ARTICLE IN PRESS. JID:COMAID AID:1153 /FLA [m3g; v 1.79; Prn:21/02/2009; 14:10] P.1 (1-13) Computer Aided Geometric Design ( )

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1 COMAID:5 JID:COMAID AID:5 /FLA [mg; v 79; Prn:/0/009; 4:0] P -) Computer Aded Geometrc Degn ) Content lt avalable at ScenceDrect Computer Aded Geometrc Degn wwwelevercom/locate/cagd Fat approach for computng root of polynomal ung cubc clppng Lgang Lu a,b,, Le Zhang a,b, Bnbn Ln a,b,guojnwang a,b a Department of Mathematc, Zhejang Unverty, Hangzhou, Chna b State Key Lab of CAD&CG, Zhejang Unverty, Hangzhou, Chna artcle nfo abtract Artcle htory: Receved 0 January 008 Receved n reved form 5 February 009 Accepted 6 February 009 Avalable onlne xxxx Keyword: Root fndng Polynomal Quadratc clppng Cubc clppng Th paper preent a new approach, called cubc clppng, for computng all the root of a gven polynomal wthn an nterval In every teratve computaton tep, two cubc polynomal are generated to encloe the graph of the polynomal wthn the nterval of nteret A equence of nterval then obtaned by nterectng the equence of trp wth the abca ax The equence of thee nterval converge to the correpondng root wth the convergence rate 4 for the ngle root, for the double root and uper-lnear 4 for the trple root Numercal example how that cubc clppng ha many expected advantage over Bézer clppng and quadratc clppng We alo extend our approach by enclong the graph of the polynomal ung two lower degree k < n polynomal by degree reducton The equence of nterval converge to the correpondng root of multplcty wth convergence rate k+ 009 Elever BV All rght reerved Introducton Computng the root of ytem of) polynomal equaton an mportant ue n varou geometrc problem Elber and Km, 00; Reuter et al, 007), uch a curve and urface nterecton Sederberg and Nhta, 990; Patrkalak and Maekawa, 00), urface renderng by ray-tracng Nhta et al, 990; Efremov et al, 005), collon detecton Ln and Gottchalk, 998; Cho et al, 006), etc Varou technque have been developed for olvng polynomal equaton, uch a ung Decarte rule ee, eg Colln and Akrta 976); Rouller and Zmmermann 004)) and Sturm theorem ee, eg Hook and McAree 990)) We refer the reader to McNamee 99 00) for a collecton of related reference Sederberg and Nhta Nhta et al, 990) propoed a technque, called Bézer clppng, to fnd the root of polynomal The bac trategy to ue the convex hull property of Bézer curve to dentfy regon of the polynomal whch do not nclude the part of the root Sederberg and Nhta, 990) Combned wth ubdvon, the proce convergng at a quadratc rate for the ngle root and wth a guarantee of robutne Recently, Bartoň and Jüttler preented a new numercal technque, called quadratc clppng to compute all the root of a unvarate polynomal equaton wthn an nterval Bartoň and Jüttler, 007) The bac dea to ue a quadratc polynomal to approxmate the orgnal polynomal baed on degree reducton Then two quadratc polynomal are obtaned to bound the orgnal polynomal and ther root bound the root to be computed Combned wth ubdvon, the technque of quadratc clppng generate a equence of nterval that contan the root of the orgnal polynomal The equence of nterval how a fater convergence rate than the technque of Bézer clppng, whch for ngle root and uper-lnear for double root In th paper, we generalze the approach of quadratc clppng to cubc clppng, whch alo baed on degree reducton The key pont that we ue an optmal cubc polynomal to approxmate the orgnal polynomal and generate a equence * Correpondng author at: Department of Mathematc, Zhejang Unverty, Hangzhou, Chna E-mal addre: lganglu@zjueducn L Lu) /$ ee front matter 009 Elever BV All rght reerved do:006/jcagd Pleae cte th artcle n pre a: Lu, L, et al Fat approach for computng root of polynomal ung cubc clppng Computer Aded Geometrc Degn 009), do:006/jcagd009000

