Research Article Efficient Recursive Methods for Partial Fraction Expansion of General Rational Functions

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1 Journal of Applied atheatic Volue 24, Article ID 89536, 8 page Reearch Article Efficient Recurive ethod for Partial Fraction Expanion of General Rational Function Youneng a, Jinhua Yu,,2 and Yuanyuan Wang Departent of Electronic Engineering, Fudan Univerity, Shanghai 2433, China 2 Key Lab of edical Iaging Coputing and Coputer Aited Intervention of Shanghai, Shanghai 2433, China Correpondence hould be addreed to Jinhua Yu; jhyu@fudan.edu.cn Received 6 June 24; Accepted 6 Augut 24; Publihed 2 October 24 Acadeic Editor: Anuar Iha Copyright 24 Youneng a et al. Thi i an open acce article ditributed under the Creative Coon Attribution Licene, which perit unretricted ue, ditribution, and reproduction in any ediu, provided the original wor i properly cited. Partial fraction expanion (pfe) i a claic technique ued in any field of pure or applied atheatic. The paper focue on the pfe of general rational function in both factorized and expanded for. ovel, iple, and recurive forula for the coputation of reidue and reidual polynoial coefficient are derived. The propoed pfe ethod require only iple pure-algebraic operation in the whole coputation proce. They do not involve derivative when tacling proper function and require no polynoial diviion when dealing with iproper function. The ethod are efficient and very eay to apply for both coputer and anual calculation. Variou nuerical experient confir that the propoed ethod can achieve quite deirable accuracy even for pfe of rational function with ultiple high-order pole or oe tricy ill-conditioned pole.. Introduction Partial fraction expanion (pfe) ha been a powerful tool widely ued in the field of calculu, differential equation, control theory, and oe other area of pure or applied atheatic. It i alo a claic topic tudied by any cholar over the tie. Though a ole olution i guaranteed, it i not an eay ta to perfor the coputation of pfe effectively, epecially when the function to be expanded contain highorder or ill-conditioned pole. any exiting ethod are difficult to apply and can lead to coniderable error. We aitoreearchthiclaictopicandpropoeeveralnovel ore efficient and ipler pfe ethod for general rational function in both factorized for and expanded for. Suppoe R() = Q()/P() i the rational function to be expanded inpfe. The polynoial, P() and Q(), are denoinator and nuerator of R(), repectively. Generally, apolynoial(letuayq()), can be written in either expanded for Q() = i a i i or factorized for Q() = i ( z i ) n i in ter of it zero. Though theoretically equivalent, a ethod baed on one of thee for ay exhibit ignificant difference fro the other in nuerical accuracy and coputational efficiency [].In pfe proble,a the pole of R() have to be nown, P() iotlywritteninfactorized for. Though Q() can be written in either factorized or expanded for, the latter ee to be ore in favor in practice.otcurrentavailablepfeethodaredeigned for R() with Q() written in expanded for. However, one ay till be otivated to deign pfe ethod for factorized R(). For one thing, any function are originally written in factorized for. A pfe ethod for factorized R() can deal with uch function directly and ore efficiently, therefore, it i ore preferable. For another, under any practical circutance, even R() i originally written in expanded for, the zero and pole of R() have to be obtained. In other word, R() need to be rewritten in factorized for. Thi can be often een in the field of yte analyi and deign, pfe ethod are widely ued in the derivation of Invere Laplace tranforation of tranfer function. A the zero and pole of the tranfer function alot characterize the whole yte, they are often obtained. The zero-pole plot of the tranfer function i alot ued a a routine in the analyi of ignal and yte. Conidering that ethod for factorized function can be ore eaily perfored, thu they can erve a an alternative chee under uch circutance. Soe other practical reaon for the developent of pfe ethod

2 2 Journal of Applied atheatic for factorized R() are alo entioned in [2]. In a word, it i of ignificance to deign ethod for the pfe of R() in both factorized and expanded for. The ot well-nown pfe ethod i Heaviide coverup forula. It i often introduced a a textboo ethod. It provide an elegant copact olution to pfe proble andoftenerveaabaiforotherpfeethod.however, thi claic ethod require ucceive differential when applying to function with high-order pole, which give rie to increaingly higher order polynoial. Evaluation of thee high-order polynoial can eventually lead to large nuerical error [3]. Another tandard pfe ethod i the ethod of undeterined coefficient. Thi ethod require the contruction and olution of a yte of equation. It can alo be very coplicated and inconvenient when tacling function with high-order pole. Beide the tandard claic ethod,anypfeethodarealopropoedduringthe pat year. Soe of thee ethod tend to perfor better than claical ethod under certain condition. However, any of the are only uitable for all-cale proble or oe particular cae and are difficult to fit into coputer progra. Two deirable ethod are pecially deigned for the pfe of function in factorized for [3, 4]. In [4], Brugia developed an efficient noniterative ethod for pfe of function with high-order pole by deriving the high-order derivative of the rational function directly without paing it lower order one. Copared with the claic ethod and any other tricy ethod, Brugia ethod i ore applicable and efficient. It can be eaily prograed for coputer coputation. Linnér [3] iplified Brugia ethod by uing operator and extended it to iproper function by Laurent expanion. In [5],aodificationwaaloadetoBrugia ethod, which lead to relative ipler ipleentation and a review of any nice aged ethod can alo be found in [5]. ot of exiting ethod aue that the rational function i written in expanded for. In [6], Karni propoed an algebraic procedure for pfe. Karni algorith, unfortunately, i liited to jut thoe cae the degree of the nuerator i no ore than the nuber of pole. Karni ethod wa further developed yet with uch ipleentation cot in [7 9]. Pottle [] propoed an iterative ethod for the digital coputer ue, but thi ethod i ubject to intolerable error when applied to function with high-order pole. Uraz and agy [] propoed a ethod calculating the reidue and coefficient utilizing atrix algebra. Beide the above relatively aged ethod, any ethod are alo propoed in the recent year, uch a the ethod in [2 7]. any of theeethodarealoonlyuitableforoeparticularcae and can becoe very coplicated for large-cale proble. In [7], a et al. propoed an efficient pure algebraic ethod for pfe of general rational function with high-order pole. It exhibit quite good perforance and avoid long diviion when dealing with iproper function. The ethod doe not involve derivative and can be coded conveniently. The calculation coplexity of oe pfe ethod i dicued in [8, 9]. In thi paper, we develop efficient recurive algebraic pfe ethod for rational function in both factorized and expanded for. Siple, elegant, and pure-algebraic forula are propoed to copute the pfe coefficient. The propoed ethod can be ued for pfe of both proper and iproper rational function with coplex pole. They do not involve polynoial diviion or differentiation or the olution of a yte of linear equation for the pfe of a general rational function. The reainder of the paper i organized a follow. In Section 2, we focu on the pfe of function in factorized for. Brugia and Linnér ethod are further developed and iplified for uch eaier ipleentation and extended to ore general cae. We alo develop novel iple algebraic ethod to copute the coefficient of thereidualpolynoialofiproperfunctionthatavoid polynoial diviion. In Section 3, we focu on the pfe of rational function in expanded for. Two novel efficient ethod are propoed. Several ueful, copact forula are derived for the coputation of reidue. Two efficient ethod that avoid polynoial diviion are developed to copute the coefficient of the reidual polynoial. In Section 4, nuerical exaple are provided to illutrate the uageandvalidityofthepropoedethod,followedbythe concluion in Section Partial Faction Expanion of Rational Function in Factorized For Let the factorized rational function be which i expanded a R () = R () = ( z j) nj ( j), () j E h= e h ( c ) h + = L= c L. (2) L ( ) Here c L are the reidue; z j and j are the zeroe and pole of R(), repectively. The pole and zeroe are aued ditinct in the whole paper; n j and j are the ultiplicitie of z j and j,repectively;ithenuberofzeroand i the nuber of pole; e h are the coefficient of the reidual polynoial which will be zero for a proper R(); c i a contant introduced to ae the reidual polynoial ore general. c i often et a zero in other article. Fro () and (2), we can oberve that the degree of the nuerator of R() i V = n j; the degree of the denoinator i K = j; E = V K.Inapfeproble,c L, e h are the unnown coefficient to be calculated. We firt propoe aiplerecuriveforulatoobtainthereidueandthen provide two algebraic ethod for obtaining coefficient of the reidual polynoial for an iproper function. 2..CoputingReidueofFactorizedFunction. Brugia [4] developed a claic pfe ethod for factorized rational function.hiethodioreefficientandeaiertoapplyfor both anual and coputer calculation copared with any other ethod. Here we are to further develop and iplify Brugia ethod. To allow for context and illuination of difference and to ae thi paper elf-contained, part of

