Higher-Order Newton-Cotes Formulas

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Journal of Mathematics and Statistics 6 (): 19-0, 010 ISSN 159-6 010 Science Publications Higher-Order Newton-Cotes Formulas Pedro Americo Almeida Magalhaes Junior and Cristina Almeida Magalhaes Pontificia Universidade Catolica de Minas Gerais, Av. Dom Jose Gaspar 500, 055-610, Belo Horizonte, Minas Gerais, Brazil Abstract: Problem statement: The present work offers equations of Newton-Cotes Integration until twenty segments. Approach: It shows Newton-Cotes closed and open integration formulas. The new type Newton-Cotes semi-closed or semi-open was proposed. Results: An analysis of error in the technique was made. To estimate the error of integration in discrete data, we propose apply different rules or mix several rules. Conclusion/Recommendations: The difference in the result of each formula provides an approximation of the error. Computational routines to generate Newton-Cotes integration rules were presented. Key words: Numerical methods, numerical integration, quadrature, Newton-cotes, higher-order, analysis of error INTRODUCTION The evaluation of integrals, a process known as integration or quadrature, is required in many problems in engineering and science (Sermutlu, 005). The function f(x), which is to be integrated, may be a known function or a set of discrete data. Some known functions have an exact integral, in which case can be evaluated exactly in closed form. Many known functions, however, do not have an exact integral and an approximate numerical procedure is required to evaluate. In many cases, the function f(x) is known only at a set of discrete points, in which case an approximate numerical procedure is again required to evaluate (Simos, 008). Numerical integration (quadrature) formulas can be developed by fitting approximating functions (e.g., polynomials) to discrete data and integrating the approximating function: xn x 1 + (N 1)h (1) I = f(x)dx P(x)dx x1 x1 When the function to be integrated is known at equally spaced points (Δx = h = constant) and N is number of points with x ranging x 1, x 1 +h, x 1 +h,, x 1 +(N-1)h. The polynomial can be fit to the discrete data with much less effort, thus significantly decreasing the amount of effort required (Simos, 008). The resulting formulas are called Newton-cotes formulas. The distance between the lower and upper limits of integration is called the range of integration. The distance between any two data points is called an increment (Δx = h). A linear polynomial requires one increment and two data points to obtain a fit. A quadratic polynomial requires two increments and three data points to obtain a fit. And so on for higher-degree polynomials. The group of increments required to fit a polynomial is called an interval (Kalogiratou and Simos, 00). A linear polynomial requires an interval consisting of only one increment. A quadratic polynomial requires an interval containing two increments. And so on. The total range of integration can consist of one or more intervals. Each interval consists of one or more increments, depending on the degree of the approximating polynomial. Closed and open forms of Newton-Cotes formulas are available. The closed forms are those where the data points at the beginning and end of the limits of integration are known. The open forms have integration limits that extend beyond the range of the data (Witteveen et al., 009). MATERIALS AND METHODS Newton-cotes closed integration equation: The rule for a single interval is obtained by fitting a first-degree polynomial to two discrete points (Sermutlu and Eyyubog lu, 007). The upper limit of integration is x = x 1 +h, then the integral (I) have the formula: h I = ( y1+ y) or I = 0.5hy1+ 0.5hy Error O( h ) () Corresponding Author: Pedro Americo Almeida Magalhaes Junior, Pontificia Universidad Catolica de Minas Gerais, Av. Dom Jose Gaspar 500, 055-610, Belo Horizonte, Minas Gerais, Brazil Tel: +55 1 1-681 19