2 COMAID:5 JID:COMAID AID:5 /FLA [mg; v 79; Prn:/0/009; 4:0] P -) L Lu et al / Computer Aded Geometrc Degn ) of nterval that bound the root by computng the root of the cubc polynomal To analyze the convergence rate for cubc clppng, we conder a more general root fndng method, called degree reducton clppng, whch can be een a a unfed form of quadratc clppng and our cubc clppng The concluon about the convergence rate for the degree reducton clppng algorthm mmedately how that cubc clppng ha fater convergence rate than quadratc clppng, whch 4 for ngle root, for double root and uper-lnear 4 for trple root To get the numercal performance of cubc clppng, we make comparon wth Bézer clppng and quadratc clppng through ome example The numercal reult how that cubc clppng ha many advantage over the other two technque when computng the root of ome polynomal Th paper organzed a follow In Secton, we gve a quck ntroducton about polynomal wth Bernten Bézer repreentaton, degree reducton and the Cardano formula for computng the root of cubc polynomal Secton decrbe the degree reducton baed clppng algorthm and the cubc clppng algorthm and the convergence rate are analyzed Secton 4 provde a comparon reult of Bézer clppng, quadratc clppng and cubc clppng The robutne of cubc clppng alo condered Fnally, we conclude th paper wth future work n Secton 5 Prelmnare We decrbe ome prelmnare n th ecton More detal can be found n Bartoň and Jüttler 007) Polynomal pace wth Bernten ba Let Π n be the lnear pace of polynomal of degree n Any polynomal p Π n n a certan nterval [α,β] R can be repreented by t Bernten Bézer BB) repreentaton a n pt) = b B n t;α,β), =0 t [α,β], ) where ) n t α) B n t;α,β)= β t) n, β α) n = 0,,,n, ) are the Bernten ba n [α,β] and b R = 0,,,n) are the BB coeffcent The L norm and the maxmum L ) norm for a polynomal pt) Π n wth repect to an nterval [α,β] are repectvely defned a p [α,β] = p, p [α,β], h p [α,β] = max pt), ) t [α,β] where h = β α the length of the nterval and p, p [α,β] = β α pt)pt) dt the L nner product Furthermore, we defne the maxmum norm of BB coeffcent for p a p [α,β] BB, = max b =0,,n 4) All the norm are nvarant under affne tranformaton of the t-ax Bartoň and Jüttler, 007) More precely, gven any affne tranformaton A : t A 0 + A t 5) wth A 0, the norm of p wth repect to the nterval [α,β] and of p A wth repect to the nterval A[α,β]) are dentcal, whch mean p [α,β] = p A A[α,β] 6) Degree reducton and dual ba The optmal degree k polynomal approxmaton q of a degree n polynomal p n L norm can be deduced by the technque of dual ba developed by Jüttler 998) The polynomal q obtaned by applyng degree reducton to p wth repect to a gven nterval may be computed by multplyng the row vector of the BB coeffcent of p by ome precomputed matrx n a look-up table Bartoň and Jüttler, 007) Pleae cte th artcle n pre a: Lu, L, et al Fat approach for computng root of polynomal ung cubc clppng Computer Aded Geometrc Degn 009), do:006/jcagd009000

3 COMAID:5 JID:COMAID AID:5 /FLA [mg; v 79; Prn:/0/009; 4:0] P -) L Lu et al / Computer Aded Geometrc Degn ) Cardano formula Gven a cubc polynomal repreented by the Bernten ba ) n the nterval [α,β] a gt) = B 0 t;α,β)d 0 + B t;α,β)d + B t;α,β)d + B t;α,β)d, 7) the well-known Cardano formula Borwen and Erdély, 995) gve all t root a t = τ )α + τ β =,, ), andτ gven by τ = v + + v a, τ = ω v + + ω v a, τ = ω v + + ω v a, 8) where ω = e π a complex, and u = d 0 d d 0 d + d d + d d d ) d /D, v = d 0 d d + d 0 d d + d 0 d d + d d d 6d 0 d d 0d d 0 d + d d + d d 6d d d ) d /D, = d 0 d + 4d 0d d d + ) 4d d 6d 0 d d d /4D 4, a = d 0 6d + d )/D, 9) wth D = d 0 + d d + d It known that t a real root, whle t and t are both real root f 0 or conjugate complexe f >0 Computng root va cubc clppng In th ecton, we propoe a novel approach, called cubc clppng, for computng the root of a polynomal baed on degree reducton Computng root va degree reducton The approach of cubc clppng hare the ame procedure a quadratc clppng Bartoň and Jüttler, 007) to olate all the root of the gven polynomal wth a equence of nterval Indeed, t can be alo generalzed to quartc clppng, and even more generalzed cae Therefore, we conder an extenve clppng method baed on degree reducton to compute