Fractional Newton-Raphson Method Accelerated with Aitken s Method

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1 Frctionl Newton-Rphson Method Accelerted with Aitken s Method Fernndo Brmbil-Pz Deprtmento de Mtemátics - UNAM fernndobrmbil@gmil.com Ursul Iturrrán-Viveros Deprtmento de Mtemátics - UNAM ursul@ciencis.unm.m Anthony Torres-Hernndez Deprtmento de Físic - UNAM nthony.torres@ciencis.unm.m Reyn Cbllero-Cruz Deprtmento de Físic - UNAM rcbllero@ciencis.unm.m rxiv: v [mth.na] 19 Aug 018 Abstrct The Newton-Rphson N-R) method is chrcterized by the fct tht generting divergent sequence cn led to the cretion of frctl [10] on the other hnd the order of the frctionl derivtives seems to be closely relted to the frctl dimension [3] bsed on the bove method ws developed tht mkes use of the N-R method nd the frctionl derivtive of Riemnn-Liouville R-L) tht hs been nmed s the Frctionl Newton-Rphson F N-R) method [9]. In the following work we present wy to obtin the convergence of the F N-R method which seems to be t lest linerly convergent for the cse where the order α of the derivtive is different from one simplified wy to construct the frctionl derivtive nd frctionl integrl opertors of R-L is presented n introduction to the Aitken s method is mde nd it is eplined why it hs the cpcity to ccelerte the convergence of itertive methods to finlly present the results tht were obtined when implementing the Aitken s method in F N-R method. Key words: Newton-Rphson Method Frctionl Clculus Frctionl Derivtive of Riemnn-Liouville Aitken s Method. 1. Newton-Rphson Method For the one-dimensionl cse the N-R method is one of the most used methods to find the roots of function f : R R with f C R) i.e. { R : f ) 0 } this due to its esy implementtion nd its rpid convergence. The N-R method is epressed in terms of n itertion function Φ : R R s follows [1] n+1 : Φ n ) n f n) n 01 1) Df n ) where D d/d. condition 0 is close enough to the root the sequence { n } should be convergent to tht root. [] This method is chrcterized by hving convergence order t lest qudrtic for the cse in which Df ) 0 on the other hnd if f ) is polynomil tht presents root with certin multiplicity m with m N i.e. f ) ) m g) g ) 0 the N-R method converges t lest linerly [1].. Frctionl Clculus Frctionl clculus is brnch of mthemticl nlysis whose pplictions hve been incresing since the lte XX century nd erly XXI century [3]. It rises round 1695 due to the nottion of Leibniz for derivtives of the integer order d n dn f ) n N Figure 1: Illustrtion of N-R method [1] The N-R method is bsed on creting succession { n } of points by mens of the intersection of the tngent line of the function f ) t the point n with the is if the initil Thnks to this nottion L Hopitl ws ble to sk Leibniz in letter bout the interprettion of tking n 1/ in derivtive becuse t tht time Leibniz could not give physicl or geometric interprettion to this question he limited himself to nswering L Hopitl in letter... this is n pprent prdo from which one dy useful consequences will be etrcted" [4]. The nme of frctionl clcultion comes from historicl question since in this 1