3 Journal of Applied atheatic 3 Brugia ethod i briefly introduced below. According to Heaviide forula, we have K= Land c L = R (K) = K!, (3) R () = ( z j) n j,j= ( j). (4) j A hown in [4], the Kth derivative of R () are K>and D (i) R (K) =(R D ) (K ) K = = ( )i i! ( i= n j ( z j ) i+ C i K R(K i ) D (i), (5) j= j ). i+ (6) ( j ) Here C i =!/( i)!i!, i = i(i ),and! =. In the above equation, we denote R () by R for iplicity of expreion. Siilar notation are alo ued for other function inthefollowingpart.thiibrugia ainreult;oredetail can be found in [4]. Equation (5) and(6) cangeneratea linear yte of equation and then Graer ethod can uedtoolvetheequationtoobtainthederivativeofr (). It repreent a very efficient pfe ethod and the coputer ipleentation i eaier to be accoplihed copared with any other exiting tricy ethod. Here we propoe a ipler ethod baed on (3), (5), and (6). According to (3), we can obtain R (i) =i!c = ( i). (7) aing oe atheatical tranforation on (6) yield D (i) =i!( j= j ( j ) i+ n j ). i+ (8) (z j ) Here we ention that the odification on (6) to obtain(8) i relatively little, but in practice it can reduce coputation cot and facilitate ipleentation to a coniderable extent. Cobining (5)and(3), we have c L = K! K i= C i K R(K i ) D (i) =. (9) Uing (7)and(8) to ubtitute the derivative of R and D in the uation of (9)yield c L = K K c ( (K i )) i= ( ( j ) i+ j= j n j (z j ) i+ ) =. () otice K= L.Equation()furtheryield c L = L c L (L+i) λi =, (L = : : ), i= (a) λ i = ( ( j ) i j= j n j (z j ) i ). (b) Here we ention that in (a) L= : :ean L=, 2, 3,...,2,. Thi i a atlab denotation which will alo be ued in the following part for the iplicity of expreion. Obviouly c = R =.Byuing(a) and (b) and decreaing progreively the value of L fro to, we can then derive other reidue ucceively. Equation (a) and(b) can be ipleented recurively in atraightforwardway.itcanbeachievedwithouttheaidofa coparatively coplex table a done in [3]. It i uch eaier to apply for anual calculation than the Brugia ethod [4] and it odification [3, 5]. eanwhile, it can be prograed ore eaily and reduce calculation cot Coputing Reidual Polynoial Coefficient of Factorized Function. Generally, a practiced by ot current article, long diviion or polynoial diviion can be ued to copute the coefficient of the reidual polynoial, naely, e h,in (2). However, a error will propagate at each tep of the polynoial diviion, thi chee i not quite uitable for large-cale proble. oreover, for a given factorized rational function, the long diviion require both the nuerator and denoinator be rewritten in expanded for, which can lead to additional error and uch coputation cot. Here we provide two novel algebraic ethod to copute e h that avoid long diviion and the reforation of R(). The firt ethod decribed in Section 2.2. i obtained by further developing Linnér ethod. The otherethodintroducedin Section i baed on derivative Laurent Expanion Applied to Iproper Rational Function for Any c. Brugia ethod i liited to proper function. In [3], Linnér propoed an excellent procedure to copute the reidual polynoial coefficient of iproper function by Laurent expanion. Here we will further develop Linnér ethod and obtain uch ipler and ore efficient calculation forula. Linnér ethod deand that c in (2) not be equal to zero or pole of R(), which can caue inconvenience in practice. We will extend c to any value. Linnér ethod i briefly introduced to allow for context, copletene, and illuination of difference. With the ubtitution c =/t,()tranforinto R (t) =t E W (t z j) ni (t j) j, (2)

