CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED NEWTON-COTES QUADRATURES

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1 Filomt 27:4 (2013) DOI /FIL M Published by Fculty of Sciences nd Mthemtics University of Niš Serbi Avilble t: CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED NEWTON-COTES QUADRATURES Mohmmd Msjed-Jmei Grdimir V. Milovnović b M. A. Jfri c Deprtment of Mthemtics K. N. Toosi University of Technology P. O. Box Tehrn Irn b Mthemticl Institute of the Serbin Acdemy of Sciences nd Arts Knez Mihil Beogrd Serbi c Deprtment of Finncil Sciences University of Economic Sciences P.O. Box: Tehrn Irn Abstrct. In this short note we derive closed expressions for Cotes numbers in the weighted Newton- Cotes qudrture formule with equidistnt nodes in terms of moments nd Stirling numbers of the first kind. Three types of equidistnt nodes re considered. The corresponding progrm codes in Mthemtic Pckge re presented. Finlly in order to illustrte the ppliction of the obtined qudrture formuls few numericl exmples re included. 1. Introduction We consider the weighted qudrture formuls on the finite intervl [ b] f (x)w(x) dx = k S n W k f (x k ) + R n ( f ) (1) where w is given weight function on ( b) nd the nodes x k re equidistntly distributed with the step h = (b )/n in the following three cses: 1 x k = + kh S n = { n}; 2 x k = + kh S n = {1... n 1}; 3 x k = + (k 1 2 )h S n = {1... n}. In the first cse the formul (1) is of the closed type nd in other ones we hve formuls of the open type. Such qudrture formuls re clled the weighted Newton-Cotes formuls if they re interpoltory i.e. R n ( f ) = 0 whenever f P d where P d is the spce of ll lgebric polynomils of degree t most 2010 Mthemtics Subject Clssifiction. Primry 65D30; Secondry 11Y35; 41A55; 65D32. Keywords. Weighted Newton-Cotes qudrture formul Cotes coefficients equidistnt nodes Stirling numbers Newton interpoltion divided differences. Received: 15 December 2011; Accepted: 15 My 2012 Communicted by Drgn Djordjević This reserch of the first uthor ws in prt supported by grnt from Bonyde Mellie Nokhbegn No. PM/1184. The second uthor ws supported in prt by the Serbin Ministry of Eduction nd Science (Project: Approximtion of integrl nd differentil opertors nd pplictions grnt number #174015). Emil ddresses: mmjmei@kntu.c.ir (Mohmmd Msjed-Jmei) gvm@mi.snu.c.rs (Grdimir V. Milovnović) m..jfri@ues.c.ir (M. A. Jfri)

2 M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) d = d n = crd S n 1. Thus the formule (1) re exct on the liner spce P d. Tking ny bsis in this spce e.g. {u 0 u 1... u d } the coefficients W k in (1) must stisfy the following system of liner equtions k S n u ν (x k )W k = u ν (x)w(x) dx ν = d n. For exmple for u ν (x) = x ν it reduces to k S n x ν k W k = x ν w(x) dx ν = d n. The weight coefficients known s the Cotes numbers cn be lso expressed using the Lgrnge interpoltion in the form W k = 1 b Ψ(x)w(x) Ψ dx k S n (2) (x k ) x x k where Ψ is the node polynomil defined by Ψ(x) = (x x k ). k S n (3) In [2] Gutschi considered the numericl construction of these coefficients ssocited by the weight function w nd the nodes x k in two wys: () using (2) nd the brycentric formul for the elementry Lgrnge interpoltion polynomils l k (x) = Ψ(x) (x x k )Ψ (x k ) = ν S n \{k} x x ν x k x ν = λ k x x k ν S n \{k} λ ν x x ν (x x k ) where λ k re the uxiliry quntities λ k = ν S n \{k} 1 x k x ν k S n ; (b) using moment-bsed moments tking certin orthogonl polynomils s bsis functions in {u 0 u 1... u d }. For some remrks on Newton-Cotes rules with Jcobi weight functions see [ ]. An interesting connection between closed Newton Cotes differentil methods nd symplectic integrtors hs been considered in [3]. As we know numericl integrtion begins by Newton s ide from In modern terminology for given distinct points x k nd corresponding vlues f (x k ) Newton constructs the unique polynomil which t the points x k ssumes the sme vlues s f expressing this interpoltion polynomil in terms of divided differences. However tody in lmost ll pplictions Cotes numbers re written in the Lgrnge form (2). In this pper we directly follow Newton s pproch in order to obtin pproprite closed-form expressions for Cotes numbers in ech of cses 1 3 (Section 2). A similr pproch with geometric distributed nodes hs been recently obtined in [4]. The corresponding progrm codes in Mthemtic Pckge for clculting nodes nd weights re lso included in Section 2. In Section 3 we give few numericl exmples in order to illustrte the ppliction of these qudrture formuls. We think tht this pproch my be useful in pplictions tht require explicit expressions for the Cotes numbers.