the root of a polynomal, namely degree reducton clppng method ee k-clp n Algorthm ) The algorthm of k-clp an teratve algorthm to fnd all the root of a polynomal wthn a gven nterval Next, we gve explanaton about ome tep of the algorthm n more detal: In lne of the algorthm, the bet degree k polynomal approxmaton q can be generated ung the technque of dual ba Jüttler, 998; Bartoň and Jüttler, 007) Th acheved by multplyng the row vector of BB coeffcent of p wth ome precomputed degree reducton matrx that tored n a lookup-table Jüttler, 998) Intead of ung the maxmum L )norm p q [α,β] = max pt) qt) t [α,β] n lne, we ue the maxmum L ) norm of BB coeffcent to compute the error bound δ a 0) δ = p q [α,β] BB, = max b c, =0,,n ) where b, c are repectvely the BB coeffcent of p and q of degree n wth repect to [α,β] Hence, from lne 4 and 5, we have mt) pt) Mt), forallt [α,β] The trp encloed by m and M n lne 6 9 bound part of) the gven polynomal and t nterecton wth t-ax cont of one, two, or three nterval that contan the root The equence of the nterval are contructed n a mlar way wth the technque of quadratc clppng Bartoň and Jüttler, 007) In lne 9, the number l of nterval could be,,,ork Pleae cte th artcle n pre a: Lu, L, et al Fat approach for computng root of polynomal ung cubc clppng Computer Aded Geometrc Degn 009), do:006/jcagd009000

4 COMAID:5 JID:COMAID AID:5 /FLA [mg; v 79; Prn:/0/009; 4:0] P4 -) 4 L Lu et al / Computer Aded Geometrc Degn ) : f length of nterval [α,β] ε then : q generate a polynomal of degree k by applyng degree reducton wth repect to the L nner product on [α,β] to p : δ compute bound on p q [α,β] by comparng the Bernten Bézer repreentaton of p and q 4: m q δ {lower bound} 5: M q + δ {upper bound} 6: f the trp encloed by m, M doe not nterect the t-ax wth [α,β] then 7: return ) 8: ele 9: Fnd nterval [α,β ], =,,l, by nterectng m, M wth the t-ax 0: f max =,,l α β α β then : return k-clpp, [α, α + β)]) k-clpp, [ α + β),β]) : ele : S 4: for =,,l do 5: S S k-clpp, [α,β ]) 6: end for 7: return S) 8: end f 9: end f 0: ele : return [α,β]) : end f Algorthm k-clpp, [α, β]) {Degree reducton clppng} Wth a precrbed tolerance ε, the call k-clpp, [α, β]) wll return ome nterval contanng all the root n [α,β] Th technque guarantee to fnd all the root of a polynomal wthn a gven nterval However, mlar to Bézer clppng and quadratc clppng, the approach may produce fale potve anwer e, nterval not contanng any root) f the graph of the polynomal get very cloe to the t-ax It can be ealy een that Algorthm actually the quadratc clppng algorthm Bartoň and Jüttler, 007) when k = Convergence rate In th ecton, we gve the convergence rate for the root fndng by the algorthm k-clp We tart wth two techncal lemma Although the proof of the lemma ue the mlar way a n Bartoň and Jüttler 007), we tll preent the detal n order to make th paper elf-contaned Lemma Gven a polynomal p wth degree n, there ext a contant C 0p dependng olely on p, uch that for all nterval [α,β] [0, ] the bound δ n lne of the algorthm k-clp atfe δ C 0p h k+,whereh= β α Proof Due to the equvalence of norm n fnte-dmenonal real lnear pace, there ext contant C and C uch that r [α,β] BB, C r [α,β] and r [α,β] C r [α,β] ) for all r Π n The contant C and C do not depend on the gven nterval [α,β], nce all the norm are nvarant wth repect to affne tranformaton of the t-ax Therefore, δ = p q [α,β] BB, C p q [α,β] C p Q α [α,β] C C p Q α [α,β] k + )! C C max p k+) t 0 ) h k+, ) t 0 [0,] where Q α the degree k Taylor polynomal at t = α to p and p k+) the k + )-th dervatve Lemma Gven a polynomal p wth degree n, there ext k + contant C p, C p,,c k+)p, dependng olely on p, uch that for all nterval [α,β] [0, ] the degree k polynomal q obtaned by applyng degree reducton to p atfe p q [α,β] C p h k+, p q [α,β] C p h k,, wth h = β α Proof We contruct a new norm n [α,β] a r [α,β] = r [α,β] p k) q k) [α,β] C k+)ph, 4) + h r [α,β] + +h k r k) [α,β] 5) Pleae cte th artcle n pre a: Lu, L, et al Fat approach for computng root of polynomal ung cubc clppng Computer Aded Geometrc Degn 009), do:006/jcagd009000

