010 Americn Control Conerence Mrriott Wterront, Bltimore, MD, USA June 30-July 0, 010 FrC07.4 On Second Derivtive-Free Zero Finding Methods Mohmmed A. Hsn Deprtment o Electricl & Computer Engineering University o Minnesot Duluth E.mil:mhsn@d.umn.edu Abstrct: High order root-inding lgorithms re constructed rom ormuls or pproximting higher order logrithmic nd stndrd derivtives. These ormuls re ree o derivtives o second order or higher nd use only unction evlution nd/or irst derivtives t multiple points. Richrdson extrpoltion technique is pplied to obtin better pproximtions o these derivtives. The proposed pproches resulted in deriving mily o root-inding methods o ny desired order. The irst member o this mily is the squre root itertion or Ostrowski itertion. Additionlly, higher order derivtives re pproximted using multipoint unction evlutions. We lso derived procedure or ourth order methods tht re dependents only on the unction nd its irst derivtive evluted t multiple points. Keywords: Zeros o polynomils, Zeros o nlytic unctions, derivtive ree methods, Root itertions, root-inding, order o convergence, Hlley s Method, Newton s Method, Squre root itertion, higher order methods, Ostrowski method 1 Introduction Methods or computing zeros o nlytic unctions my be clssiied into one-point nd multi-point zero-inding methods. In one-point zero-inding methods, new pproximtions in ech itertion re ound by using the vlues o nd perhps its derivtives t only one point. In multipoint methods, new pproximtions re obtined by using the vlues o nd sometimes its derivtives t number o points. Newton s nd Hlley s methods re exmples o onepoint methods, while the secnt nd Muller s methods re exmples o multi-point methods. Anlysis relted to onepoint zero-inding methods ppers in [1]-[], while multipoint methods re nlyzed in [3]-[4]. Good tretments o generl root-inding methods cn be ound in [5]-[6] nd the reerences therein. In this pper, optiml pproximtion o derivtives using multi-point computtion o the originl unction is pplied in the development o derivtive-ree methods. Other methods in this work re bsed on pproximting the irst nd higher order derivtives o log((x)), the nturl logrithm o (x). These pproximtions re then utilized or developing derivtive-ree multi-point root itertion methods. Preliminries Assume tht ech zero o is is rth order i For given lgorithm, the order o convergence is deined s ollows: Let x k be sequence o complex numbers nd λ C. I there is rel number r 1 nd constnt C r IR, such tht x k+1 λ C r x k λ r s k whenever x 0 is suiciently ner λ, then the sequence x k is sid to be order r convergent to λ. I r = 1, we urther require tht C r < 1 nd we cll C r the symptotic liner convergence constnt or the sequence i it is the smllest such constnt. Alterntively, ssume tht the sequence x k is generted by the ixed point itertion x k+1 =Φ(x k ) where Φ is nlytic in bounded neighborhood V r o root ξ o polynomil hving only simple roots. I or some ξ we hve Φ(ξ) =ξ, Φ (ξ) =0,, Φ (r 1) (ξ) =0ndΦ (r) (ξ) 0, then the rootinding lgorithm is t lest rth order convergent. Here, Φ (x), Φ (x), Φ (x),, Φ (r) (x), denote the irst, second, third, nd rth derivtives o Φ evluted t the complex number z. We lso use the convention tht Φ (k) =Φi k =0. Most multi-point itertions hve rctionl order o convergence, however it is oten the cse tht one-point methods hve integer order o convergence. The ollowing result provides