3 M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) Closed-form expressions for weighted Cotes numbers Let the nodes x k be given s in 1 i.e. x k = + kh k = n nd h = (b )/n. We strt this section with the Newton interpoltion formul [1 pp ] where f (x) = b 0 + b 1 (x x 0 ) + b 2 (x x 0 )(x x 1 ) + + b n (x x 0 )(x x 1 ) (x x n 1 ) + r n+1 ( f ; x) (4) b 0 = f [x 0 ] b 1 = f [x 0 x 1 ] b 2 = f [x 0 x 1 x 2 ]... b n = f [x 0 x 1... x n ] respectively denote divided differences nd r n+1 ( f ; x) is the corresponding interpoltion error r n+1 ( f ; x) = f [x 0 x 1... x n x]ψ n+1 (x) where the node polynomil Ψ n+1 (x) is defined s in (3) i.e. Ψ n+1 (x) = (x x 0 )(x x 1 ) (x x n ). In the sequel we use the Stirling numbers of the first kind s(m ν) which re defined by the coefficients in the expnsion (x) m = x(x 1) (x m + 1) = s(m ν)x ν. (5) For m = 0 we hve (x) 0 = 1 nd s(0 0) = 1. In generl the following recurrence reltion s(m + 1 ν) = s(m ν 1) ms(m ν) 1 ν < m holds with the following initil conditions s(m 0) = 0 nd s(1 1) = 1. Theorem 2.1. Let n N h = (b )/n x k = + kh k S n = { n} nd µ ν ( h) = ( ) x ν w(x) dx ν = (6) h Then the coefficients W k in the qudrture formul (1) re given by W k = ( 1) k n ( m k m=k ) A m ( h) k S n (7) where A m ( h) = ( 1)m m! s(m ν)µ ν ( h) m = n (8) nd s(m ν) re Stirling numbers of the first kind defined in (5). Proof. Let Ψ 0 (x) = 1 nd Ψ m+1 (x) = Ψ m (x)(x x m ) 0 m n nd x = + th where h = (b )/n. Since x x ν = h(t ν) we hve n n Ψ m (x)w(x) dx = h m+1 (t) m w( + th) dt = h m+1 s(m ν)t ν dt 0 0

4 M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) where the Stirling numbers of the first kind re defined in (5). Further it gives Ψ m (x)w(x) dx = h m+1 s(m ν) = h m s(m ν) n = h m s(m ν)µ ν ( h) = ( 1) m m! h m A m ( h) 0 t ν w( + th) dt ( ) x ν w(x) dx where we introduced the nottions (6) nd (8). Now integrting (4) with respect to the weight w(x) over ( b) we obtin f (x)w(x) dx = n b m m=0 According to the generl identity f [x 0 x 1... x m ] = f (x k ) Ψ m+1 (x k) h Ψ m (x)w(x) dx + r n+1 ( f ; x)w(x) dx. (9) in which Ψ m+1 (x k) is the derivtive of the polynomil Ψ m+1 (x) t x = x k we get b m = ( 1) m k f (x k ) k!(m k)!h m = ( 1)m m!h m ( m ( 1) k k ) f (x k ) becuse Ψ m+1 (x k) = m νk (x k x ν ) = ( 1) m k k!(m k)!h m. Therefore (9) reduces to n ( ) m f (x)w(x) dx = A m ( h) ( 1) k f (x k ) + R n ( f ) k = = m=0 n ( 1)k n ( ) m A m ( h) k f (x k) + R n ( f ) m=k n W k f (x k ) + R n ( f ) where the coefficients W k re given by (7) nd R n ( f ) = r n+1( f ; x)w(x) dx. A progrm code in the Mthemtic Pckge for the nodes nd weights (Cotes numbers) from Theorem 2.1 cn be done by the following procedure: NC1[n b_w_]:= Module[{h =(b-)/nmuxkmnuanodesweights} mu=tble[integrte[((x-)/h)ˆnu w[x]{xb}]{nu0n}]; A=Tble[(-1)ˆm/m! Sum[StirlingS1[mnu] mu[[nu+1]]{nu0m}]{m0n}]; nodes = Tble[+k h{k0n}]; weights = Tble[(-1)ˆk Sum[Binomil[mk] A[[m+1]] {mkn}]{k0n}]//simplify; Return[{nodesweights}];]