4 4 Journal of Applied atheatic z j =/(z j c ), j =/( j c ),andw i a contant. ow write R (t) =t E R (t), (3) R(t) doe not include the factor t. In(3), applying a Taylor expanion to R(t) at t=,wecanobtain R () = H h= R (h) () ( h! c ) E h + o [( c ) E H ], (4) H i an integer. ow expand R() a R () = E h= e h ( c ) h +R 2 (), (5) R 2 () i the proper fraction, which can be expanded a the econd part on the right-ide of (2). Coparing (4)and (5), we have e h = R (E h) () (E h)!. (6) Thi i Linnér ain reult. In the ethod, the derivative of R(t) are calculated uing equation lie (5)and(6)anditalo require c =z j and c = j, which can caue inconvenience in practice. In the following part, we firt how how to extend c to zero and pole of R(). Wethenobtainuchipler recurive forula for the calculation of e h. Suppoe c i the th root of R(),.Wecanrewrite R () =( ) R (), (7) R () i a defined in (4). With the ubtitution = /t in R (),wehave R (t) =t E R (t), (8) R (t) =W (t z j) n i Here z j =/(z j ), j =/( j ),and W =, j= (t j). (9) j ( z j ) n j,j= ( j ). (2) j By Taylor expanding R (t) at t=in (8), we can obtain H R (h) () R () = ( h! ) E h+ + o [( ) E H+ ]. h= (2) Fro (7)and(2), we have H R (h) () R () = ( h! ) E h + o [( ) E H ]. (22) h= Coparing (5)and(22), we can oberve that R (E h) () e h = (E h)!. (23) Thu c i extended to the th root of R(),.Theae procedure can be ued to extend c to the th zero of R(), z. We can eaily obtain e E by etting h = E in (23) orby inpection: e E =. (24) In Section 2., we have iplified Brugia ethod and derived iple recurive forula for the calculation of the reidue. otice that, iilar to R (), R (t) and R(t) are both of factorized for and that (3) and(23), which calculate c ij and e h,repectively,areoftheiilarfor.therefore,the aforepropoed procedure in Section2. for the calculation of c ij i alo uitable for the calculation of e h here. Tae c =, foranintance.itcanbeproventhat D (i) R (K) =( R D ) (K ) K = =i!( j= Fro (23), we have j i= ( j t) i+ C i K (K i ) R D (i), (25) n j ). i+ (26) ( z j t) R (i) () =i!e E i. (27) According to (23)and(25), we have e h = E h C i E h (E h)! i= R (E h i ) D (i) t=. (28) Uing (26)and(27) to ubtitute the derivative of D and R in (28)andthenevaluating(28)att=,wecanobtain e h = E h (E h) i= e (h+i) ( j ( j ) i j= n j (z j ) i ), (h =E : :). (29) The ae procedure can be ued to copute e h when c =. In uary, e h canbecalculatedaccordingto (h =E), E h e h = e (E h) (h+i) η i i= (h=e : :), (3a)

5 Journal of Applied atheatic 5 η i = j ( j c ) i j= j ( j c ) i n j (z j c ) i ( c = ), j ( j c ) i n j (z j c ) i ( c =z ), j= n j (z j c ) i (otherwie). (3b) Equation (3a) and(3b) areuchaipleandcopact forula that it ipleentation i even very eay for hand calculation. It i alo uch ore convenient for coputer progra and require le coputation cot than the original ethod in [3]. We jut need firt calculate η i uing (3b). Then e h can be iediately obtained uing (3a) Coputing e h through Derivative. We provide another iple way to copute e h which alo avoid the unfavorable polynoial diviion. That e E = i obviou. Therefore, we jut need to obtain the reaining coefficient e,e,e 2,...,e E.oticethatathereiduec L can be firtly obtained in Section 2., theycanbeeena nown quantitie here. Three condition are conidered a below regarding the different value of c. () c =z and c =.Wefirtdicuthecalculationofe h when c =z and c =.Herez and are the th zero and the th pole, repectively. Fro () and(2), we oberve that the coefficient e,e,e 2,...,e E can be obtained according to e h = h! (R(h) +T (h) ) =c, (3) T () = = L= c L. (32) L ( ) Here R = R() a defined in () and T = T(). The hth derivative of T() can be eaily obtained a T (h) = ( ) h P h h+j c j, (33) j+h ( ) = P n =!/( n)!. AR() i of factorized for, it derivative can be derived uing function lie (4) and (5). Thu the proble i olved. To further iplify the ipleentation, we introduce an auxiliary variable e h = h! R(h) = c. (34) Equation (34) haaiilarforulationa(3); thu, e h can be calculated and uing the procedure we copute c ij in Section2.. Accordingly, we can finally obtain R( c ) (h =), h e h = h h i λ i i= e (h=:e), λ i = j ( j c ) i n j (z j c ) i. (36a) (36b) Here we ention that in (36a) h = : E ean h =, 2, 3,..., E. Thi i a atlab denotation which will alo be ued in the reaining part for iplicity of expreion. We can firt obtain e h uing (36a)and(36b); then, e h can be eaily obtained uing (35).oticetheevaluationof(36b) require c =z and c =. Thu other two condition reain to be dicued. (2) c =z. Here we dicu the condition when c i one of the zero of R() (Let u ay z ). Actually it will be een that it i ore preferable to let c =z.firtwerewriter() a n i the ultiplicity of z and R () = ( z ) n G (), (37) G () =,j= ( z j) n j ( j). (38) j Coputing the hth derivative of both ide of (37) uing Leibniz forulaandthenetting a z yield R (h) (z )= (h < n ) (z ) (h n ). P n h G(h n ) (39) A G () i of factorized for, it derivative can be coputed recurively G (h n ) = G (h = n ), h n P i i) h n λ i i= (h > n ), λ i = j ( j ) i j= n j (z j ) i, (4a) (4b) Fro (3)and(34), we can eaily oberve that e h = e h + h! T(h) = c. (35) D (i) =i!( j ( j ) i+ j= n j (z j ) i+ ). (4c)

6 6 Journal of Applied atheatic Cobining (3) and(39), we have T (h) (z ) e h = h! G (h n ) (z ) (h n )! + T(h) (z ) h! (h = : n ), (h = n :E). (4) A we did in Section 2.2.2(), to further iply the ipleentation, we can alo introduce an auxiliary variable e h which atifie e h =e h T(h) (z ). (42) h! AndthenuingtheprocedurethatwepropoedinSection 2. lead to (h = : n ), G (z ) (h = n ), e h = h e h n h i λi c (h = n =z +:E ), i= (43) λ i i a defined in (4b).Itcanbeeailyobervedthat the larger n i, the le coputation load the ethod will involve. Thu c canbechoenaz = axz,z 2,...,z }. Particularly, if n E,wehave e h = T(h) (z ). (44) h! (3) c =. Lat, let u dicu the condition when c =. Here i the th pole of R(). ultiplying both ide of (2) with ( ) give R () = E h= e h ( ) h+ ( ) T (), (45) R () and T() are a defined in (4) and(32), repectively. According to (45), we have e h = R(h+ ) () +[( ) T()] (h+ ). (46) (h + )! = BaedonLeibniz forula,wecanobtain Let [( ) T()] (h+ ) = T (h) = ( ) h e h = i= i= R (h+ ) i =P h+ T (h) =, (47) P h h+j c ij ( i ) j+h. (48) () = (h + )!. (49) Thenaccordingto(46), (47), and (49), we have e h = e h + T(h). (5) h! = otice that (49) ha the iilar forulation a (3) othat itcanalobetacleduingtheprocedurewepropoedin Section 2..Wecanfinallyobtain e h = h (h + ) ( i= e (h i) λi + +h i=h+ c (i h) λi ) = (h =:E ), (5) λ i are a defined in (b)andc (i h) are reidue related to the th pole. It i worthy to ention that we do not have to calculate all the λ i here if we tored the when calculating the reidue in Section 2.. Till thi point, the ethod for the coputation of e h i dicued for any c. Generally, it i ore preferable to let c = z in practice a thi condition require the leat coputation cot. 3. Partial Fraction Expanion of Rational Function in Expanded For Let the rational function in expanded for be R () = Q () P () = = a ( ) i= ( i) i = E h= i e h ( c ) h + i= c ij ( i ) j, (52) c ij, e h are the reidue and coefficient of reidual polynoial, repectively; ithenuberofpole; i are the pole of R(); i are the ultiplicitie of i ; a are the polynoial coefficient of the nuerator; and c are contant which are often zero in practice and they are introduced to ae thepolynoialoregeneral.obviouly,thedegreeofthe nuerator i, and the degree of the denoinator i K= j, E= K.Foraproperfunction,e h are zero. It can be oberved that R() canalobeexpreeda R () = = a r (), (53) ( r () = ) i= ( i). (54) i Suppoe r () can be expanded in pfe a r () = K h= i e h ( c ) h +. j (55) ( i ) i= c ij