5 COMAID:5 JID:COMAID AID:5 /FLA [mg; v 79; Prn:/0/009; 4:0] P5 -) L Lu et al / Computer Aded Geometrc Degn ) 5 Due to the affne nvarance of the norm, there ext a contant C, whch doe not depend on the nterval [α,β], uch that Bartoň and Jüttler, 007) r [α,β] C r [α,β] Conequently, 6) p q [α,β] = p q [α,β] + h p q [α,β] + +h k p k) q k) [α,β] C p q [α,β] C p Q α [α,β] C C p Q α [α,β] k + )! C C max p k+) t 0 ) h k+, 7) t 0 [α,β] where Q α the k-th Taylor polynomal at t = α to p Clearly, th mple 4) Baed on the two lemma above, we get the followng theorem decrbng the convergence rate when computng root of a polynomal ung the algorthm k-clp Theorem If the polynomal p ha a root t n [α,β] and provded that th root ha multplcty wth k, then the equence of the length of the nterval whch contan that root ha the convergence rate d = k+ Proof We analyze the equence of generated nterval: [α,β ] ) =0,,, 8) wth the length h = β α whch contan the root t It oberved that the length of the nterval reduce at leat a half n each teraton ee lne 0 and of Algorthm ), hence the length of nterval [α,β ] tend to be 0 A the root t ha multplcty of, we have pt ) = p t ) = = p ) t ) = 0, p ) t ) 0 9) We aume that the -th dervatve p ) t )>0 otherwe we conder the polynomal p ntead of p) Let and x = t t, px) = pt) = a n x n + a n x n + +a x, 0) q x) = q t) = b k x k + +b x + b 0, ) where a =! p) t ),, a n = n! pn) t ), and a > 0, a n 0, ) b 0 = q t ), b = q t ), b = q t ),, b k = k! qk) t ) ) The multplcty of the root can be ether odd or even We dcu the two cae n the followng repectvely I The multplcty odd Let M t) = q t) + δ, m t) = q t) δ 4) Let t and t be the root of M t) and m t) repectvely That, M t ) = q t ) + δ = 0, m t ) = q t ) δ = 0 5) And t bounded by t and t, e, t [t, t ],eefg Let x = t t > 0 and x = t t < 0 From we have M t ) = q x ) + δ = b k x k + +b x + b 0 + δ = 0, 6) Pleae cte th artcle n pre a: Lu, L, et al Fat approach for computng root of polynomal ung cubc clppng Computer Aded Geometrc Degn 009), do:006/jcagd009000

6 COMAID:5 JID:COMAID AID:5 /FLA [mg; v 79; Prn:/0/009; 4:0] P6 -) 6 L Lu et al / Computer Aded Geometrc Degn ) Fg Root t wth odd multplcty bounded by [t, t ] where t and t are repectvely the root of M and m b x bk x k + +b +x + + b x + +b x + b 0 + δ 7) Gven a contant ε atfyng p ) t )>ε > 0ay,ε = p) t )>0), nce p ) contnuou and due to Lemma, the nequalte p ) p ) t ) [α,β ] ε and ) q p ) [α,β ] ε 8) hold for uffcently large Thee two nequalte mply ) q p ) t ) [α,β ] ε, hence, we have a bound etmaton about q ) a q ) t) p ) t ) ε > p) t )>0, t [α,β ], 0) for uffcently large Therefore, we have 9) b a > 0 ) Secondly, by Lemma and Lemma, we have b 0 + δ = M t ) = M t ) pt ) M t ) q t ) + q t ) pt ) p q [α,β ] + δ C 0p + C p )h k+, ) b = q t ) p t ) p q [α,β ] C p h k,, b C p h k + ) Thrdly, we have b + = + )! q+) t ) + )! p+) t ) + + )! q+) t ) + )! p+) t ) a )! C +)ph k { max a +, } := D +)p,, b k D kp, where D +)p,,d kp are contant olely dependent on p Therefore, we have the bound etmaton for b x a follow b x bk x k + +b +x + + b x + +b x + b 0 + δ x + bk x k + +b + + b h + + b h + b 0 + δ x D kp x k + + D+)p x ) + C 0p + C p + +C p )h k+ 5) A x h and h tend to 0, then D kp x k + + D +)p x tend to 0 Hence, we have Pleae cte th artcle n pre a: Lu, L, et al Fat approach for computng root of polynomal ung cubc clppng Computer Aded Geometrc Degn 009), do:006/jcagd )