conditions or given one-point itertion to be o given order. Theorem 1[7]. Let be polynomil o degree n with zeros ξ 1,,ξ n. Let g be nlytic unction in neighborhood o ξ k, k =1,,n. simple. Then the itertion (x) g(x) g (i) (ξ j)= (i+1) (ξ j ) i+1 or j =1,,n nd i =0,,r 1. Hence i ξ is simple zero o, then the Tylor expnsion o g round z = ξ is given s: g(x) = r 1 k=0 (k+1) (ξ) (k + 1)! (x ξ)k + O((x ξ) r ) (1) Additionlly, method is o ininite order i (k+1) (ξ) g(x) = (k + 1)! (x ξ)k = (x) z ξ. k=0 Hence i g cn be expressed s r 1 g = + h k k + O( r ), (1b) (1c) 978-1-444-745-7/10/$6.00 010 AACC 6507
where {h k } r 1 re nlytic unctions round neighborhoods o the zeros o, then Φ is t lest rth order ixed point unction. For r =3, we hve h 1 = (x), (x) h = (x) (x) 4 (x) 3. 6 (x) Proo. A version o this result is stted in [7]. The proo ollows by showing tht Φ(ξ) = ξ, nd Φ (k) (ξ) = 0, or k =1,,r 1. 3 Approximtion o Logrithmic derivtives Let h Csuch tht h 0 nd h is suiciently smll. Let be polynomil o degree n nd consider the unction F deined s F (x, h) =(x + h)(x h) (x). () Clerly, F is n even unction o h nd hence its Tylor expnsion round x contins only even powers o h. Speciiclly, F (x, h) =(x + h)(x h) (x) = h ( (x) (x) (x)) + O(h 4 ). Additinlly, it cn be shown tht (x) (x + h)(x h) (x) = h { } + h 4 3 4 + (4) 1(x) + O(h 6 ). Note tht the expression (x) (x) (x) is the numertor o ( (x) (x) ). Thereore, i x is n pproximtion o simple zero o, then one cn show tht h (x) F (x, h) = (x) (x) (x) (x) + O(h ), (4) or (x) (x) F(x, h) (x) = + O(h ). (x) Now considering the expression nd compring tht with the term in Ostrowski method, we obtin h(x) (x) (x + h)(x h), (5) which is n pproximted squre root itertion. Speciiclly, it cn be shown tht (5) is symptoticlly o order 3 s h 0. The min dvntge o this itertion is tht it only requires unction computtion t three points x, x+h, x h, nd without clculting ny derivtives. Similrly, the second derivtive o log( ) cn be obtined rom the expression (x + h)(x + wh)(x + w h) (x) 3, where w is primitive cube root o 1, i.e., ω = 1+j 3, or ω = 1 j 3. It is esy to veriy tht F 3(x, h) = F 3(x, wh) = F 3(x, w h). This implies tht (3) F 3(x, h) =G 3(x, h 3 ) or some unction G 3. Using this symmetric property nd ter lgebric simpliictions, the expression F 3(x, h) =(x + h)(x + wh)(x + w h) (x) 3 cn be written s F 3(x, h) =h 3 ( (x) 3 3 (x) (x) (x)+ 1 (x) (x)) + O(h 6 ) (6) or F 3(x, h) = 1 (x)3 ( (x) (x) ) h 3 + O(h 6 ). Thus using the cubic root itertion ormul [8] with r = 3, we obtin h(x) φ(x) =z (x + h)(x + wh)(x + w h) (x), (7) 3 3 which is symptoticlly ourth order itertion ner simple zero o. 3.1 Richrdson Extrpoltion This procedure hs been pplied to mny numericl methods in order to improve the ccurcy o the results [9],[10],[11]. Richrdson extrpoltion cn be pplied to ny numericl method which hs n error expnsion in the orm C 1h + C h + C 3h 3 +, where h is prmeter o the method, e.g., intervl width, nd C 1,C, re constnts independent o h. It consists o combining successive pproximtions, using dierent vlues o h, to obtin pproximtions o higher ccurcy. Richrdson extrpoltion hs been pplied in Romberg numericl integrtion [9] nd in Grgg s method or solving ordinry dierentil equtions. As shown erlier i is suiciently smooth, then (x + h)(x h) (x) (x) = h (x) (x) (x) +O(h 4 ), (x) (8) By evluting this expression t h we obtin (x +h)(x h) (x) =(h) (x) (x) (x) (x) (x) + O(h 4 ). (9) Richrdson extrpoltion my be pplied to obtin more ccurte computtion o ( ) which cn be shown to be ( ) = 15(x) 16(x + h)(x h)+(x +h)(x h) 1h (x) + O(h 4 ) Thus vrint but more ccurte representtion o the squre root itertion cn be expressed s Φ(x) =x 1h(x) 15(x) +16(x + h)(x h) (x +h)(x h) (10) which is third order s h 0. 6508