7 Journal of Applied atheatic 7 Inerting (55) into(53) andthencopare(53) with(52), we can oberve that the reidue of R() can be obtained a c ij = = a c ij, (56) and the reidual polynoial coefficient can be calculated according to E h e h = a K+h+i e (K+h+i)h, (h =:E). (57) i= In the following part, we will firt how how to copute c ij andthentocalculatee h. 3.. Coputing Reidue of Function in Expanded For. The ethod for coputation of reidue can be divided into two tep: firt, calculate the reidue of r ();econd,calculatethe reidue of R(). Firt, let u uppoe r () can be expanded in pfe a i r () =. j (58) i= ( i ) otice that r () i a function of factorized for. Thu it can be expanded uing the ethod we propoed in Section 2.. We jut need to et the nuerator a in (); naely, all the n i = in (b). Referring to (a) and(b), the following forula i iediately obtained c ij = i j c i j i(j+) λ =i, = λ = L= L=i c ij L ( L ). (j = i : :), (59a) (59b) It ha been proved in [7] thatthereidueofr () and thoe of r () atify c ij =c ( )i(j+) + ( i ) c ( )ij. (6) Baedontheaboveforula,allthereidueofr () can be obtained. Then uing (56), the reidue of R() can be obtained. Here we develop novel ipler forula to derive the reidue of R() baed on the reidue of r (). Itcanbe oberved that the nuerator of R() canberewrittena Q () =a +( )(a +( ) (a 2 +( ) ( a 2 +( ) (a +a ( )) ))). (6) According to (52)and(6), we can denote f = a P (), (62a) f L = a L P () +( )f L, (L =:). (62b) By the above denotation, we have f = R().Suppoef L can be expanded in pfe a f L = L Κ = i e L ( c ) +. j (63) ( i ) i= d Lij otice that r () = /P(). Thu according to (58), we eaily obtain the reidue of f dij =a cij. (64a) If we have already obtained the reidue of f L, d (L )ij, referring to [7], the reidue of ( )f L canbeobtained a d (L )i(j+) +( i ) d (L )ij. Therefore, according to (62b) we have d Lij =a L c ij +d (L )i(j+) +( i ) d (L )ij, (64b) (L =:). otice that c ij =d ij.ac ij can already be calculated uing (59a) and(59b), we can finally obtain the reidue of R() uing forula (64a) and(64b) ucceively by increaing the value of L fro to. Equation(64a) and(64b) canbe further developed and yield another beautiful forula: d Lij = L = a (L ) = C ( i ) c i(j+ ). (65) In (64a), (64b), and (65), ifj+> i or j+ > i,the reidue c i(j+) and c i(j+ ) do not exit and we can iply et the a zero. Forula (65) can be proved by induction a below. Proof of Forula (65). When L=,(65) i reduced to (64a). Thuititrue.Suppoe(65)itruewhenL=n;naely, d nij = n = a (n ) = C ( i ) c i(j+ ). (66) Thenaccordingto(64b), when L=n+,wehave d (n+)ij =a (n+) c ij +d ni(j+) + ( i ) d nij. (67) Applying (66) to(67) yield d (n+)ij =a (n+) c ij n a (n ) = = + +( i ) = C ( i ) c i(j++ ) n a (n ) = C ( i ) c i(j+ ). (68)

8 8 Journal of Applied atheatic Fro (68), we have Λ= d (n+)ij =a (n+) c ij +Λ, (69) n a (n ) = = + C ( i ) c i(j++ ) = Equation (7a)canfurtherleadto Λ= n+ = a (n+ ) C ( i ) + c i(j+ ) }. = C ( i ) c i(j+ ). Fro (69)and(7b), we can finally obtain d (n+)ij = n+ a (n+ ) = = (7a) (7b) C ( i ) c i(j+ ). (7) Equation (7) i(65) withl replaced by n+;thu,by induction, it follow that (6) i true. Setting L=in (65)yield c ij = a = = C ( i ) c i[j+ ]. (72) Equation (72) preent a direct relation between the reidue of r () and the reidue of R(). It can be ueful when calculating a particular reidue of R() Coputing Reidual Polynoial Coefficient of Function in Expanded For. We will propoe novel ethod to calculate the reidual polynoial coefficient, naely, e h in (52). ot exiting ethod calculate uch coefficient uing polynoial diviion. An algebraic procedure to calculate uch coefficient that avoid polynoial diviion can be found in [7]. Here we propoe two novel ipler ethod which alo avoid polynoial diviion. We firt propoe a ethod through Laurent expanion and then propoe a ethod through derivative Coputation of e h through Laurent Expanion. A hown in (53), R() i a u of r (), if we can obtain the reidual polynoial coefficient of r (), e h, e h can be then obtained uing (57). Obviouly, r () will contribute to the e h of R() only when K= j.thu iauedtobeno le than K in thi part. The proble i then reduced to how to calculate the e h of r () ( =K:). otice all the r () are in factorizedfor;thu,theycanbeexpandeduingtheethod we propoed in Section 2. However,itinotadviabletodo o,forthicanleadtocuberoeipleentation.actually, e h Table : The relationhip between the e h (<)ande h. h 2 E 2 E E e e e 2 e (E 2) e (E ) e E e e 2 e 3 e (E ) e E 2 e 2 e 2 e 4 e E K+2 e (E 2) e (E ) e E K+ e (E ) e E K e E ipler ethod can be obtained. Referring to Section 2.2., we can prove that e h = r( K h) (), (h =: K), (73) h! W r (t) = (t j). j (74) Here j =/( j c ) and W i a contant. Let = +in (73), we have e (+)h = r(+ K h) (), (h =:+ K). (75) h! Fro (73)and(75), we can oberve that e h =e (+)(h+), (h =: K). (76) otice that e (+)(h+) are the reidual polynoial coefficient of r + (). It i clear that if we have already obtained the coefficient of r + (), we can iediately obtain the coefficient of r () by (76).Itherebyfollowthatifwehavealreadyobtained the coefficient of r (), e h,wecanucceivelyobtain e ( )h,e ( 2)h,...,e Kh.Fro(76), the following forula can be obtained: e (K+i)h =e (E i+h), (i =:E,h=:i). (77) Equation (77) preent the relationhip between the e h ( < ) ande h,whichcanbeeenoreclearlyintable. Here we ention that we calculate the e h with decreaing fro to K, while the ethod in [7] calculatee h with increaing fro K to. Itwillbeeenthattheethod propoed here i ipler and ore convenient. According to (57)and(77), we have E h e h = a K+h+i e (E i), (h =:E). (78) i= Frotheaboveanalyi,thewholeprobleireducedto how to obtain e h.ar () i of factorized for, e h can becalculateduingtheethodwepropoedinsection 2.2. through Laurent expanion. Thu, we have, (h =E), E h e h = e (E h) (h+i) η i, i= (h =E : :), (79a)