7 COMAID:5 JID:COMAID AID:5 /FLA [mg; v 79; Prn:/0/009; 4:0] P7 -) L Lu et al / Computer Aded Geometrc Degn ) 7 b x b x + C 0p + C p + +C p )h k+ 6) for uffcently large Let D 0p = C 0p + C p + +C p Then D 0p h k+ b x 4 a x 7) Therefore we have x 4D 0p a k+ h 8) Smlarly, we have the bound for the root t a follow x 4D 0p a k+ h 9) A x < 0, x > 0, we have h + = t t =x x 4D 0p + D 0p) a Hence, the equence {h } =0,,, ha the convergence rate k+ for the cae of odd II The multplcty even k+ h 40) A t a root wth even multplcty, t mut be bounded by the two root x and x of m, for all but fntely many value of, eefg Let m t) = q t) δ 4) From m x ) = q x ) δ = 0, we have b x bk x k + +b +x + + b x + +b x + b 0 δ 4) LkentheproofofcaeI,wehave x = o h k+ ), and by conderng m x ) = q x ) δ = 0) x = k+ ) o h Therefore we have h + = t t = x k+ ) x =o h Thu, we complete the proof, and the equence h ) =0,,, ha the convergence rate k+ 4) 44) 45) Fg Root t wth even multplcty bounded by [t, t ] where t and t are the root of m Pleae cte th artcle n pre a: Lu, L, et al Fat approach for computng root of polynomal ung cubc clppng Computer Aded Geometrc Degn 009), do:006/jcagd009000

8 COMAID:5 JID:COMAID AID:5 /FLA [mg; v 79; Prn:/0/009; 4:0] P8 -) 8 L Lu et al / Computer Aded Geometrc Degn ) Table Convergence rate of root fndng approache baed on degree reducton root multplcty ngle root double root trple root quadruple root, etc k+ k+ k+ k-clp k CubClp 4 QuadClp BezClp Table Number of operator per teraton for varou value of the degree degree n ± arctan nco) 4 CubClp QuadClp BezClp CubClp QuadClp BezClp CubClp QuadClp BezClp Cubc clppng algorthm When we et k = n Algorthm, then we have the approach of cubc clppng CubClp for computng the root of a polynomal We ue the Cardano formula 8) to compute the root of m and M n lne 6 n the algorthm From Theorem, t ealy concluded that cubc clppng algorthm CubClp ha fater convergence rate than quadratc clppng, whch 4 for ngle root, for double root and 4 for trple root In the remander of th paper, we refer BezClp a Bézer clppng and QuadClp a quadratc clppng The convergence rate of dfferent clppng method are ummarzed n Table 4 Expermental reult and comparon We ue fve crtera to compare cubc clppng wth Bézer clppng and quadratc clppng a Bartoň and Jüttler 007) Thee nclude convergence rate, number of operaton per teraton tep, tme per teraton tep, number of teraton needed to acheve a certan precrbed accuracy, and computng tme needed to acheve a certan precrbed accuracy 4 Convergence rate, number of operaton and tme per teraton tep The convergence rate of the three algorthm are ummarzed n Table It how that cubc clppng ha hgher convergence rate than Bézer clppng and quadratc clppng n all the cae of the ngle root, double root and trple root Neverthele, the computaton effort per teraton tep alo mportant to evaluate an algorthm Bartoň and Jüttler, 007) Table how the number of operaton needed per teraton, where t aumed that one new nterval generated e, l = n lne 9 of Algorthm ) and that th nterval ha a progrevely length hrunk by more than ee Bartoň and Jüttler, 007) In the cae of cubc clppng, nce the root of the cubc equaton have three dtnct expreon ee Cardano formula 8)), the operator needed n computaton dffer a lot, eg, f <0, then we need nco) operator to compute the real root x, x and x, whle f >0, the only real root x wthout any nco) operator Here, we aume the new nterval obtaned by the root x or x wth the larget computaton effort For large degree n, both algorthm how comparable computaton cot We mplement the two algorthm on a PC wth IntelR) PentumR) M CPU 7 GHz) wth 5 MB of RAM Table how the computaton tme cot n each teraton It reaonable that cubc clppng take more tme nce addtonal operaton are needed at each teraton 4 Number of teraton and computng tme v accuracy To analyze the relaton between the computatonal effort and the dered accuracy, we provde fve numercal example, whch repreent polynomal wth a ngle root, a double root, two root whch are very cloe near double root ) and three root are very cloe near trple root ) wthn an nterval of nteret In all the example, we compute the root wthn the Pleae cte th artcle n pre a: Lu, L, et al Fat approach for computng root of polynomal ung cubc clppng Computer Aded Geometrc Degn 009), do:006/jcagd009000

