotopy functon H(x, t) mentoned on above used the standard homotopy functon. Let us consder the standard homotopy functon where H(x, t) = (1 t)g(x) + tf(x) (26) G(x) = F(x) = g 1 (x 1, x 2,, x n ) g 2 (x 1, x 2,, x n ). g n (x 1, x 2,, x n ) f 1 (x 1, x 2,, x n ) f 2 (x 1, x 2,, x n ). f n (x 1, x 2,, x n ) and t s an arbtrary parameter whch can vary from 0 to 1,.e. t [0, 1]. Thus, we wll have H(x, 0) = G(x). (29) (27) (28) H(x, 1) = F(x). (30) The homotopy functon cannot stand alone; t must be followed by a method whch s called homotopy contnuaton method. Accordng to Burden and Fares [1], the formula of Newton HCM s as follows where x +1 = x [D x H(x, t)] 1 H(x, t), = 1, 2,..., k. (31) D x H(x, t) =. H 1 H 1 x 1 H 2 H 2 x 1 H n x 1 x 2 x 2....... H n x 2 D x H(x, t) s called the Jacoban matrx. To facltate better understandng, let us consder the 2-dmensonal of homotopy functon whch s n = 1. H(x, t) = (1 t)g(x) + tf(x) (32) where g(x) s an auxlary homotopy functon and f(x) s a gven functon whch s scalar. H 1 x n H 2 x n H n x n Example 2.1. Consder the followng smple equaton defned by Kotsreas [14]. f(x) = (x 2 1 4 )(x2 4) = 0 (33) and the auxlary homotopy functon s g(x) = x 2 1. So that, the homotopy functon (30) wll be H(x, t) = (1 t)(x 2 1) + t(x 2 1 4 )(x2 4) (34)
164 Hafzudn Mohamad Nor, Ahmad Izan Md. Ismal and Ahmad Abdul Majd Snce we need to solve H(x, t) = 0, there are a set equatons of H (x, t /k ) = 0 and there are a set solutons for x where = 0, 1, 2,, k and k = number of teratons. If there are k teratons, the number of equaton nvolved s k+1. Let k =10, so that the equaton vares from 0 untl 1. Graphcally, t can be represented as Fgure 1. Fgure 1: Homotopy Path of Equaton (34) Fgure 1 shows the movement of H(x 0, 0) = x 2 1 = g(x) to H(x 10, 1) = (x 2 1 4 )(x2 4) = f(x) wth t beng unformly ncreased by 0.1 n equaton (34). x 0 = 1 wll move to the x 10 = 2.000618931 and x 10 = -2.000618931 when x 0 = 1. Both approxmatons wll have the same value of f(x 10 ).e. f(x 10 ) = 9.29 10 3 (35) However, ths approxmaton can be mproved by usng a new homotopy functon whch wll be called the Quadratc Bezer Homotopy Functon. Ths wll be dscussed n the next secton. 3 Quadratc Bezer Homotopy Functon Suppose that we want to solve followng a system of n polynomal equatons F(x) = 0. (36) where x = {x 1, x 2, x 3,, x n 1, x n }. We ntroduce a new homotopy functon vz H 2 (x, t) = (1 t) 2 G(x) + 2t(1 t)[(1 t)g(x) + tf(x)] + t 2 F(x). (37) Ths new homotopy functon (37) can also be wrtten as H 2 (x, t) = (1 t) 2 G(x) + 2t(1 t)h(x, t) + t 2 F(x) (38)
Quadratc Bezer Homotopy Functon for Solvng System of Polynomal Equatons 165 where H(x, t) s the standard homotopy functon. Ths new homotopy functon fulflls the two boundary condtons.e. (29) and (30) are stll satsfed when we substtute t = 0 and t = 1 respectvely nto (38). It s nterestng to note that there s a smlar functon of homotopy functon between (26) and (38) when t = 1 2 where H(x, 1 2 ) = H 2(x, 1 2 ) = 1 2 G(x) + 1 2 F(x). (39) Snce we want to solve F(x) = 0, therefore H 2 (x, t) s set to zero by varyng the parameter t from 0 to 1. In other words, we start from auxlary functon, G(x 0 ) = 0 and fnsh when F( x) = 0. As dscussed before, the soluton moves from x 0 untl x and the curves wll move from H 2 (x 0, 0) untl H 2 ( x, 1). The dea of ths new homotopy functon comes from De Casteljau Algorthm [25]. It s known that, De Casteljau Algorthm descrbes the movement of pont n a curve. Homotopy