2 UNAM Frctionl Newton-Rphson Method Fcultd de Ciencis brnch of mthemticl nlysis we study the derivtives nd integrls of rbitrry order α with α R or C. Currently the frctionl clcultion does not hve unified definition of wht is considered frctionl derivtive this becuse one of the conditions tht is sked to consider n epression s frctionl derivtive is tht the results of the conventionl clcultion re recovered when the order α n with n N [5] mong the most common definitions of frctionl derivtives re the frctionl derivtive of Riemnn-Liouville R-L) nd the frctionl derivtive of Cputo [6] the ltter is usully the most studied since the frctionl derivtive of Cputo llows to give physicl interprettion to problems with initil conditions this derivtive complies with the property of the clssicl clculus tht the derivtive of constnt is null regrdless of the order α of the derivtive however this does not occur with the frctionl derivtive of R-L. Unlike the frctionl derivtive of Cputo the frctionl derivtive of R-L does not llow to give physicl interprettion to the problems with initil condition becuse its use induces frctionl initil conditions however the fct tht this derivtive does not cncel the constnts for α with α N llows to obtin spectrum" of the behvior tht the constnts hve for different orders of the derivtive which is not possible with the conventionl clculus. It must be tken into ccount tht those functions tht do not hve clssicl derivtive or wek derivtive these cn present frctionl derivtive [7] however tht is subject tht eceeds the purpose of this document..1. Frctionl Derivtive of Riemnn-Liouville The frctionl derivtive opertor R-L is constructed in simplified wy tking into ccount tht the integrl opertor is defined for loclly integrble function s I f ) : f t)dt pplying twice the integrl opertor we obtin I f ) 1 f t)dt ) d 1 I 1 f 1 )d 1 mking integrtion by prts tking u I 1 f 1 ) nd dv d 1 I f ) 1 I 1 f 1 ) 1 f 1 )d 1 I f ) I f ) t)f t)dt repeting the previous process pplying three times the integrl opertor we obtin I 3 f ) I 1 f 1 )d 1 mking integrtion by prts tking u I 1 f 1 ) nd dv d 1 I 3 f ) 1 I 1 f 1 ) 1 I 1 f 1 )d 1 I f ) I I f )) 1 ) I 1 f 1 )d 1 relizing gin integrtion by tking prts u I 1 f 1 ) nd dv 1 )d 1 I 3 f ) 1) I 1 f 1 ) + 1 ) f 1 )d 1 1 t) f t)dt repeting the bove process n times it is possible to obtin the epression known s n-fold iterted integr [6] I n f ) 1 t) n 1 f t)dt n 1)! to mke generliztion of the previous epression it is enough to tke into ccount the reltionship between the gmm function nd the fctoril function Γ n) n 1)!) nd doing n α with α CRe α) > 0) the epression for the frctionl integrl opertor of R-L is obtined [6] I α f ) 1 t) α 1 f t)dt ) Γ α) tking into ccount tht the differentil opertor is the inverse opertor on the left of the integrl opertor i.e. D n I n f ) f ) we cn consider etending this nlogy to frctionl clculus using the epression D α f ) I α f ) however this would cuse convergence problems becuse the gmm function is not defined for α Z to solve this is defined D α f ) : D n I n I α f ) D n I n α f ) 3) tking α CRe α) > 0) with n Re α) +1 we get the epression for the frctionl derivtive opertor of R-L [6] D α 1 d f ) n Γ n α) d n t) n α 1 f t)dt the frctionl derivtive of R-L in its unified version with the frctionl integrl of R-L is given by the following epression

3 UNAM Frctionl Newton-Rphson Method Fcultd de Ciencis D α f ) 1 t) α 1 f t)dt Re α) < 0 Γ α) 1 d n Γ n α) d n t) n α 1 f t)dt Re α) 0 where n Re α) + 1. The frctionl derivtive of R-L of monomil of the form f ) c) m with m N nd c R through the eqution 4) is epressed s D α 1 d f ) n Γ n α) d n t) n α 1 t c) m dt 5) tking the vrible chnge t c + c)u in the integrl 4) 3. Frctionl N-R Method The N-R method is useful to find the roots of polynomil of degree n with n N however it is limited to not being ble to find comple roots of the polynomil if rel initil condition is tken to solve this problem nd to develop method tht is ble to find both the rel nd comple roots of polynomil regrdless of whether the initil condition is rel is mde use of the N-R method with the implementtion of the frctionl derivtive of R-L. Tking into ccount tht polynomil of degree n is composed of n + 1 monomios of the form m with m N we cn tke the eqution 6) with 0 getting 0 Dα m Γ m + 1) Γ m α + 1) m α 7) nd using the eqution 1) we cn define the F N-R method for f ) P n [] s follows t) n α 1 t c) m dt 1 c) m+n α 1 u) n α 1 u m du c c the previous result cn be epressed in terms of the Bet function nd the incomplete Bet function [8] f n+1 : Φα n ) n n ) n 01 0 Dα f ) n where 0 < α <. 8) 1 c) m+n α 1 u) n α 1 u m du c c 1 c c) m+n α 1 u) n α 1 u m c du 1 u) n α 1 u m du 0 0 c) m+n α [ Bn αm + 1) B c n αm + 1)] c so the eqution 5) cn be rewritten s D α Γ m + 1) d f ) n { )} Γ m + n α + 1) d n c) m+n α c G c where G ) c c : 1 B c c n αm + 1) Bn αm + 1) when c we hve to G0) 1 D α ) m Γ m + 1) d n Γ m + n α + 1) d n )m+n α tking into ccount tht in the conventionl clculus d n d n k k! Γ k n)! k n k + 1) Γ k n + 1) k n we get the frctionl derivtive of R-L for monomil of the formf ) ) m D α ) m Γ m + 1) Γ m α + 1) )m α. 6) Figure : Illustrtion of some lines generted by the F N-R method To understnd why the F N-R method hs the bility to enter comple spce unlike the clssicl N-R method it is enough to observe the frctionl derivtive of R-L with α 1/ of the constnt function f 0 ) 1 0 nd the identity function f 1 ) 1 0 D1/ f 1 ) 0 D1/ f 0 ) Γ ) Γ 3/) 1/ Γ 1) Γ 1/) 1/ for polynomils of degree n 1 n initil condition must be tken 0 0 this s consequence of the frctionl derivtive of R-L order α N of the constnts re of the form α. When we tke α 1 you hve two cses: i) When we tke n initil condition 0 > 0 the sequence { n } generted by the eqution 8) cn be divided into three prts this hppens becuse there my be N N for which { n } N 1 n0 R+ nd N R consequently the succession { n } n N +1 C. 3