J. Math. & Stat., 6 (): 19-0, 010 Simpson s 1/ rule is obtained by fitting a seconddegree polynomial to three equally spaced discrete points. The upper limit of integration is x, then: h I = ( y1+ y + y) or Error O( h ) I = 0.hy1 + 1.hy + 0.hy () Simpson s /8 rule is obtained by fitting a thirddegree polynomial to four equally spaced discrete points. The upper limit of integration is x, then: h I = y + y + y + y or Error O h 8 I = 0.75hy + 1.15hy + 1.15hy + 0.75hy ( 1 ) ( ) 1 () Boole s rule is obtained by fitting a fourth-degree polynomial to five equally spaced discrete points. The upper limit of integration is x 5, then: h 6 I = (7y 1+ y + 1 y + y +7 y 5) or Error O( h ) 5 I = 0.111111111111111111111111111111111111111hy1 + 1.hy + 0.5hy 1.hy + + 0.111111111111111111111111111111111111111hy 5 (5) Generally, have the closed formulas, where N is number of points, ci are integer coefficients and cr are real coefficients: num I= h(ciy 1 1+ciy +ciy +ci y +... den +cin y N +cin 1y N 1+ciNy N) or cr 1 y1+ cr y + cr y +... I= h + cr N yn + cr N 1yN 1+ cr Ny N (6) upper limit of integration is x = x 1 +h, then the integral (I) have the formula: ( ) ( ) I = h y or I = hy Error O h (7) For three intervals, the rule is obtained by fitting a first-degree polynomial to four discrete points. The upper limit of integration is x = x 1 +h, then the integral (I) have the formula: h I = ( y + y) or I = 1.5hy + 1.5hy Error O( h ) (8) For N = 5, the rule is obtained by fitting a seconddegree polynomial to four equally spaced discrete points. The upper limit of integration is x 5, then: h I = ( y y+ y) or Error O( h ) I =.666666666666666666666666666666666666667hy 1.hy +.666666666666666666666666666666666666667hy (9) For N = 6, the rule is obtained by fitting a thirddegree polynomial to five equally spaced discrete points. The upper limit of integration is x 6, then: 5h I = (11 y + y + y +11 y 5) or Error O( h ) I =.91666666666666666666666666666666666667hy + 0.08hy + 0.08hy.91666666666666666666666666666666666667hy + 5 (11) Generally, have the open formulas, where N is number of points, ci are integer coefficients and cr are real coefficients: num I= h(ciy +ciy +ci y +...+cin y N +cin 1y N 1) den or I = h cr y + cr y +... + cr y + cr y ( ) N N N 1 N 1 (1) RESULTS AND DISCUSSION The Table 1 and shows of coefficients for higher order closed integration formulas. Figure 1 shows the generation of rules in Maple 1.0. Newton-cotes open integration equation: In the open integration formulas, the first (y 1 ) and last (y N ) point does not appear in equation. The midpoints rule for a Fig. 1: Verification Newton-cotes closed integration double interval is obtained by fitting a zero-degree formulas with maple 1.0. (number of points polynomial to three discrete points (Espelid, 00). The N =..105) 19