s a movement of a curve to another curve [14,23]. Therefore, we beleve that there s relaton between the De Casteljau algorthm and homotopy concepts. In short, lnear and quadratc curves for De Casteljau and homotopy can be formed as n Table 1. Table 1: Lnear and Quadratc for De Casteljau and Homotopy De Casteljau, P(t) Homotopy, H(x, t) Lnear (1 t)p 0 + tp 1 (1 t)g(x) + tf(x) Quadratc (1 t) 2 P 0 + 2t(1 t)p 1 + t 2 P 2 (1 t) 2 G(x)+ 2t(1 t)[(1 t)g(x) + tf(x)] + t 2 F(x). Let s observe how we obtan (38) wth three reference curves. The recursve constructon of quadratc Bezer homotopy functon s llustrated n Fgure 2. We note that A(x, t) = (1 t)g(x) + th(x, t), B(x, t) = (1 t)h(x, t) + tf(x) Therefore H 2 (x, t) = (1 t)a + tb (40) Then, we have H 2 (x, t) = (1 t)[(1 t)g(x) + th(x, t)] + t [(1 t)h(x, t) + tf(x)] = (1 t) 2 G(x) + 2t(1 t)h(x, t) + t 2 f(x) = H 0 B0(t) 2 + H 1 B1(t) 2 + H 2 B2(t) 2 2 = H B 2 (t) (41) =0
166 Hafzudn Mohamad Nor, Ahmad Izan Md. Ismal and Ahmad Abdul Majd Fgure 2: Recursve Constructon of Quadratc Bezer Homotopy Functon where H 0 = G(x) for t = 0 H 1 = H(x, t) for t (0, 1) H 2 = F(x) for t = 1. B 2 (t) s a Bernsten functon whch s defned as [25] ( ) B 2 2 (t) = (1 ) 2 t 2! =! (2 )! (1 )2 t (42) where : 0, 1, 2, t [0, 1]. An nterestng property of Bernsten functon for standard homotopy functon and QBHF s that the summaton s equal to one. Lnear : Quadratc : 1 B 1(t) =0 = (1 t) + t = 1, 1 B 2(t) = (1 t)2 + 2t(1 t) + t 2 =0 = 1. (43) (44) Both (43) and (44) show that the sum of bnomal expanson always fulfll one of the convex hull property [25]. Another convex hull property s B 2 (t) 0, : 0, 1, 2, t [0, 1]. (45)
Quadratc Bezer Homotopy Functon for Solvng System of Polynomal Equatons 167 Ths contnuaton technque cannot stand alone n that t must be combned wth other methods such as Newton, secant, Adoman method and so on. By these combnatons, the name of the newly developed method wll then change to Newton-Homotopy, Secant- Homotopy, Adoman-Homotopy and so on. Now, we wll consder several examples that compares the standard and new homotopy functons. The method chosen s Newton- Homotopy contnuaton method (NHCM). The formula of classcal Newton method s well-known. The formula of NHCM s x +1 = x [D x H 2 (x, t)] 1 H 2 (x, t), = 1, 2,...,k. (46) where H 2 (x, t) s Quadratc Bezer homotopy functon as (38). 4 Numercal Experments and Dscusson Example 4.1. Consder the followng system of equatons [2] : f 1 (x, y) = x 2 2x y + 1 2 = 0, f 2 (x, y) = x 2 + 4y 2 4 = 0. (47) The auxlary homotopy functon s g 1 (x) = x, g 2 (y) = y and the ntal value (x 0, y 0 ) = (0, 0)are used for NHCM. The results are shown n Table 2 by varyng the number of teratons. Table 2: Comparson between Standard and New Homotopy Functons for Equaton (47) Number of teratons Standard Homotopy Functon 10 f 1 = 4.90 10 3 f 2 = 2.93 10 2 100 f 1 = 3.83 10 5 f 2 = 2.96 10 4 1000 f 1 = 3.72 10 7 f 2 = 2.96 10 6 Quadratc Bezer Homotopy Functon CPU Tme, second f 1 = 2.38 10 4 f 2 = 2.63 10 3 0.0070004 f 1 = 3.28 10 8 f 2 = 2.63 10 7 0.0300017 f 1 = 3.33 10 12 f 2 = 2.66 10 11 0.2180124 where ( x, ỹ) = (1.90067672637127,0.31121856542355) Example 4.2. Consder the followng example [26]: f 1 (x, y, z) = x 2 + y 2 + z 2 1 = 0 f 2 (x, y, z) = 2x 2 + y 2 4z = 0 f 3 (x, y, z) = 3x 2 4y 2 + z 2 = 0 (48) The auxlary homotopy functon, g 1 (x) = x, g 2 (y) = y and g 3 (z) = z and ntal value (x 0, y 0, z 0 ) = (0, 0, 0) are used for NHCM. The results are shown n Table 3.