4 UNAM Frctionl Newton-Rphson Method Fcultd de Ciencis if ξ is root of f ) Φαξ) ξ DΦαξ) 1 Df ξ) D n 1 gαξ) α 1 0 α 1 Figure 3: Frctionl Derivtive of R-L with α [01] of the function f 1 ) D Φαξ) Df ξ)dn gαξ) D n 1 gαξ) ) D f ξ) D n 1 gαξ) α 1 D f ξ) Df ξ) α 1 performing Tylor series epnsion of the itertion function 9) round ξ nd ssuming tht ξ << 1 we cn despise superior terms of order two Figure 4: Frctionl Derivtive of R-L with function f 0 ) lph in[0 1] of the ii) When we tke n initil condition 0 < 0 the succession { n } n 1 C Convergence of the F N-R method Considering f ) s function with roots of multiplicity one nd defining the complementry function gα) D 0 I n α f ) g1) f ) the eqution 8) cn be written s Φα) f ) D n 1 gα) 9) clculting the first nd second derivtives of 9) we get DΦα) 1 Df ) D n 1 gα) + f )Dn gα) D n 1 gα) ) Φα) Φαξ) + DΦαξ) ξ) + D Φαξ) ξ) Φα) Φαξ) DΦαξ) D Φαξ) ξ + ξ from the previous results we get Φαξ) ξ DΦαξ) D Φαξ) ξ + ξ α 1 D Φαξ) ξ α 1 mking the ssumption tht ξ << ξ cn be considered Φαξ) ξ DΦαξ) ξ α 1 D Φαξ) ξ α 1 10) whereby the F N-R method converges t lest linerly when α 1 nd t lest qudrticlly when α 1 [1]. 4. The Aitken s Method The Aitken s method or lso known s the method of Aitken [] is one of the first nd simplest methods to ccelerte the convergence of given convergent sequence { i } N i1 i.e. D Φα) D f ) D n 1 gα) + Df )Dn gα) D n 1 gα) ) + f )Dn+1 gα) D n 1 gα) ) f ) Dn gα)) D n 1 gα) ) 3 lim i ξ i this method llows to trnsform the succession { i } N i1 to succession { i } N i1 which in generl converges fster 4