J. Math. & Stat., 6 (): 19-0, 010 Table 1: Newton-cotes closed integration formulas with points =.1 Rule Integer coefficient Real coefficient Trapezoidal rule num = 1 Points (N) = den = Segments (N-1) = 1 ci 1 = ci = 1 cr 1 = cr = +0.5000000000000000000000000000000000000000 Error»O (h ) Simpson s 1/ rule num = 1 Points (N) = den = Segments (N-1) = ci 1 = ci = 1 cr 1 = cr = +0. Error»O (h ) ci = cr = +1. Simpson s /8 rule num = Points (N) = den = 8 Segments (N-1) = ci 1 = ci = 1 cr 1 = cr = +0.750000000000000000000000000000000000000 Error»O (h ) ci = ci = cr = cr = +1.150000000000000000000000000000000000000 Boole s rule num = Points (N) = 5 den = 5 Segments (N-1) = ci 1 = ci 5 = 7 cr 1 = cr 5 = +0.111111111111111111111111111111111111111 Error»O (h 6 ) ci = ci = cr = cr = +1. ci = 1 cr = +0.5 Points (N) = 6 den = 88 Segments (N-1) = 5 ci 1 = ci 6 = 19 cr 1 = cr 6 = +0.98611111111111111111111111111111111111 Error»O (h 6 ) ci = ci 5 = 75 cr = cr 5 = +1.008 ci = ci = 50 cr = cr = +0.8680555555555555555555555555555555555556 num = 1 Points (N) = 7 den = 10 Segments (N-1) = 6 ci 1 = ci 7 = 1 cr 1 = cr 7 = +0.98571857185718571857185719 Error»O (h 8 ) ci = ci 6 = 16 cr = cr 6 = +1.58571857185718571857185719 ci = ci 5 = 7 cr = cr 5 = +0.198571857185718571857185719 ci = 7 cr = +1.98571857185718571857185719 num = 7 Points (N) = 8 den = 1780 Segments (N-1) = 7 ci 1 = ci 8 = 751 cr 1 = cr 8 = +0.057070707070707070707070 Error»O (h 8 ) ci = ci 7 = 577 cr = cr 7 = +1.90160707070707070707070707 ci = ci 6 = 1 cr = cr 6 = +0.55975000000000000000000000000000000000 ci = ci 5 = 989 cr = cr 5 = +1.108175959595959595959595959 num = Points (N) = 9 den = 1175 Segments (N-1) = 8 ci 1 = ci 9 = 989 cr 1 = cr 9 = +0.790889165795590889165795591 Error»O (h 10 ) ci = ci 8 = 5888 cr = cr 8 = +1.66151675850088181516758500881815 ci = ci 7 = -98 cr = cr 7 = -0.61869885615508186988561550819 ci = ci 6 = 1096 cr = cr 6 = +.96181516758500881815167585008818 ci 5 = -50 cr 5 = -1.811877795160811877795160811 num = 9 Points (N) = 10 den = 89600 Segments (N-1) = 9 ci 1 = ci 10 = 857 cr 1 = cr 10 = +0.86975685718571857185718571 Error»O (h 10 ) ci = ci 9 = 1571 cr = cr 9 = +1.58117185718571857185718571 ci = ci 8 = 1080 cr = cr 8 = +0.10881857185718571857185719 ci = ci 7 = 19 cr = cr 7 = +1.9057185718571857185718571 ci 5 = ci 6 = 5778 cr 5 = cr 6 = +0.58079685718571857185718571 Points (N) = 11 den = 9976 Segments (N-1) = 10 ci 1 = ci 11 = 16067 cr 1 = cr 11 = +0.68186196197091781695597075 Error»O (h 1 ) ci = ci 10 = 10600 cr = cr 10 = +1.775591801719060809196980859 ci = ci 9 = -855 cr = cr 9 = -0.810570669096079570669096079 ci = ci 8 = 700 cr = cr 8 = +.59688796161959688796161959 ci 5 = ci 7 = -60550 cr 5 = cr 7 = -.51551655165516551655165517 ci 6 = 768 cr 6 = +7.17660197097967660197097968 num = 11 Points (N) = 1 den = 8709100 Segments (N-1) = 11 ci 1 = ci 1 = 17165 cr 1 = cr 1 = +0.76550015990597687156199710 Error»O (h 1 ) ci = ci 11 = 18659 cr = cr 11 = +1.700897795160811877795161 ci = ci 10 = -711 cr = cr 10 = -0.088615971775619870661510875955 ci = ci 9 = 56685 cr = cr 9 = +.186080770899708997089970899709 ci 5 = ci 8 = -95955 cr 5 = cr 8 = -1.11958980905097001766680509700 ci 6 = ci 7 = 159566 cr 6 = cr 7 = +1.956905157885199118165788515 195