9 Journal of Applied atheatic 9 j ( j c ) i (z j c ), ( c = ), j= η i = j ( j c ) i, ( c = ), j ( j c ) i (z j c ), (otherwie). (79b) In uary, the reidual polynoial coefficient can be obtained by firt uing (79a)and(79b)andthenuing(78) Coputation of e h through Derivative. A we dicued insection 3.2.,wejuthavetocalculatethereidualpolynoial coefficient of r (), e h andthenwecanobtaine h.a r () ioffactorizedfor,itialopoibletouetheethod we propoed in Section to calculate e h.insection 2.2.2, three condition are conidered. Here we only dicu the ot favorable condition, c = (naely, c i a zero of r ()) becaue firtly, thi condition i ot uitable a it require the leat coputation and econdly, c = = in ot practical cae. otice that >K.Thu we can ue a forula lie (44) tocalculatee h when i not equal to any pole of r (). When i equal to a pole of r (), wejutneedto ae a iple odification. e E =i obviou. Uing the procedure in Section 2.2.2(2), we can finally have T (h) () = e h = T(h) (), (h =:E ), (8) h! = c ( ) h i P h h+j c ij i=( i ), j+h i= ( = ), ( ) h i P h h+j c ij, (otherwie). j+h i=( i ) (8) 3.3.RecoendationontheChoiceofProcedure. In Section 3. and 3.2, everal procedure are propoed for the coputation of both e h and c ij.athecalculationofe h can have a relation with that of c ij, we recoend two ethod copried of everal aforepropoed procedure for iproper rational function. We only ugget two typical ethod which we conider to be preferable and uitable a below. ethod ue (8)and(8)tocalculatee h.awecanee fro (8)and(8), the ethod in Section require that we firt obtain the reidue of r (). Therefore, we ue (6) (rather than (64a) and(64b)) to obtain the reidue becaue the reidue of r () are obtained in the calculation proce. ethod 2 ue (59a), (59b), (64a), and (64b) toobtainthe reidue, (79a) and(79b), and (78) tocalculatee h.itiore preferable to ue (79a) and(79b) tocalculatee h a it i not baedonthereidueofr (). ethod. The following procedure coprie ethod. (a) Calculate the reidue of r (): =,=i ( i ), (i=:;j= i ) c ij = i j (82) c i j i(j+) λ =i, = (i=:;j= i : :), λ = L=,L=i ( L/( L ) ). (b) Calculate the reidue of r () uing c ij =c ( )i(j+) +( i ) c ( )ij, (=:;i=:;: i ), (83) andthenobtainthereidueofr() uing c ij = = a c ij. (c) Calculate the reidue of r () uing (8)and(8). (d) Calculate the reidual polynoial coefficient uing (78). ethod 2. The following procedure coprie ethod 2. (a) Calculate the reidue of r () theaewayathefirt tep of ethod. (b) Calculate the reidue of R(). Herenoticethatc ij = d ij. Conider a c ij, (L =) d Lij = a L c ij +d (L )i(j+) +( i ) d (L )ij, (L =:). (84) (c) Calculatethereidualpolynoialcoefficientofr () uing (79a)and(79b). (d) Calculate the reidual polynoial coefficient uing (78). 4. Exaple and Dicuion We provide ix exaple to illutrate the uage and validity ofthepropoedethod.weueiilarpracticeaued in [7] to validate the efficiency of the propoed ethod for large-cale proble. atlab code that perfor the propoed ethod can be eaily obtained. The calculation of the nuerical exaple i perfored by atlab (2b) on a PC of 32-bit word length. In thoe exaple, c and are et a zero, if not entioned particularly. For iplicity of expreion, for large-cale function of R(), we denote the coefficient of R() a follow: array of pole, S = [, 2,..., ]; array of the ultiplicitie of pole, = [, 2,..., ];arrayofzero,z=[z,z 2,...,z ];array of the ultiplicitie of zero, n=[n,n 2,...,n ];polynoial coefficient of the nuerator of R() in expanded for, A= [a,a,...,a ]. The reader can refer to () and(52) toee clearly the eaning of thoe coefficient.

10 Journal of Applied atheatic 4.. Uage and Validation of ethod for Factorized Function Exaple. The following iple rational function i providedaanintancetoillutratetheuageoftheethod propoed in Section 2.TheexapleialouedbyLinnér in [3]. One ay refer to [3] to ee the difference between the propoed ethod and Linnér ethod: R () = (+3)( )3 ( 2) 3 ( 3) (+) 4. (85) (+2) Obviouly, the deired expanion i given by R () =y() T(), (86) y () =e +e ( c )+e 2 ( c ) 2, Thu we have T () = [ (+) (+) (+) (+) ] Coputing Coefficient of the Reidual Polynoial (9) (a) ethod Baed on Laurent Expanion. Hereweuethe ethod propoed in Section Let c be, the econd pole of R(). Referring to (3a)and(3b), we have η = ( ( 2 + ) + (+)) ( ( 3 + ) +3 (+) T () = [ c + c 2 (+) + c 22 (+) 2 + c 23 (+) 3 + c 24 (+) 4 + c 3 +2 ]. (87) η 2 = (+) +3 (2+) + (3+)) = 7, (( 2) ) = 57, (92) 4... Calculation of Reidue. The following reidue can be calculated in a direct way: c = R() = = 36, c 24 =(+) 4 R() = = 728, (88) c 3 = ( + 2)R() = 2 = 432. Uing (a)and(b), we can calculate thereainingreidue recurively at =.oticethat itheecondpoleofr(). Firt, calculate λ i uing (b) λ =( ) ( ) = 9 4, λ 2 =( ( ) ) ( ( 2) ) = 29 48, λ 3 =( ( ) ) ( ( 2) ) = (89) Then uing (a), c 23 =c 24 λ = 3888; c 22 = 2 (c 23 λ +c 24 λ2 ) = 4896; c 2 = 3 (c 22 λ +c 23 λ2 +c 24 λ3 ) = (9) Thu, we have e 2 =, e =e 2 η = 7, e = 2 (e η +e 2 η 2 ) = 6. y () = 6 7 (+) + (+) 2. (93) (b) ethod through Derivative. Here we ue the ethod propoed in Section to copute e h. Obviouly, the econd zero of R(),,hathelargetultiplicityof3.Thuit i choen a c here. A 2=E<3=n 2,wecanue(44) to derive the reidual polynoial coefficient Thu we have e =T() = 86, e =T () () = 3. (94) y () =86 3( ) + ( ) 2. (95) The coplete expanion i now given by R () =86 3( ) + ( ) (+) (+) (+) (+) (96) Exaple 2. We conider here a large-cale factorized iproper R() with S = [,2,...,9,]; = [,,...,, ]; Z = [.5,.5,..., 8.5, 9.5]; n = [,,...,, ]. In thi cae, the degree of the nuerator i. The degree of the denoinator reache. It i quite a large-cale proble. Here, one ay ee the ignificance of developing