9 COMAID:5 JID:COMAID AID:5 /FLA [mg; v 79; Prn:/0/009; 4:0] P9 -) L Lu et al / Computer Aded Geometrc Degn ) 9 Table Tme per teraton n mcroecond for varou degree n degree of the polynomal CubClp 54 7 QuadClp BezClp Table 4 Example ngle root): Number of teraton N and computng tme t n μ for varou value of degree n and accuracy ε The tme for more than 6 gnfcant dgt hown n talc) have been obtaned by extrapolaton We ue bez a the abbrevaton for BezClp, quad for QuadClp, and cub for CubClp degree n ε cub quad bez cub quad bez cub quad bez cub quad bez cub quad bez 4 N t N t N t Table 5 Example double root): See capton of Table 4 degree n ε cub quad bez cub quad bez cub quad bez cub quad bez cub quad bez 4 N t N t N t nterval [0, ] The number of teraton were obtaned from an mplementaton n Maple whch can provde arbtrary precon, whle the computng tme were meaured wth the help of the mplementaton n C wth double arthmetc precon Lke n Bartoň and Jüttler 007), the runnng tme for accuracy below 0 6 were obtaned by multplyng the number of teraton wth the tme per teraton ee Table ) However, for the accuracy above 0 6, there poble propagaton of error n the floatng arthmetc We do not gve a thorough analy on the accumulaton of error n the computaton here, but jut compre the reult obtaned wth Maple and C by extrapolaton, f there a remarkable dfference Example Sngle root) We appled algorthm CubClp, QuadClp and BezClp to the three polynomal f 4 t) = t ) t)t + 5), f 8 t) = t ) t) t + 5) 4, f 6 t) = t ) t) 5 t + 5) 0 n order to compute the ngle root n the nterval [0, ] Table 4 report the number of teraton and the computng tme for varou value of the dered accuracy ε For thee three polynomal, cubc clppng doe not how evdent advantage over quadratc clppng and Bézer clppng, and even perform wore ometme Th due to the lght dfference wth repect to the number of teraton between the two algorthm, but cubc clppng ha more computaton effort n each teraton Fg a) and Fg 4a) llutrate the relaton between computng tme and dered accuracy for polynomal of degree 8 and 6 repectvely Example Double root) We appled algorthm CubClp, QuadClp and BezClp to the three polynomal f 4 t) = t ) 4 t)t + 7), f 8 t) = t ) 4 t) t + 5) t + 7), Pleae cte th artcle n pre a: Lu, L, et al Fat approach for computng root of polynomal ung cubc clppng Computer Aded Geometrc Degn 009), do:006/jcagd009000

10 COMAID:5 JID:COMAID AID:5 /FLA [mg; v 79; Prn:/0/009; 4:0] P0 -) 0 L Lu et al / Computer Aded Geometrc Degn ) Fg Computng tme t n μ v accuracy for the polynomal of degree 8 n Example,,, 4 and 5 Fg 4 Computng tme t n μ v accuracy for the polynomal of degree 6 n Example,,, 4 and 5 Pleae cte th artcle n pre a: Lu, L, et al Fat approach for computng root of polynomal ung cubc clppng Computer Aded Geometrc Degn 009), do:006/jcagd009000

11 COMAID:5 JID:COMAID AID:5 /FLA [mg; v 79; Prn:/0/009; 4:0] P -) L Lu et al / Computer Aded Geometrc Degn ) Table 6 Example trple root): See capton of Table 4 degree n ε cub quad bez cub quad bez cub quad bez cub quad bez cub quad bez 4 N t N t N t Table 7 Example 4 near double root): See capton of Table 4 degree n ε cub quad bez cub quad bez cub quad bez cub quad bez cub quad bez 4 N t N t N t f 6 t) = t ) 4 t) 7 t + 5) 6 t + 7) n order to compute the double root n the nterval [0, ] Table 5 report the number of teraton and the computng tme for varou value of the dered accuracy ε For thee three polynomal, cubc clppng ha an ncreangly better performance a rang the degree and reducng the accuracy Fg b) and Fg 4b) how the relaton between computng tme and dered accuracy for polynomal of degree 8 and 6 repectvely Example Trple root) We appled algorthm CubClp, QuadClp and BezClp to the three polynomal f 4 t) = t ) t 5), f 8 t) = t ) t 5) + t), f 6 t) = t ) t 5) 7 + t) t + 7) 4 n order to compute the trple root n the nterval [0, ] Table 6 report the number of teraton and the computng tme for varou value of the dered accuracy ε For thee three polynomal, cubc clppng how clearly the uperorty wth repect to quadratc clppng and Bézer clppng Th due to the uper-lnear convergence rate of cubc clppng Fg c) and Fg 4c) llutrate the relaton between computng tme and dered accuracy for polynomal of degree 8 and 6 repectvely Example 4 Near double root) We appled algorthm CubClp, QuadClp and BezClp to the three polynomal f 4 t) = t 04)t )t + ) t), f 8 t) = t )t )t + 5) t + 7), f 6 t) = t )t )6 t) 7 t + 5) 6 t + 7) n order to compute the two near root n the nterval [0, ] Table 7 report the number of teraton and the computng tme for varou value of the dered accuracy ε Fg d) and Fg 4d) llutrate the relaton between computng tme and dered accuracy for polynomal of degree 8 and 6 repectvely Pleae cte th artcle n pre a: Lu, L, et al Fat approach for computng root of polynomal ung cubc clppng Computer Aded Geometrc Degn 009), do:006/jcagd009000