J. Math. & Stat., 6 (): 19-0, 010 Table 1: Continued num = 1 Points (N) = 1 den = 55550 Segments (N-1) = 1 ci 1 = ci 1 = 16651 cr 1 = cr 1 = +0.596789959565785099816707101 Error»O (h 1 ) ci = ci 1 = 990168 cr = cr 1 = +1.888100709576181756671899590 ci = ci 11 = -758786 cr = cr 11 = -1.86565006197991699577851 ci = ci 10 = 57510 cr = cr 10 = +6.7979867751967891565580701061 ci 5 = ci 9 = -519195 cr 5 = cr 9 = -9.7980676669075105078195609789 ci 6 = ci 8 = 8751688 cr 6 = cr 8 = +16.651160680155985888971696876 ci 7 = -8779716 cr 7 = -16.706557557557557557557 num = 1 Points (N) = 1 den = 061000 Segments (N-1) = 1 ci 1 = ci 1 = 818190909 cr 1 = cr 1 = +0.65186660650979081711700018578 Error»O (h 1 ) ci = ci 1 = 568079661 cr = cr 1 = +1.818891085509856597001678106 ci = ci 1 = -168557 cr = cr 1 = -1.01051798908059558680 ci = ci 11 = 156071795 cr = cr 11 = +5.0650005170768199551851069551 ci 5 = ci 10 = -151659575 cr 5 = cr 10 = -.900009611759190875751715688715 ci 6 = ci 9 = 06687987 cr 6 = cr 9 = +6.67779000198699008817766770 ci 7 = ci 8 = -1119961 cr 7 = cr 8 = -1.99168701559917715501868891 Table : Newton-cotes closed integration formulas with points = 15.1 Rule Integer coefficient Real coefficient num = 7 Points (N) = 15 den = 50198000 Segments (N-1) = 1 ci 1 = ci 15 = 901897 cr 1 = cr 15 = +0.585970117768090887761918008816 Error»O (h 16 ) ci = ci 1 = 71098686 cr = cr 1 = +1.989915698615007066510566890818 ci = ci 1 = -77070657 cr = cr 1 = -.156585879515698911799011957 ci = ci 1 = 50178 cr = cr 1 = +9.7968761675850975178007955675 ci 5 = ci 11 = -665096 cr 5 = cr 11 = -18.55966790515559565711086809850 ci 6 = ci 10 = 16011616 cr 6 = cr 10 = +5.708856100907779901559917981696 ci 7 = ci 9 = -1680707 cr 7 = cr 9 = -7.01010850797091556780865608711 ci 8 = 19586 cr 8 = +5.65781975980118776056857 Points (N) = 16 den = 68881618 Segments (N-1) = 15 ci 1 = ci 16 = 5100 cr 1 = cr 16 = +0.56099657891551155115511551 Error»O (h 16 ) ci = ci 15 = 6555865 cr = cr 15 = +1.976106801610777615190095618667078 ci = ci 1 = -96065 cr = cr 1 = -1.690857589578079118081056596 ci = ci 1 = 107777585 cr = cr 1 = +7.60560615508585968799771657 ci 5 = ci 1 = -15680685 cr 5 = cr 1 = -11.9676610517851785178518 ci 6 = ci 11 = 6188669 cr 6 = cr 11 = +17.8700560198506850956578788699 ci 7 = ci 10 = -000805 cr 7 = cr 10 = -1.50078177071809999585857095 ci 8 = ci 9 = 1018807605 cr 8 = cr 9 = +7.95585075971008889565807687 num = 8 Points (N) = 17 den = 886975 Segments (N-1) = 16 ci 1 = ci 17 = 15061177 cr 1 = cr 17 = +0.6815866909995710907787767189 Error»O (h 18 ) ci = ci 16 = 176660659 cr = cr 16 = +.0905906086859899795111505 ci = ci 15 = -17971170 cr = cr 15 = -.9616890897881576688196097 ci = ci 1 = 81185560 cr = cr 1 = +1.69908706959767066087766 ci 5 = ci 1 = -1999860750 cr 5 = cr 1 = -1.60118577116098809101515167 ci 6 = ci 1 = 17758889696 cr 6 = cr 1 = +68.099756710660518989696087555 ci 7 = ci 11 = -680650796 cr 7 = cr 11 = -111.76918975800018568658017916 ci 8 = ci 10 = 9688750180 cr 8 = cr 10 = +15.7768869867878558697185776 ci 9 = -108970 cr 9 = -167.61560116666709070885809 num = 17 Points (N) = 18 den = 7661017980000 Segments (N-1) = 17 ci 1 = ci 18 = 5597087657 cr 1 = cr 18 = +0.959765097715667085870170106559560 Error»O (h 18 ) ci = ci 17 = 50185515685 cr = cr 17 = +.011578691950976767809965859 ci = ci 16 = -50700885 cr = cr 16 = -.66671650597095010788790111018 ci = ci 15 = 866557660 cr = cr 15 = +10.967581986979986980569906758 ci 5 = ci 1 = -76891680010 cr 5 = cr 1 = -1.5665587107171889655759185558 ci 6 = ci 1 = 885516686898 cr 6 = cr 1 = +9.979151889870187076607881 ci 7 = ci 1 = -1090571859796660 cr 7 = cr 1 = -9.6118687716986678507585897 ci 8 = ci 11 = 10069615750186 cr 8 = cr 11 = +5.5750111880597001687787857 ci 9 = ci 10 = -7597859705070 cr 9 = cr 10 = -16.971889577517060976986516686 num = Points (N) = 19 den = 5850000 196