11 Journal of Applied atheatic f() 6 Relative error Ref. pfe (a) (b) Figure : Coparion of pfe with reference olution (a) and relative error of pfe (b) in Exaple 2. Reidual polynoial coefficient are calculated uing the ethod propoed in Section a ethod which can deal with factorized R() directly. If we ue a ethod deigned for rational function in expanded for, the reforulation of the R() into expanded for canleadtohugecoputationcot.itcanalointroduce coniderable additional error. A done in [7], in the largecale experient, we alo do not diplay the expanion coefficient becaue they are too any and too tediou to diplay and reference of the coefficient are very difficult to find. Intead, we plot the figure of R() and that of it pfe, pfe(), together to validate the accuracy of the expanion reult.andwealocalculatetherelativeerrorofpfe() at different value of a (R() pfe())/r() (R() = ). The reult are hown in Figure and 2. The figure of the function are plotted within the interval [25, ]. The reidue are calculated uing the ethod propoed in Section 2.. InFigure, the reidual polynoial coefficient are calculated by the ethod we propoed in Section oticethat,inthifigure,thecurveofpfe,pfeiinperfect agreeent with that of R() (the reference olution), deontrating the high accuracy of the expanion reult. And the relative error i nearly negligible. Figure 2 preent the relative error when reidual polynoial coefficient arecalculateduingtheethodpropoedinsection Three experient regarding the value of c are conidered: (a) c = 8 (pole of R(), ethodinsection 2.2.2(3) ued); (b) c = 3.5 (zero of R(), ethod in Section 2.2.2(2) ued); (c) c =(neither pole nor zero of R(), ethodin Section 2.2.2()ued).Awecanee,therelativeerrorare aloquiteall,deontratingthegoodperforanceof thoe propoed ethod. Fro Figure and 2, one ay notice that the ethod through Laurent expanion ee to perfor better than the ethod through derivative to oe extent. Thi i becaue the forer calculate the reidue without the uage of the reidue. Exaple 3. In thi exaple, we conider a large-cale factorized iproper rational function with ill-conditioned pole. S = [,,., 8, 9]; = [8,7,7,6,6]; Z = [, 2, 7,, 2, 99]; n = [8,8,8,8,8,8]. The degree of the nuerator i 34. The degree of the denoinator i 48. R() contain three ill-conditioned high-order pole: two highorder pole ( and.), very cloe to each other, and an 8th pole () uch larger than the other pole. Thi proble can be rather tricy and the ill-conditioned pole ay often lead to intolerable large error. In thi exaple, the reidual polynoial coefficient are calculated uing the ethod propoed in Section We find the ethod propoed in Section i not quite uitable here a oe of the reidue are extreely large (nearly, 5 ), the anipulation of which will inevitably introduce large error. A can be een in Figure 3, our ethod can provide quite atifactory reult. The figure of pfe function i in perfect agreeent with the reference olution. And the nearly negligible relative error well deontrate the ethod good perforance in tacling function with ill-conditioned pole.

12 2 Journal of Applied atheatic Relative error Relative error Relative error (a) (b) (c) Figure 2: Relative error of pfe with reidual polynoial coefficient calculated uing the ethod in Section 2.2.2(3) (a), 2.2.2(2) (b), and 2.2.2() (c) in Exaple 2. f() Ref. pfe (a) Relative error (b) Figure 3: Coparion of pfe with reference olution (a) and relative error of pfe (b) in Exaple 3.

13 Journal of Applied atheatic 3 Table 2: The reidue of r (), c ij in Exaple 4. c c 2 c 22 c 3 c Uage and Validation of ethod for Function in Expanded For Exaple 4. The following iple rational function i provided to illutrate the uage of the ethod we propoed for the pfe of function in expanded for propoed in Section 3: R () = (+) 2 (+2) 2. (97) It can be etiated that R () =e +e +e 2 2 +e c + c 2 (+) + c 22 (+) 2 + c 3 (+2) + c 32 (+2) 2. (98) otice the polynoial coefficient of the nuerator are A= [8,3,,4,5,,,2,].Wewilluethetwoethodtoolve thi proble. The procedure of the ethod are decribed in Section 3.3. ethod. (a) Ue (59a)and(59b) to expand r () r () = (+) 2 (+2) 2 =.25 + (+) + (+) (+2) +.5 (+2) 2. (99) (b) Ue (6)tocalculatethereidueofr (), c ij baed on the reidue of r ().ThereultarehowninTable 2. Then ue (56)to obtain the reidue ofr(). c =2, c 2 = 4, c 22 = 7, c 3 = 69, c 32 = 59. (c) Fro Table 2, we can find the reidue of r 8 (), c 8ij. r 8 () = 8 (+) 2 (+2) 2 =e 8 +e 8 e T 8 (), () () Table 3: The reidual polynoial coefficient of r (), e h in Exaple 4. e h h T 8 () = [ (+) + (+) (+2) + 28 (+) 2 ]. (2) Uing (8) and(8), we have e 8 =T 8 () = 72, e 8 =T () 8 () = 23, e 82 =T (2) 8 () = 4, e 83 =. (3) Then referring to (78) ortable,wecanobtainallthee h a hown in Table 3. (d) Uing (78), we have e = 32, e =2, e 2 = 4, e 3 =. ethod 2. (a) Ue (59a)and(59b) to expand r (). (b) Uing (64a) and(64b) byincreaingthevalueofl fro to, we can then obtain the reidue of R() a c =2, c 2 = 4, c 22 = 7, c 3 = 69, c 32 = 59. (4) (c) Obtain the reidual polynoial coefficient of r () (here, ==8), e h uing (79a)an(79b) r 8 () = 8 (+) 2 (+2) 2 = 7 (+) 2 (+2) 2 = T 8 (), (5) T 8 () i proper reidual fraction. Then uing (78) or Table,wecanobtainallthee h a hown in Table 3. (d) Finally, uing (78), we have e = 32, e =2, e 2 = 4, e 3 =. Thu the deired expanion i R () = (+) + 7 (+) (+2) + 59 (+2) 2. (6) Exaple 5. In thi exaple, we conider a large-cale rational function in expanded for. The exaple ha alo been ued in [7]. Thu one can ae a coparion between the perforance of the propoed ethod and the ethod