For the cubic root itertion, consider the expression F 3(x, h) = 1 (x)3 ( (x) (x) ) (h) 3 + O(h 6 ). The Richrdson extrpoltion procedure yields 64F 3(x, h) F 3(x, h) 56h 3 (x) 3 = 1 ( (x) (x) ) + O(h 6 ). (11) nd 3 56h(x) 3 64F3(x, h) F 3(x, h) is ourth order ixed point itertion. 4 Approximtions Bsed On the Trpezoidl Rule From the trpezoidl o numericl integrtion it ollows tht (x + h) =(x)+h (x + h)+ (x) + O(h 3 ), nd (x h) =(x) h (x h)+ (x) + O(h 3 ). Hence, (x + h)(x h) (x) = h(x) (x + h) (x h) h 4 { (x + h)+ (x)}{ (x h)+ (x)} + O(h 3 ). (1) Thus the squre root itertion my be pproximted s h(x) G(x, h), (13) where G(x, h) =h { (x + h)+ (x)}{ (x h)+ (x)} h{{ (x + h) (x h)}. Note tht s h 0, (13) tends to converge s third order itertion tht is ree o second nd higher order derivtives. 5 Itertions Bsed on Derivtive Approximtions The irst nd higher order derivtives o the unction my be pproximted using orwrd, bckwrd, or centrl dierences. It is known tht orwrd, bckwrd dierences re o order O(ɛ) while the centrl dierence pproximtion is o order O(ɛ ). Speciiclly, orwrd, bckwrd, nd centrl dierences re respectively given by: (x) (x ɛ) = (x)+o(ɛ), ɛ (14) (x + ɛ) (x) = (x)+o(ɛ), ɛ (14b) (x + ɛ) (x ɛ) = (x)+o(ɛ ). (14c) ɛ Thus i the derivtive (x) in Newton s method is replced with the pproximtion (14c) we obtin the ixed point unction h(x) (x + h) (x h). I h = α(x), then h is smll ner zero o nd we obtin the ollowing second order itertion: α(x) (x + α(x)) (x α(x)). (15) Similrly, replcing the derivtive (x) in Newton s method with the orwrd dierence pproximtion (14b) yields h(x) Φ(x) =x (x + h) (x) = x h(x) (x)h + h (x) + O(h 3 ) (x) = x (x)+ h (x) + O(h ). (16) I h is chosen to be h = (x), then method o third (x) order cn be obtined: (x) (x) (x) (x) + O( (x) ). 5.1 Multi-Point Approximtion Similr nlysis my be pplied to derive multi-point pproximtion o irst order derivtive s ollows. Let multipoint pproximtion be given by (x) = α k ((x + β k ɛ) (x β k ɛ)) + O(ɛ s ). (17) or some nonzero prmeters α k nd distinct prmeters β k. The number s is represents some integer. Tylor expnsion will be used to ind β 1,β, nd α 1,α, nd the highest integer r such tht the β k s re distinct nd ech o α k is nonzero. The prmeters β k must solve polynomil eqution whose coeicients re determined rom the ollowing equtions: α k β k = 1, α k β 3 k =0, α k βk 5 =0. Additionlly, we cn incorporte the equtions (18) 6509
α k = γ 0, α k βk = γ, α k βk 4 = γ 4,, (18b) where γ 0,γ,γ 4, re ree prmeters. This will result in multi prmeter set o solutions o α k,β k. For the cse r =, the two numbers β 1,β re the zeros o the qudrtic polynomil β + 1β + = 0, where 1, re determined s [ 1 γ0 1 γ [ ] Thus = 1 [ 4γ 4γ 0 γ 1 γ 4γ 0 γ 1 ] 1 [ ] [ ] γ = 0 1 ]. (18c). Note tht γ 0, or otherwise β 1 = β = 0, which is not cceptble solution. I we ssume tht γ 0 = 0 then [ ] [ γ ] [ ] = 1/4 4γ γ 1 =, (18d) γ 1/ i.e., β 1β =4γ nd β 1 + β = γ. This leds to complex solutions or β 1 nd β. Another pproch or pproximting (x) is to consider vrition o (17) given by: (x) = α k (x + β k ɛ) (x β k ɛ) β k ɛ + O(ɛ s ). (19) or some integer s. The vribles α