168 Hafzudn Mohamad Nor, Ahmad Izan Md. Ismal and Ahmad Abdul Majd Table 3: Comparson between Standard and New Homotopy Functons for Equaton (48) Number of teratons Standard Homotopy Functon 10 f 1 = 2.12 10 3 f 2 = 1.21 10 3 f 3 = 1.75 10 3 100 f 1 = 1.82 10 5 f 2 = 1.04 10 5 f 3 = 1.56 10 5 1000 f 1 = 1.79 10 7 f 2 = 1.02 10 7 f 3 = 1.54 10 7 Quadratc Bezer Homotopy Functon f 1 = 1.36 10 4 f 2 = 7.23 10 5 f 3 = 1.13 10 4 f 1 = 1.59 10 8 f 2 = 9.02 10 9 f 3 = 1.36 10 8 f 1 = 1.61 10 12 f 2 = 9.14 10 13 f 3 = 1.38 10 12 where ( x, ỹ, z) = (0.69828860997219,-0.62852429796055,0.34256418968992) CPU Tme, second 0.0200011 0.1260072 1.1730670 Example 4.3. Consder the followng example [6]: f 1 (w, x, y, z) = x + 10y = 0 f 2 (w, x, y, z) = 5(z w) = 0 f 3 (w, x, y, z) = (y 2z) 2 = 0 f 4 (w, x, y, z) = 10(x w) 2 = 0 (49) The auxlary homotopy functon, g 1 (w) = w 1, g 2 (x) = x 4, g 3 (y) = y 1, g 4 (z) = z 2 and ntal value s (w 0, x 0, y 0, z 0 ) = (1, 4, 1, 2) are used. The results are shown n Table 4. The results n Table of Eq. (47), (48), and (49) show that Quadratc Bezer Homotopy Functon has better performance than the standard homotopy functon for solvng the consdered system of nonlnear algebrac equaton. Ths has been ascertaned by usng the followng stoppng crtera F( X k+1 ) < ε (50) where ε = 10 20. CPU tme s assumed not to be an mportant consderaton. 5 Concluson From the partcular set of examples chosen, the use of Quadratc Bezer Homotopy Functon (QBHF) s better than the use of standard homotopy functon. It should be noted though that for the partcular set of examples consdered, the accuracy of approxmate solutons ncreases when the numbers of teratons ncrease. Acknowledgments Part of ths research has been supported by a scholarshp from the School of Mathematcal Scences USM and Kementeran Pengajan Tngg Malaysa.
Quadratc Bezer Homotopy Functon for Solvng System of Polynomal Equatons 169 Table 4: Comparson between Standard and New Homotopy Functons for Equaton (49) Number of teratons Standard Homotopy Functon 10 f 1 = 2.78 10 17 f 2 = 7.14 10 16 f 3 = 2.93 10 2 f 4 = 7.21 10 2 100 f 1 = 6.37 10 16 f 2 = 2.72 10 15 f 3 = 2.88 10 3 f 4 = 5.61 10 3 1000 f 1 = 6.67 10 16 f 2 = 2.67 10 15 f 3 = 2.86 10 4 f 4 = 5.78 10 4 Quadratc Bezer Homotopy Functon f 1 = 3.89 10 16 f 2 = 4.97 10 16 f 3 = 1.79 10 2 f 4 = 2.27 10 2 f 1 = 6.14 10 16 f 2 = 4.09 10 16 f 3 = 1.32 10 4 f 4 = 2.60 10 4 f 1 = 4.20 10 16 f 2 = 4.00 10 16 f 3 = 1.33 10 6 f 4 = 2.65 10 6 CPU Tme, second 0.1240071 0.4270244 3.4601979 where ( w, x, ỹ, z) = (0.00050521152754,0.00142117446223,-0.00014211744622,0.00050521152754) and the actual soluton s (w, x,y, z) = (0,0,0,0) as stated n Morgan [6] References [1] R. L. Burden, J. D. Fares, Numercal Analyss, 9 th Internatonal Edton, Brooks/Cole, Cencag Learnng. 2011. [2] Rafq, A. and Awas, M. Convergence on the homotopy contnuaton method. Int. J. of Appl. Math. And Mech. 2008. 4(6) :62 70. [3] Alexander, J. C. and Yorke, J. A. The homotopy contnuaton method: numercally mplementable topologcal procedures, Trans. Amer. Math. Soc. 1978. 242: 271 284. [4] Garca, C. B. and Zangwll, W. I. Fndng all solutons to polynomal systems and other systems of equatons, Mathematcal Programmng. 1979. 16 :159 176. [5] Melhem, R. C. and Rhenboldt, W. C. A comparson of methods for determnng turnng ponts of nonlnear equatons, Computng. 1982. 29 : 201 226. [6] Morgan, A. P. A method for computng all solutons to systems of polynomal equatons, ACM Transacton on Mathematcal Software. 1983. 9(1) :1-17. [7] Watson, L. T. Numercal lnear algebra aspects of globally convergent homotopy methods, Techncal Report. 1986. [8] Watson, L. T. Globally convergent homotopy methods: A tutoral, Elsever Scence Publshng Co. Inc. 1989: 369 396.
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