14 4 Journal of Applied atheatic f() 6 Relative error 8 Relative error Ref. ethod ethod 2 (a) (b) (c) Figure 4: Coparion of expanion reult of the propoed ethod with reference (a) and relative error of ethod (b) and ethod 2 (c) in Exaple 5. in [7]. S = [8,7,6,5,4,3,2,]; = [,2,3,4,5,6,7,8]; A = [.,.2,..., 5.9, 6]. Inthicae,thedegreeofthe denoinator i 36, and the degree of the nuerator i 59. The two ethod propoed in Section 3 arevalidated.thereult are hown in Figure 4.Ahowninthefigure,theethodwe propoed can provide quite good reult for uch a large-cale proble. The relative error i deirable for both the ethod, deontrating it good perforance. Fro Figure 4(b) and 4(c),wecaneethatethod2perforbetterthanethod. Exaple 6. In thi exaple, we conider aproperrational function in expanded for with ill-conditioned pole. Thi exaple i alo ued in [7]. S = [, 3, 2,,.]; = [7,3,3,6,7]; A = [.,.2,..., 2.2]. ThuR() i a function containing three ill-conditioned high-order pole: two cloe high-order pole ( and.) and a 7th pole () uch larger than the other pole. Again we validate the two ethod propoed in Section 3.AhowninFigure 5,ourethodcan provide quite atifactory reult and the relative error are all enough. While the relative error i coparable to the ethod in [7], we find that the propoed ethod are ore robut. A entioned or hown in [8, 9, 7], the input order of pole can have ignificant influence on the accuracy of the expanion reult epecially when the rational function to be expanded contain ill-conditioned pole. However, the accuracy of our ethod tend to be le affected by the order of the pole. In thi Exaple, we can rearrange the equence of pole S a [,., 2, 3, ] or other equence and achieve coparable accuracy a the equence ued thi Table 4: Count of long operation above one-degree coplexity for a proper function. ethod for factorized function (Section 2.2.) ethod for expanded function Count of long operation above one-degree coplexity /2(H K) ethod : /2(H K) VK ethod 2: /2(H K) + 2 +VK exaple,whiletheethodin[7]doenotperforquitewell under all thee condition Coputational Efficiency. To provide an inight into the calculation efficiency of the foregoing ethod, we count the nuber of long operation (ultiplication and diviion) of the propoed ethod. Suppoe that the zeroe and pole are nown and the function to be expanded are proper. We only calculate the ain part of operation. Operation of contant and O(n) coplexity are not included. Let V be the degree of nuerator of R(), K= i= i the degree of denoinator, a nuber of pole, H= i= 2 i,anda the nuber of zeroe. Table 4 preent count of long operation above onedegree coplexity for a proper function.

15 Journal of Applied atheatic f().6.8 Relative error 2.5 Relative error Ref. ethod ethod 2 (a) (b) (c) Figure 5: Coparion of expanion reult of the propoed ethod with reference (a) and relative error of ethod (b) and ethod 2 (c) in Exaple 6. Table 5: Calculation tie of large-cale exaple. Exaple 2 Exaple 3 Exaple 5 Exaple 6 Calculation tie () (ethod ).72 (ethod 2).6 (ethod ).55 (ethod 2) The calculation tie of the exaple can alo provide an inight into the efficiency of the ethod. The calculation tie of the exaple i diplayed in Table 5.Overall,wecan eaily notice that the coputing tie for each ethod i very all, which deontrate their good coputational efficiency. For factorized Exaple (Exaple 2 and 3), we only provide the reult when the reidual polynoial coefficient are calculated by the ethod we propoed in Section 2.2. in Table 5. If the reidual polynoial coefficient are calculated uing the ethod in Section 2.2.2, the calculation tie i between.6 and.7 for Exaple 2. In Exaple 3,ethodinSection perfor poorly and are not applicable. Obviouly ethod in Section 2.2. i ore coputational efficient and accurate than the ethod in Section Thu for factorized function, we recoend pfe algorith (ee Algorith ) uing (a) and (b) to calculate reidue and (3a) and(3b) tocalculatereidue to achieve the bet perforance. For function in expanded for, both ethod perfor well for the proper function (Exaple 6). The calculation tie i le than.2 for uch a large-cale tricy proble, which prove their good calculation efficiency. In Exaple 5 which involve an iproper function, though both perfor well, ethod 2 i obviouly ore efficient a it ue le than /4 of the tie of ethod. eanwhile, fro the expanion reult, we can alo find ethod 2 can produce ore accurate reult than ethod for iproper function. Thu it i ore preferable in practical ue. The algorith for ethod 2 i provided in Algorith Dicuion. We would lie to dicu oe other iue about the uage and nuerical validation of the propoed ethod. Firtly, in the experient, we define relative error a (R() pfe())/r() to etiatequantitatively theaccuracy of the expanion reult. Thi criterion i valid becaue the original function and it pfe are uppoed to be theoretically equivalent.thereinodoubtthattheorecoherenttheirfigure are the ore accurate the expanion reult are. However, incoherence between the figure doe not alway indicate low-accuracy reult. The error calculated by thi definition are not the exact error of the expanion ethod. The actual error can be uch aller than thoe hown in the figure, epecially when ill-conditioned pole are involved. Thi i becaue that in floating point arithetic, evaluation of partial fraction function itelf (even when the expanion reult i % correct) can lead to coniderably large, even

16 6 Journal of Applied atheatic Step. calculate c directly For =: c = ( z j ) nj ( j ). j j= Step 2. calculate c L (L = : : ) uing (a) and (b) For =: For i=: λ i =( ( j ) i j= j For L= : : c L = For i=: L c L =c L +c (L+i) λi c L = c L ( L) n j (z j ) i ) Step 3. Calculate e h uing (3a) and (3b) Step 3.. e E = Step 3.2. calculate η i uing (3b) For i=:e η i = j ( j c ) i j = n j (z j c ) i ( c = ), j ( j c ) i n j (z j c ) i ( c =z ), j = n j (z j c ) i (otherwie). j ( j c ) i Step 3.3. Calculate e h (h = E : : ) uing (3a) For h=e : : e h = (E h) E h i= e (h+i) η i Algorith : Algorith for factorized function. intolerable error when function with ill-conditioned highorder pole [2]. Hence, it doe not necearily indicate inaccurate expanion reult even when the relative error are coparatively large at oe value of. On the other hand, all relative error can guarantee good expanion reult. When uing the defined relative error to validate a pfe ethod, we ugget that hould not have the value that ae R() have an extree all (or extree large) abolute value. Otherwie the relative error will inevitably grow large due to the propertie of floating point arithetic of the coputer. Secondly, we provide everal ethod to calculate the reidual polynoial coefficient of iproper R() in Section 2.2. In ter of accuracy, thoe ethod perfor alot equally for all- or edian-cale proble and oe large-cale proble. But when R() contain ill-conditioned pole and when the nuerator of R() ha uch larger degree than it denoinator, the ethod propoed in Section 2.2. perfor better than the ethod propoed Section 2.2.2,a it doe not ue the reidue a a bai. Thirdly, though we propoed ethod for R() in both factorized and expanded for, for a given factorized or