k nd β k stisy the ollowing equtions: α k =1, α k βk =0, α k βk 4 =0. (0) Since the sums r α kβ k, r α kβk, 3 re unknown, one my incorporte the ollowing equtions α k β k = γ 1 α k βk 3 = γ 3 α k βk 5 = γ 5, or some nonzero prmeters γ 1, γ 3, nd γ 5. I r = 3, the numbers β 1,β,β 3 cn be determined s the roots o the eqution β 3 + 1β + β + 3 = 0, where 1,, 3 re determined rom the ollowing eqution [ 1 γ1 ] 0 1 [ ] γ3 [ ] 3 γ 1 0 γ 3 0 =. (1) 0 γ 3 0 γ 5 1 Hence 3 = γ 3 γ 1γ 3, γ 3 = γ5, γ 3 1 = γ 1γ 5 γ3γ 1. γ3 These vlues o 1,, 3 will be used to ind the roots o the eqution β 3 + 1β + β + 3 = 0. Then the quntities α 1,α,α 3 re computed by solving liner system o equtions. 6 Fourth Order Methods In this section we derive ourth order methods tht do not depend on second nd higher order derivtives. Thus we consider the ollowing itertion unction: r α k (x β k ) (x γ ). () where α k,β k,γ k re prmeters to be determined. Now, nd + 1 α k (x β k )= α k α k βk + O( 3 ), α k β k (x γ )= γ + γ + O( 3 ). Consequently, r α k (x β k ) = { α k (x γ k ) + 1 α k β k + } { γ + γ + } 1 = 1 { α k α k β k + 1 α k β k α k βk + } 6510
+ γ(γ {1+γ γ + 3 γ + } 4 = 1 { α k ( α k β k γ α k ) α k α k β k ) r ( α kβk 3 γ α k ) + O( 3 )} To derive ourth order methods, Theorem 1 requires tht α k =1, α k β k γ γ(γ 1 α k α k = 1, α k β k = 1 4, α k β k γ r α k = 1 6. (3) I r = 1, then the choice α 1 = 1, nd β 1 γ = 1 yields third order method. For rbitrry prmeters β 1 nd γ, nd α 1 = 1 the method is generlly second order. We next present n itertion method tht is ree o second nd higher order derivtives. The ixed point unction is o the orm α 1(x) r β (x) k(x γ k ). (4) (x) From Theorem 1 it ollows tht the coeicients β k,γ k must stisy the ollowing equtions: β 1 + β =0, β 1γ 1 + β γ = α 1, (5) β 1γ1 + β γ = α 1. For the cse r =, the two numbers γ 1,γ re the zeros o the qudrtic polynomil γ + 1γ + = 0, where 1, re determined s [ ] 1 [ ] 0 α1 α1 = α 1 γ α 1 [ ] = 1 [ ] α1(γ α 1), (6) α 1 The prmeter γ is n rbitrry number given by γ = β 1γ 3 1+ β γ 3. 7 Methods Bsed on Numericl Integrtion Mny multipoint methods cn be developed using the representtion x (x) =()+ (t)dt. (7) The deinite integrl (x) =()+ x (t)dt cn be pproximted using some qudrture ormuls such s Simpson s, Boole s, or Gussin qudrture. These methods pproximte the integrl by weighting sum o the vlues o the unction t vlues o x [, b]. Cuchy integrl ormul sttes tht (k) (x 0) k! = 1 πj C (t) dt (8) (t x 0) k+1 where is nlytic in the region consisting o simple closed contour C, positively oriented, nd ll points in the interior o C. The point x 0 is in the interior o the contour C. Remrk: It is pprent rom Cuchy integrl ormul tht the vlues o (k) (x 0) cn be computed rom the vlues o round the contour C. Similrly, one cn veriy tht b (x)dx = 1 πj C (t) log( t )dt (9) t b is n exct ormul or computing deinite integrl using complex integrtion round closed contour C. Here C is contour contining the intervl [ b]. This ormul is new up to the uthor knowledge. This ormul shows tht b (x)dx my be dependent on vlues o t points tht lie outside the intervl [, b]. The integrl ormul (9) my be used to derive better pproximtion o deinite integrls by incorporting points outside the intervl [, b]. This ide will be explored in orthcoming pper. 8 Exmples In the ollowing three exmples we show tht the proposed methods cn lso be pplied to entire unctions. Exmple 1: Let (x) = sin(x), then h sin(x) sin(x + h) sin(x h) = x h sin h tn(x). (30) As h 0, the quntity h 1, thus the bove ixed point sin h itertion trnsorms into tn(x), which is third order itertion. Exmple : Let (x) = cos(x), then h cos(x) cos(x + h) cos(x h) = x + h sin h cot(x), (31) is third order itertion s h 0. Clerly s h 0, this itertion trnsorms into Φ(x) =x + cot(x). 6511