5 UNAM Frctionl Newton-Rphson Method Fcultd de Ciencis to the vlue ξ tht the originl sequence under certin circumstnces the Aitken s method cn ccelerte the liner convergence of method to qudrtic convergence so it is usully used to ccelerte the eisting methods used to find the roots of function [1] []. To illustrte Aitken s method suppose tht the succession { i } N i1 converges to the vlue ξ s geometric sequence with fctor k with k < 1 such tht lim i i ξ to note tht the sequence { i converges more quickly i1 thn the sequence { i } N i1 we cn consider in the eqution 11) } N i+1 ξ k i ξ) i 01 11) where the vlue of ξ cn be used by using the system of equtions k k 0 + δ i lim δ i 0 k < 1 i i+1 ξ k i ξ) 1) i+ ξ k i+1 ξ) 13) subtrcting the eqution 1) to 13) nd clering k k i+ i+1 i+1 i on the other hnd clering ξ from 1) ξ k 1 + 1) i i+1 k 1 i i+1 i k 1 substituting the vlue of k in the previous epression i i+1 ξ) i ξ) k 1) i ξ) nlogously i+1 k 1) i+1 ξ) k [ 1) i+1 i ) + k + 1) i ξ) k 1) + k + 1) ] i ξ) whereby i k 1) i ξ) [ k 0 1) + µ i ] i ξ) where lim i µ i 0 of the previous results nd the eqution 15) we obtin ξ i i+1 i ) i+1 i ) i+ i+1 ) i+1 i ) i i+1 i ) i+ i+1 + i defining the difference opertor so tht with which i : i+1 i i ) i i+1 i i i+ i+1 + i ξ i i) i 14) the method is nmed considering the eqution 14) therefore the method of Aitken consists in generting new sequence i i i) i i+1 i ) 15) i i+ i+1 + i thus i ξ i ξ) [k δ i ) i ξ)] [k 0 1) + µ i ] i ξ) i ξ i ξ 1 k δ i ) k 0 1) + µ i i lim ξ i i ξ 0 which shows tht in generl the speed of convergence of the succession { i succession. } N i1 is greter thn tht of the originl 5. Results of the F N-R Method The following results using the F N-R method with the Aitken s method were performed with Python using the softwre Ancond Nvigtor ) Selected polynomil: f )

6 UNAM Frctionl Newton-Rphson Method Fcultd de Ciencis roots obtined with the F N-R method for fied initil condition 0 9 nd different orders α from the derivtive ) Without Aitken s method f ) Iter e e e e e e e e e e e e b) With Aitken s method f ) Iter e e e e e e e e e e e e ) Selected polynomil: f ) roots obtined with the F N-R method for fied initil condition nd different orders α from the derivtive ) Without Aitken s method f ) Iter e e e e e e e e e e e b) With Aitken s method f ) Iter e e e e e e e e e e e ) Selected polynomil: f ) roots obtined with the F N-R method for fied initil condition 0 3 nd different orders α from the derivtive ) Without Aitken s method f ) Iter e e e e e e e e e e b) With Aitken s method f ) Iter e e e e e e e e e e Conclusions Using the proposl tht is eposed in this pper to obtin the convergence of the F N-R method llows us to understnd the reson for the lrge number of itertions tht were needed to obtin the results eposed in [9] considering tht 6

7 UNAM Frctionl Newton-Rphson Method Fcultd de Ciencis in generl the clssicl N-R method presents qudrtic convergence. Tking into ccount the results in this pper it is noted the gret efficiency of the Aitken s method to ccelerte the convergence of the F N-R method since it eposes the possibility tht the method is t lest linerly convergent for the cse in which the order of the derivtive α is different from one. So in conjunction with the Aitken s method it is concluded tht the F N-R method becomes efficient to clculte the mjority of the roots of polynomil using only rel initil conditions. References [1] Robert Plto. Concise numericl mthemtics. Number 57. Americn Mthemticl Soc [] Josef Stoer nd Rolnd Bulirsch. Introduction to numericl nlysis volume 1. Springer Science & Business Medi 013. [3] Fernndo Brmbil editor. Frctl Anlysis - Applictions in Physics Engineering nd Technology. InTech 017. [4] Kenneth S Miller nd Bertrm Ross. An introduction to the frctionl clculus nd frctionl differentil equtions. Wiley-Interscience [5] Keith Oldhm nd Jerome Spnier. The frctionl clculus theory nd pplictions of differentition nd integrtion to rbitrry order volume 111. Elsevier [6] Rudolf Hilfer. Applictions of frctionl clculus in physics. World Scientific 000. [7] Sbir Umrov. Introduction to frctionl nd pseudodifferentil equtions with singulr symbols volume 41. Springer 015. [8] G Arfken nd HJ Weber. Mthemticl Methods for Physicists [9] Fernndo Brmbil nd Anthony Torres. Frctionl newton-rphson method. rxiv preprint rxiv: [10] SG Tthm. Frctls derived from newton-rphson itertion. See sgtthm/newton/