17 Journal of Applied atheatic 7 Step. calculate reidue of r () uing (59a)(59b). Step.. calculate c ii directly For i=: c ii = ( i j ). j j=i Step.2. calculate c ij (j = i : :) uing For i=: For =: i λ i = L= L=i L ( L i ) For j= i : : c ij = For =: i j c ij =c ij +c i(j+) λ c ij = c ij ( i j) Step 2. calculate the reidue of R() uing (64a) and (64b) d ij =a c ij. (i = :, j = : i ) For L=: For i=: For : i d Lij =a L c ij +d (L )i(j+) +( i ) d (L )ij Step 3. calculate the reidual polynoial coefficient of r (),e h uing (79a)(79b). Step 3.. e E = Step 3.2. calculate η i uing (79b) For i=:e j ( j c ) i (z j c ) ( c = ), j = η i = j ( j c ) i ( c = ), j ( j c ) i (z j c ) (otherwie). Step 3.3. Calculate e h (h = E : : ) uing (79a) For h=e : : e h = (E h) E h i= e (h+i) η i Step 4. calculate the reidual polynoial coefficient of R() uing e h = E h i= a K+h+ie (E i) (h = : E). Algorith 2: Algorith for function in expanded for. polynoial function, it i not neceary to tranfor the function into another for and then perfor pfe. Whether in factorized or expanded for, the expanion reult uing the propoed ethod are coparable. eanwhile, we ention that the ethod for factorized R() can be ore eaily perfored by anual calculation. Finally, and c in (2)and(52)areetazeroinalot alltheexitingarticle.inourpaper,theyareintroducedto

18 8 Journal of Applied atheatic generalize a polynoial. They can facilitate ipleentation and reduce calculation in any practical ituation when and c of R() are not zero. oreover, though a careful choice of and c ay iprove the accuracy to oe extent in certain cae, it will not lead to a ignificant iproveent of accuracy in ot cae. Thu, and c can generally be iply etazeroorothervalueatyourconvenienceinpractice. 5. Concluion In thi paper, we developed efficient recurive ethod for the pfe of general rational function in both factorized and expanded for. Siple, elegant, recurive forula that decribe the relation of the reidue and the coefficient of the reidual polynoial are obtained. Thee ethod tend to be ipler and ore applicable than any exiting ethod for pfe of rational function with ultiple high-order pole. They can be eaily prograed for coputer ue with deirable efficiency and accuracy. They are alo very table whoe accuracy i le affected by the input-order of pole. The ethod are alo very uitable for anual calculation. Appendix See Algorith and 2. Conflict of Interet The author declare that there i no conflict of interet regarding the publication of thi paper. Acnowledgent Thi wor i upported by The ational atural Science Foundation of China (849, 6277), Shanghai Pujiang Talent Progra (2PJ42), Doctoral Fund of initry of Education (27 29), and Fudan Univerity ASIC and Syte State Key Laboratory Project (S7). Reference [] R. J. Fatean, Rational Function Coputing with Pole and Reidue, Univerity of California, Bereley, Bereley, Calif, USA, 2, fatean/paper/pole.pdf. [2] F. O. Sion Jr. and R. C. Harden, Purely real arithetic algorith optiized for the analytical and coputational evaluation of partial fraction expanion, in Proceeding of the 3th Southeatern Sypoiu on Syte Theory, pp , 998. [3] L. J. P. Linner, The coputation of the th derivative of polynoial and rational function in factored for and related atter, IEEE Tranaction on Circuit and Syte, vol. 2, no. 2, pp , 974. [4] O. Brugia, A noniterative ethod for the partial fraction expanion of a rational function with high order pole, SIA Review,vol.7,pp ,965. [5] J.F.ahoneyandB.D.Sivazlian, Partialfractionexpanion:a review of coputational ethodology and efficiency, Journal of Coputational and Applied atheatic,vol.9,no.3,pp , 983. [6] S. Karni, Eay partial fraction expanion with ultiple pole, Proceeding of the IEEE,vol.57,pp ,969. [7]. F. oad and S. Karni, On partial fraction expanion with ultiple pole through derivative, in Proceeding of the IEEE, pp , 969. [8] F. Y. Chin and K. Steiglitz, An O( 2 )algorithforpartial fraction expanion, IEEE Tranaction on Circuit and Syte, vol.24,no.,pp.42 45,977. [9] D. Wetreich, Partial fraction expanion without derivative evaluation, IEEE Tranaction on Circuit and Syte,vol.38, no. 6, pp , 99. [] C. Pottle, On the partial fraction expanion of a rational function with ultiple pole by digital coputer, IEEE Tranaction on Circuit Theory, vol. CT-, pp. 6 62, 964. [] A. Uraz and F. L..-agy, atrix forulation for partialfraction expanion of tranfer function, Journal of the Franlin Intitute,vol.297,pp.8 87,974. [2] S. H. Kung, Partial fraction decopoition by diviion, The College atheatic Journal,vol.37,pp.32 34,26. [3] R. Witula and D. Slota, Partial fraction decopoition of oe rational function, Applied atheatic and Coputation,vol.97,no.,pp ,28. [4] A. Özyapici and C. S. Pintea, Coplex partial fraction decopoition of rational function, Journal of Coputational and Applied atheatic,vol.,article2,22. [5]. I. Garcia-Plana and J. L. Doinguez, A general approach for coputing reidue of partial-fraction expanion of tranfer atrice, WSEAS Tranaction on atheatic, vol. 2, pp , 23. [6] Y. K. an, A cover-up approach to partial fraction with linear or irreducible quadratic factor in the denoinator, Applied atheatic and Coputation, vol.29,no.8,pp , 22. [7]Y..a,J.H.Yu,andY.Y.Wang, Aneaypurealgebraic ethod for partial expanion of rational function with ultiple high-order pole, IEEE Tranaction on Circuit and Syte I,vol.6,pp.83 8,24. [8] F. Y. Chin, The partial fraction expanion proble and it invere, SIAJournalonCoputing,vol.6,no.3,pp , 977. [9] H. T. Kung and D.. Tong, Fat algorith for partial fraction decopoition, SIA Journal on Coputing, vol.6,no.3,pp , 977. [2] D. Calvetti, E. Gallopoulo, and L. Reichel, Incoplete partial fraction for parallel evaluation of rational atrix function, Journal of Coputational and Applied atheatic,vol.59,no. 3,pp ,995.

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