In the next exmple, the method described in (5) is pplied. Exmple 3: Let (x) =e x 1, then (x) (x + h)(x h) =e x (cosh(h) 1) As in (5) we obtin the ollowing itertion e x 1 h ex (cosh(h) 1) = x sinh( x )+O(h ) Note tht ner x = 0 we hve Φ(x) =x sinh( x )+O(h )= x3 4 +O(x5 )+O(h ). (3) This shows tht s h 0, the itertion converges cubiclly to x = 0 provided tht h is very smll number. As h 0, the itertion (3) simpliies to sinh( x ), (33) which is third order. A generliztion o this itertion hs the orm α sinh( x α ), where α is ny nonzero number. Clerly better convergence properties cn be obtined i α is chosen to be lrge number. 9 Miscellneous Third Order Methods We stte here ew other methods tht re ree o second nd higher order derivtives: (x), (34) (x) (x (x) ) (x) (x), (35) ( x(x (x) )) (x) (x){ 1 (x) + 1 (x (x) )}, (x) (36) (x), (x) + { (x (x) (x) )} (37) (x) 1+ (x). (38) (x), (39) The ixed point itertions (34)-(37) re third order ner simple zero o. The ixed point itertions (38) is third order i (ξ) = 1 while (39) is third order i (ξ) = 1 nd (ξ) = 0, where ξ is simple zero o. 10 Conclusion Most higher order root inding methods require evlution o unction nd/or its derivtives t one or multiple points. There re cses where the derivtives o given unction re costly to compute. In this pper, higher order methods which do not require computtion o ny derivtives higher thn two re derived. Asymptotic nlysis hs shown tht these methods re pproximtions o root itertions. One o the min etures o the proposed pproches is tht one cn develop multi-point derivtive-ree methods o ny desired order. For lower order methods, these correspond to the Newton, nd Ostrowski itertions. Severl exmples involving polynomils nd entire unctions hve shown tht the proposed methods cn be pplied to polynomil nd non-polynomil equtions. Reerences [1] A. S. Householder, The Numericl Tretment o Single Nonliner Eqution, McGrw-Hill, New York, 1970. [] P. Henrici, Applied nd computtionl complex nlysis, Vol.I, New York: John Wiley nd Sons Inc.,1974. [3] P. Jrrtt, Some Fourth Order Multipoint Itertive Methods or Solving Equtions, Mthemtics o Computtion, Vol. 0, No. 95. (Jul., 1966), pp. 434-437. [4] M. K. Jin, Fith order implicit multipoint method or solving equtions, BIT Numericl Mthemtics, Vol 5, Number 1, Mrch, 1985, Pp. 50-55. [5] A. M. Ostrowski, Solution o Equtions in Eucliden nd Bnch Spces, Acdemic, New York, 1973. [6] J. F. Trub, Itertive Methods or the Solution o Equtions, Prentice-Hll, Englewood Clis, NJ, 1964. [7] M. A. Hsn, On the Derivtion o Higher Order Root- Finding Methods, Americn Control Conerence, 007, pp:38-333. [8] M. A. Hsn, Derivtive-ree mily o higher order root inding methods, Americn Control Conerence, 009. ACC 09. 10-1 June 009 Pge(s):5351-5356. [9] R. L. Burden, J. D. Fires, A.C. Reynolds, 1981. Numericl Anlysis, Second Edition, Prindle, Weber, nd Schmidt, Boston, Mss. [10] M. Mki, 1985. Richrdson Extrpoltion, Nucler Science nd Engineering. Volume 89. p. 38-383. [11] S.M Lee, 1986. Comments o Richrdson Extrpoltion, Nucler Science nd Engineering, Volume 9. p. 489. 651