CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED NEWTON-COTES QUADRATURES

Filomt 27:4 (2013) 649 658 DOI 10.2298/FIL1304649M Published by Fculty of Sciences nd Mthemtics University of Niš Serbi Avilble t: http://www.pmf.ni.c.rs/filomt 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 16315 1618 Tehrn Irn b Mthemticl Institute of the Serbin Acdemy of Sciences nd Arts Knez Mihil 36 11001 Beogrd Serbi c Deprtment of Finncil Sciences University of Economic Sciences P.O. Box: 15875-1111 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 = {0 1... 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)

M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) 649 658 650 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 = 0 1... 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 = 0 1... 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 [5 5.1.2]. 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 1676. 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.

M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) 649 658 651 2. Closed-form expressions for weighted Cotes numbers Let the nodes x k be given s in 1 i.e. x k = + kh k = 0 1... n nd h = (b )/n. We strt this section with the Newton interpoltion formul [1 pp. 96 101] 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 = {0 1... n} nd µ ν ( h) = ( ) x ν w(x) dx ν = 0 1... (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 = 0 1... 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

M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) 649 658 652 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}];]

M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) 649 658 653 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]:= 1 3 4 1 2 1 4 0 1 4 1 2 3 4 1 989 14 175 5888 928 14 175 14 175 NC181 1 w22 10 496 908 14 175 2835 10 496 928 14 175 14 175 5888 14 175 989 14 175 Out[4]= 9769 155 925 In[5]:= 15 104 33 632 69 376 51 975 155 925 155 925 NC181 1 w32 148 69 376 33 632 15 104 297 155 925 155 925 51 975 9769 155 925 Out[5]= 1249 18 900 544 116 1575 675 352 47 675 90 352 116 675 675 544 1575 1249 18 900 In[6]:= Out[6]= In[7]:= NC151 1 w42 14 947 48 253 6 67 45091252 252474335 2 48 24 57 2757750 2 252137290 2 1253 24 48 6 NC151 1 w52 1351 48 Out[7]= 7500875Π2 12Π 4 2530031Π 2 7500725Π2 6Π 5 2Π 5 3Π 5 In[8]:= 7500725Π 2 3Π 5 NC15 0 1 w6 Out[8]= 0 1 5 2 5 3 5 4 5 In[9]:= NC151 1 w72 2530031Π 2 2Π 5 1 1 054 232 480 249 7500875Π2 12Π4 6Π 5 2 783 252 1 134 032 1 440 747 1 440 747 8024 9801 290 168 1 440 747 8816 205 821 Out[9]= 34000Π2 7 680 000Π 4 32400Π2 2 560 000Π 4 31600Π2 3 840 000Π 4 31600Π2 3 840 000Π 4 32400Π2 2 560 000Π 4 34000Π2 7 680 000Π 4

M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) 649 658 654 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 = 0 1... (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:

M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) 649 658 655 In[3]:= NC281 1 w1 Out[3]= 3 4 1 2 1 4 0 1 4 1 2 3 184 212 4 189 105 488 4918 105 945 488 212 105 105 184 189 In[4]:= Out[4]= In[5]:= NC281 1 w22 11 224 9308 14 175 4725 3736 1978 945 405 3736 9308 11 224 945 4725 14 175 NC281 1 w32 Out[5]= 118 91 135 45 38 139 9 27 38 91 9 45 118 135 In[6]:= NC251 1 w42 Out[6]= In[7]:= Out[7]= In[8]:= 11811612 1013 24 8 NC251 1 w52 506Π2 Π 3 504Π2 Π 3 NC25 0 1 w6 133 8 504Π2 Π 3 39233 2 8 506Π2 Π 3 809892 24 Out[8]= 1 5 2 5 3 5 4 5 In[9]:= NC251 1 w72 14 116 6080 1323 441 4120 441 2944 1323 Out[9]= 1 1 1 1 1600Π 2 1600Π 2 1600Π 2 1600Π 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 = 0 1... (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:

M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) 649 658 656 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]:= 7 8 5 8 3 8 1 8 1 8 3 8 5 8 7 8 295 627 967 680 71 329 967 680 NC381 1 w22 17 473 35 840 128 953 967 680 128 953 967 680 17 473 71 329 295 627 35 840 967 680 967 680 Out[4]= In[5]:= Out[5]= In[6]:= 534 929 2 073 600 265 823 459 983 343 367 343 367 459 983 265 823 534 929 2 903 040 1 612 800 2 903 040 2 903 040 1 612 800 2 903 040 2 073 600 NC381 1 w32 77 437 276 480 1525 55 296 3479 5101 5101 10 240 55 296 55 296 3479 1525 77 437 10 240 55 296 276 480 NC351 1 w42 Out[6]= In[7]:= 595931305 2 42 30557252 384 96 38 1895189 2 56989945 2 64 96 NC351 1 w52 56457905 2 384 Out[7]= In[8]:= 2596001168Π 2 21Π 4 2596001072Π 2 9Π 4 96Π 5 24Π 5 15 000 Π 5 1625 Π 3 189 16Π 2596001072Π2 9Π4 24Π 5 NC35 0 1 w6 2596001168Π 2 21Π 4 96Π 5 Out[8]= 1 10 3 10 1 2 7 10 9 In[9]:= 10 NC351 1 w72 2 286 121 381 024 542 119 95 256 361 021 63 504 239 899 199 921 95 256 381 024 Out[9]= 34600Π2 33400Π2 311000Π 2 33400Π2 34600Π2 3 840 000Π 4 960 000Π 4 640 000Π 4 960 000Π 4 3 840 000Π 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 ( 1 2 1 2 ; 3 2 3 2 ; iπ )} 1.048915591526369693098789786118853446154.

M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) 649 658 657 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 2.1 2.3) 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 5 1.69( 3) 2.98( 1) 1.01( 2) 1.21( 1) 1.70( 2) 10 4.26( 9) 7.14( 6) 2.14( 6) 1.67( 2) 4.46( 3) 15 9.08( 14) 4.14( 10) 1.05( 12) 6.54( 3) 1.99( 3) 20 4.03( 21) 4.92( 17) 1.07( 17) 3.03( 3) 1.10( 3) 25 1.21( 26) 2.60( 22) 1.91( 25) 1.82( 3) 6.82( 4) 30 4.90( 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) = 0.333567469... 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(1 + 10 4 π 2 ) 2.38143139021284126073282 10 5 which integrnd is displyed in Fig. 1. 2 1 0 1 2 1.0 0.5 0.0 0.5 1.0 Figure 1: Integrnd x e x cos(100πx) on [ 1 1]

M. Msjed-Jmei G.V. Milovnović nd M. A. Jfri / Filomt 27:4 (2013) 649 658 658 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 5 1.51( 3) 1.20( 1) 3.68( 3) 10 6.68( 10) 6.71( 7) 3.34( 7) 15 3.97( 15) 1.18( 11) 2.08( 14) 20 1.79( 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 1997. [2] W. Gutschi Moments in qudrture problems Comput. Mth. Appl. 33 (1997) 105 118. [3] Z. Klogirtou T. E. Simos Newton-Cotes formule for long-time integrtion J. Comput. Appl. Mth. 158 (2003) pp. 75 82. [4] M. Msjed-Jmei G.V. Milovnović M.A. Jfri Explicit forms of weighted qudrture rules with geometric nodes Mth. Computer Modeling 53 (2011) 1133 1139. [5] G. Mstroinni nd G.V. Milovnović Interpoltion Processes: Bsic Theory nd Applictions Springer Monogrphs in Mthemtics Springer-Verlg Berlin Heidelberg 2008.