12 COMAID:5 JID:COMAID AID:5 /FLA [mg; v 79; Prn:/0/009; 4:0] P -) L Lu et al / Computer Aded Geometrc Degn ) Table 8 Example 5 near trple root): See capton of Table 4 degree n ε cub quad bez cub quad bez cub quad bez cub quad bez cub quad bez 4 N t N t N t Example 5 Near trple root) We appled algorthm CubClp, QuadClp and BezClp to the three polynomal f 4 = t 04)t )t )t + 5), f 8 = t )t )t )t + 5) t + 7), f 6 = t )t )t )6 t) 4 t + 5) 7 t + 7) n order to compute the three near root n the nterval [0, ] Table 8 report the number of teraton and the computng tme for varou value of the dered accuracy ε Fg e) and Fg 4e) how the relaton between computng tme and dered accuracy for polynomal of degree 8 and 6 repectvely Remark For near double root and near trple root, f ɛ maller than the dtance between two near root, we can alway olate the root by nterval However, f ɛ larger than the dtance, two near root may be found by one nterval Smlar effect can be oberved for quadratc clppng and Bézer clppng 4 Numercal robutne We analyze the robutne of CubClp method for computng root of polynomal n two apect: the tablty of applyng Cardano formula to compute the root of a cubc polynomal and the tablty of Bernten Bézer repreentaton 4 The tablty of Cardano formula In each teraton of CubClp, a new nterval generated by applyng Cardano formula to compute the root of a cubc polynomal However, for the lmt precon of floatng pont number, the formula 8) wll become untable f large a, v or are nvolved From the formula 8), t oberved that f one or two root of the cubc polynomal far away from the other two or one root, ome of a, v and wll be large Conequently, numercal ntablty maybe come up when applyng the Cardano formula n CubClp The ame ntablty happen to the quadratc clppng method However, n the cae p > 0 and p q, computng root of quadratc equaton x + px + q = 0, we ue another varant of the oluton formula a p + p 4 q = q, p + p 4 q whch make the computaton more table But t eem that we could hardly fnd mlar formula for Cardano formula In the tuaton of untable computaton n cubc clppng, we may reort to quadratc clppng or Bézer clppng ntead It eem dffcult to gve a defnte crtera to ndcate when Cardano formula uffer ntablty for computng the root We wll not addre the problem here and leave t a future work 4 The tablty of BB repreentaton Smlar a the quadratc clppng, we ue CubClp to compute the root of the Wlknon polynomal 0 W x) = x ) = wth Bernten Bézer repreentaton wthn the doman nterval [0, 5] wth ɛ = 0 for the fgure of th polynomal and t control polygon CubClp generate 0 nterval of length le than ɛ contanng the root, and the maxmum devaton of the center of thee nterval from the root,,,0 le than 0 4 Snce more complex operaton durng the algorthm Pleae cte th artcle n pre a: Lu, L, et al Fat approach for computng root of polynomal ung cubc clppng Computer Aded Geometrc Degn 009), do:006/jcagd009000