5 M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) We tke the following seven weight functions w1[x_]:= 1; w2[x_]:= xˆ2; w3[x_]:= Abs[x]; w4[x_]:= Exp[x]; w5[x_]:= Cos[Pi x/2]; w6[x_]:= xˆ(-1/2)log[1/x]; w7[x_]:= Cos[100 Pi x]; where the lst of them is not stndrd (nonnegtive) weight function. Remrk. Alterntively in the cses when symbolic integrtion of the moments µ ν ( h) is not possible then numericl clcultion must be included in the previous subprogrm. Using the previous procedure we obtin the following results for some selected intervls weights nd number of nodes: In[3]:= NC181 1 w1 Out[3]= In[4]:= NC181 1 w Out[4]= In[5]:= NC181 1 w Out[5]= In[6]:= Out[6]= In[7]:= NC151 1 w NC151 1 w Out[7]= Π2 12Π Π Π2 6Π 5 2Π 5 3Π 5 In[8]:= Π 2 3Π 5 NC w6 Out[8]= In[9]:= NC151 1 w Π 2 2Π Π2 12Π4 6Π Out[9]= 34000Π Π Π Π Π Π Π Π Π Π Π Π 4

6 M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) Now we consider the open Newton-Cotes qudrture formul (1) with nodes x k = +kh k = 1... n 1 given s in 2. In this cse the corresponding Newton interpoltion formul is f (x) = c 1 + c 2 (x x 1 ) + c 3 (x x 1 )(x x 2 ) + + c n 1 (x x 1 )(x x 2 ) (x x n 2 ) + ˆr n 1 ( f ; x) (10) where c 1 = f [x 1 ] c 2 = f [x 1 x 2 ] c 2 = f [x 1 x 2 x 3 ]... c n 1 = f [x 1 x 2... x n 1 ] nd the corresponding error is ˆr n 1 ( f ; x) = f [x 1 x 2... x n 1 x] Ω n 1 (x) where the node polynomil ω n 1 (x) is defined now s Ω n 1 (x) = (x x 1 )(x x 2 ) (x x n 1 ). Theorem 2.2. Let n 2 h = (b )/n x k = + kh k S n = {1... n 1} nd µ ν( h) = ( x ν 1) w(x) dx ν = (11) h Then the coefficients W k in the qudrture formul (1) re given by where n 1 ( m W k = k( 1) k k m=k A m( h) = ( 1)m m! ) A m( h) k S n (12) m 1 s(m 1 ν)µ ν( h) m = 1... n 1 (13) nd s(m 1 ν) re Stirling numbers of the first kind defined in (5). Proof. Tking Ω m (x) = (x x 1 ) (x x m ) we get m 1 Ω m 1 (x)w(x) dx = h m 1 s(m 1 ν) ( x m 1 = h m 1 s(m 1 ν)µ ν( h) = ( 1) m m! h m 1 A m( h) h ) 1 w(x) dx where µ ν( h) nd A m( h) re defined by (11) nd (13) respectively. Similrly s before we hve c m = f [x 1 x 2... x m ] = k=1 so tht (12) follows immeditely. f (x k ) Ω m(x k ) = ( 1)m m!h m 1 ( m k( 1) k k k=1 ) f (x k ) Mthemtic code of the corresponding procedure is s follows: NC2[n b_w_]:= Module[{h =(b-)/nmuxkmnuanodesweights} mu=tble[integrte[((x-)/h-1)ˆnu w[x]{xb}]{nu0n-1}]; A=Tble[(-1)ˆm/m! Sum[StirlingS1[m-1nu] mu[[nu+1]] {nu0m-1}]{m1n-1}]; nodes = Tble[+k h{k1n-1}]; weights = Tble[k(-1)ˆk Sum[Binomil[mk] A[[m]] {mkn-1}]{k1n-1}]//simplify; Return[{nodesweights}];] Tking the sme previous weight functions we get the following results:

7 M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) In[3]:= NC281 1 w1 Out[3]= In[4]:= Out[4]= In[5]:= NC281 1 w NC281 1 w32 Out[5]= In[6]:= NC251 1 w42 Out[6]= In[7]:= Out[7]= In[8]:= NC251 1 w52 506Π2 Π 3 504Π2 Π 3 NC w Π2 Π Π2 Π Out[8]= In[9]:= NC251 1 w Out[9]= Π Π Π Π 2 Finlly for n open qudrture formul with nodes given s in 3 we cn prove the following sttement: Theorem 2.3. Let n N h = (b )/n x k = + ( k 1 2) h k Sn = {1... n} nd µ ν( h) = ( x 1 ν w(x) dx ν = (14) h 2) Then the coefficients W k in the qudrture formul (1) re given by where W k = k( 1) k n ( m k m=k à m( h) = ( 1)m m! )à m( h) k S n (15) m 1 s(m 1 ν) µ ν( h) m = 1... n (16) nd s(m 1 ν) re Stirling numbers of the first kind defined in (5). Mthemtic code of the corresponding procedure is s follows:

8 M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) NC3[n b_w_]:= Module[{h =(b-)/nmuxkmnuanodesweights} mu=tble[integrte[((x-)/h-1/2)ˆnu w[x]{xb}]{nu0n}]; A=Tble[(-1)ˆm/m! Sum[StirlingS1[m-1nu] mu[[nu+1]] {nu0m-1}]{m1n}]; nodes = Tble[+(k-1/2)h{k1n}]; weights = Tble[k(-1)ˆk Sum[Binomil[mk] A[[m]] {mkn}]{k1n}]//simplify; Return[{nodesweights}];] For the sme previous weight functions we get the following results: In[3]:= NC381 1 w1 Out[3]= In[4]:= NC381 1 w Out[4]= In[5]:= Out[5]= In[6]:= NC381 1 w NC351 1 w42 Out[6]= In[7]:= NC351 1 w Out[7]= In[8]:= Π 2 21Π Π 2 9Π 4 96Π 5 24Π Π Π Π Π2 9Π4 24Π 5 NC w Π 2 21Π 4 96Π 5 Out[8]= In[9]:= 10 NC351 1 w Out[9]= 34600Π Π Π Π Π Π Π Π Π Π 4 3. Numericl exmples As first exmple we consider n integrl which vlue cn be expressed in terms of the generlized hypergeometric function p F q (; b; z) 1 0 sin πx x log 1 x dx = 4 Im { 2F 2 ( ; ; iπ )}

9 M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) In this cse we tke qudrture rules with respect to the weight function w(x) = w 6 (x) = x 1/2 log(1/x). Then for f (x) = sin πx the reltive errors in the corresponding qudrtures with equidistnt nodes given in the cses 1 3 (Theorems ) re presented in Tble 1. Numbers in prentheses indicte deciml exponents. Notice tht the corresponding number of qudrture nodes in these cses re n + 1 n 1 nd n respectively. Tble 1: Reltive errors of qudrture sums for n = 5(5)30 w(x) w 6 (x) = x 1/2 log(1/x) w 5 (x) = cos(πx/2) n Cse 1 Cse 2 Cse 3 Cse 2 Cse ( 3) 2.98( 1) 1.01( 2) 1.21( 1) 1.70( 2) ( 9) 7.14( 6) 2.14( 6) 1.67( 2) 4.46( 3) ( 14) 4.14( 10) 1.05( 12) 6.54( 3) 1.99( 3) ( 21) 4.92( 17) 1.07( 17) 3.03( 3) 1.10( 3) ( 26) 2.60( 22) 1.91( 25) 1.82( 3) 6.82( 4) ( 35) 1.99( 30) 3.56( 31) 1.13( 3) 4.67( 4) Evidently the closed rule (Cse 1 ) converges fster thn other two open rules (Cses 2 nd 3 ) with smller number of nodes. As second exmple we consider the following integrl 1 log(1 x 2 ) cos πx 1 2 dx = 4 ( ) γ Ci(π) + log(π/4) = π nd n ppliction of the previous qudrture rules with respect to the weight function w(x) = w 5 (x) = cos(πx/2) to the function f (x) = log(1 x 2 ). Regrding the logrithmic singulrities t ±1 the first rule cnnot be pplied nd the other ones show very slow convergence (of course becuse of the influece of these singulrities). The corresponding reltive errors in qudrture sums re presented in the second prt of the sme Tble 1. Finlly we consider n integrl of highly oscilltory function 1 e x e 2 1 cos(100πx) dx = 1 e( π 2 ) which integrnd is displyed in Fig Figure 1: Integrnd x e x cos(100πx) on [ 1 1]

10 M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) Tble 2: Reltive errors of qudrture sums for n = 5(5)20 in three different cses for w 7 (x) = cos(100πx) nd f (x) = e x n Cse 1 Cse 2 Cse ( 3) 1.20( 1) 3.68( 3) ( 10) 6.71( 7) 3.34( 7) ( 15) 1.18( 11) 2.08( 14) ( 23) 1.55( 19) 5.27( 20) In Tble 2 we present reltive errors in qudrture sums for ech of the obtined qudrture rules (Cses 1 3 ) with respect to the oscilltory weight function w 7 (x) = cos(100πx). As we cn see the convergence of these rules is very fst. References [1] W. Gutschi Numericl Anlysis: An Introduction Birkhäuser Boston [2] W. Gutschi Moments in qudrture problems Comput. Mth. Appl. 33 (1997) [3] Z. Klogirtou T. E. Simos Newton-Cotes formule for long-time integrtion J. Comput. Appl. Mth. 158 (2003) pp [4] M. Msjed-Jmei G.V. Milovnović M.A. Jfri Explicit forms of weighted qudrture rules with geometric nodes Mth. Computer Modeling 53 (2011) [5] G. Mstroinni nd G.V. Milovnović Interpoltion Processes: Bsic Theory nd Applictions Springer Monogrphs in Mthemtics Springer-Verlg Berlin Heidelberg 2008.