13 COMAID:5 JID:COMAID AID:5 /FLA [mg; v 79; Prn:/0/009; 4:0] P -) L Lu et al / Computer Aded Geometrc Degn ) are needed, we took 7 gnfcant dgt n Maple to olate all the 0 root If the length of the doman nterval ncreaed/decreaed, then more/le gnfcant dgt are needed 5 Concluon In th paper, we preent a new cubc clppng approach for computng all the root of a polynomal n a gven nterval wthn a certan accuracy The algorthm baed on degree reducton, and a generalzaton of the quadratc clppng The convergence rate and numercal performance were preented Cubc clppng ha fater convergence rate than quadratc clppng for the ngle, double and trple root However, becaue computng the root of cubc equaton need more computatonal effort, cubc clppng doe not how evdent uperorty n computaton tme for the polynomal of lower degree n or lower precrbed accuracy ε However, cubc clppng reduce the computaton effort much f the gven polynomal ha hgh degree and the accuracy pretty hgh, epecally for trple root and near trple root Addtonally, we generalzed cubc clppng to general degree reducton clppng The proof n Secton llutrate uch root fndng method baed on degree reducton ha precrbed convergence rate a hown n Table There tll a challengng problem ung thee root fndng method baed on degree reducton, uch a quadratc clppng and cubc clppng: thee method may uffer numercal ntablty when computng the root of the quadratc and cubc polynomal after degree reducton We have to reort to Bézer clppng when encounterng the tuaton In th paper, we gave a heurtc llutraton for the cubc clppng about when the ntablty of Cardano formula occur However, t mot gnfcant to gve a reaonable and complete analy on th problem We feel the followng prncple can be a gude to handle t: f the neghborhood of the nterval of nteret contan root, then ue cubc clppng; f t contan root, then ue quadratc clppng; f t contan only one root, then ue Bézer clppng A one of the future work, we wll focu on the analy of numercal tablty of thee root fndng method by clppng Concretely, we want to fnd a practcal crtera to judge when Bézer clppng, quadratc clppng and cubc clppng become untable, and how thee three method can be combned to form a robut numercal method for computng the root of a polynomal Another future work wll focu on the extenon of the technque to the multvarate cae Acknowledgement Th work upported by the jont grant by Natonal Natural Scence Foundaton of Chna and Mcrooft Reearch Aa No ) Reference Bartoň, M, Jüttler, B, 007 Computng root of polynomal by quadratc clppng Computer Aded Geometrc Degn 4, 5 4 Borwen, P, Erdély, T, 995 Polynomal and Polynomal Inequalte Sprnger-Verlag, New York Cho, YK, Wang, WP, Lu, Y, Km, MS, 006 Contnuou collon detecton for two movng ellptc dk IEEE Tranacton on Robotc ), 4 Colln, GE, Akrta, AG, 976 Polynomal real root olaton ung Decarte rule of gn In: Proceedng of the Thrd ACM Sympoum on Symbolc and Algebrac Computaton, New York, US, pp 7 75 Efremov, A, Havran, V, Sedel, H-P, 005 Robut and numercally table Bézer clppng method for ray tracng NURBS urface In: The t Sprng Conference on Computer Graphc ACM Pre, New York, pp 7 5 Elber, G, Km, M-S, 00 Geometrc contrant olver ung multvarate ratonal plne functon In: The Sxth ACM/IEEE Sympoum on Sold Modelng and Applcaton, Ann Arbor, Mchgan, pp 0 Hook, DG, McAree, PR, 990 Ung Sturm equence to bracket real root of polynomal equaton In: Glaner, AS Ed), Graphc Gem I, pp 46 4 Jüttler, B, 998 The dual ba functon of the Bernten polynomal Advanced n Computatonal Mathematc 8, 45 5 Ln, M, Gottchalk, S, 998 Collon detecton between geometrc model: a urvey In: Proceedng of IMA Conference on Mathematc of Surface, Brmngham, UK, pp 7 56 McNamee, JM, Bblographe on root of polynomal J Comp Appl Math 47, 9 94; 0, 05 06; 4, 4 44, avalable at wwwelevercom/homepage/ac/cam/mcnamee/ Nhta, T, Sederberg, TW, Kakmoto, M, 990 Ray tracng trmmed ratonal urface patche In: Proceedng of Sggraph ACM, pp 7 45 Patrkalak, NM, Maekawa, T, 00 Interecton problem In: Farn, G, Hochek, J, Km, M-S Ed), Handbook of Computer Aded Geometrc Degn Elever, pp Reuter, M, Mkkelen, T, Sherbrooke, E, Maekawa, T, Patrkalak, N, 007 Solvng nonlnear polynomal ytem n the barycentrc Bernten ba The Vual Computer 4 ), Rouller, F, Zmmermann, P, 004 Effcent olaton of polynomal real root Journal of Computatonal and Appled Mathematc 6 ), 50 Sederberg, TW, Nhta, T, 990 Curve nterecton ung Bézer clppng Computer-Aded Degn 9), Pleae cte th artcle n pre a: Lu, L, et al Fat approach for computng root of polynomal ung cubc clppng Computer Aded Geometrc Degn 009), do:006/jcagd009000