J. Math. & Stat., 6 (): 19-0, 010 Table : Continued Segments (N-1) = 18 ci 1 = ci 19 = 075169 cr 1 = cr 19 = +0.1117198806095876665811617501 Error»O (h 0 ) ci = ci 18 = 188701900 cr = cr 18 = +.18797879956850611711685661805 ci = ci 17 = -17795 cr = cr 17 = -.808585591189679055701595966787 ci = ci 16 = 1559801096 cr = cr 16 = +18.7956818608507697059199888 ci 5 = ci 15 = -686068580 cr 5 = cr 15 = -50.11515775515560970780775 ci 6 = ci 1 = 106805665808 cr 6 = cr 1 = +1.706000889785961958105715086506 ci 7 = ci 1 = -19868986770 cr 7 = cr 1 = -5.100595687957155608811091606 ci 8 = ci 1 = 1905787980 cr 8 = cr 1 = +77.5791780116760010588797588891 ci 9 = ci 11 = -191795111198 cr 9 = cr 11 = -96.0807017176971988111708018080 ci 10 = 6170680 cr 10 = +55.98168196766155065797598965 num = 19 Points (N) = 0 den = 577999181150000 Segments (N-1) = 19 ci 1 = ci 0 = 69087615560 cr 1 = cr 0 = +0.8781880719661998559776551718 Error»O (h 0 ) ci = ci 19 = 6065087080815 cr = cr 19 = +.1650165899858765501997507 ci = ci 18 = -9680515700955 cr = cr 18 = -.795590796708617900008 ci = ci 17 = 015815850500095 cr = cr 17 = +15.197156869170556599916 ci 5 = ci 16 = -106919788 cr 5 = cr 16 = -6.59650598990601018596617106 ci 6 = ci 15 = 6087996116 cr 6 = cr 15 = +78.9169677991769179775710 ci 7 = ci 1 = -51888111781580 cr 7 = cr 1 = -1.858905561570966950155906 ci 8 = ci 11 = 907687056515580 cr 8 = cr 1 = +155.168795055806701081960105079 ci 9 = ci 1 = -708870617985190 cr 9 = cr 1 = -11.0009159001186767986088806008 ci 10 = ci 11 = 15187059175957 cr 10 = cr 11 = +5.51780115010585607095117577606 num = 1 Points (N) = 1 den = 87050 Segments (N-1) = 0 ci 1 = ci 1 = 19701019 cr 1 = cr 1 = +0.65056980606895700559795 Error»O (h ) ci = ci 0 = 1879609080000 cr = cr 0 = +.875589199979905887519008 ci = ci 19 = -895819177500 cr = cr 19 = -.795671085986086619180670906 ci = ci 18 = 19859691590000 cr = cr 18 = +.1778696751880701691658195 ci 5 = ci 17 = -608988588975 cr 5 = cr 17 = -75.0650660957559909597860681 ci 6 = ci 16 = 170198777651750 cr 6 = cr 16 = +06.7596987960870666807797666 ci 7 = ci 15 = -789767190000 cr 7 = cr 15 = -5.1761687959059595990586115659 ci 8 = ci 1 = 6886987570000 cr 8 = cr 1 = +86.56118871090695160791598 ci 9 = ci 1 = -10599018170150 cr 9 = cr 1 = -181.505589800800901109969777170557 ci 10 = ci 1 = 16517980000 cr 10 = cr 1 = +1655.9566995701709670788179 ci 11 = -1819566078096 cr 11 = -1800.107708578915807861809576 Table : Newton-cotes open integration formulas with points =.16 Rule Integer coefficient Real coefficient Midpoint rule num = Points (N) = den = 1 Segments (N-1) = ci = 1 cr = +.0000000000000000000000000000000000000000 Error»O (h ) Points (N) = num = Segments (N-1) = den = Error»O (h ) ci = ci = 1 cr = cr = +1.5000000000000000000000000000000000000000 Points (N) = 5 num = den = Segments (N-1) = ci = ci = cr = cr = +.6666666666666666666666666666666666666667 Error»O (h ) ci = -1 cr = -1. Points (N) = 6 den = Segments (N-1) = 5 ci = ci 5 = 11 cr = cr 5 = +.916666666666666666666666666666666666667 Error»O (h ) ci = ci = 1 cr = cr = +0.08 Points (N) = 7 num = den = 10 Segments (N-1) = 6 ci = ci 6 = 11 cr = cr 6 = +.000000000000000000000000000000000000000 Error»O (h 6 ) ci = ci 5 = -1 cr = cr 5 = -.000000000000000000000000000000000000000 ci = 6 cr = +7.8000000000000000000000000000000000000000 Points (N) = 8 num = 7 den = 10 Segments (N-1) = 7 ci = ci 7 = 611 cr = cr 7 = +.970188888888888888888888888888888888889 Error»O (h 6 ) ci = ci 6 = -5 cr = cr 6 = -.008 ci = ci 6 = 56 cr = cr 5 = +.719 Points (N) = 9 num = 8 den = 95 Segments (N-1) = 8 ci = ci 8 = 60 cr = cr 8 = +.89179891798917989179891798917989 Error»O (h 8 ) ci = ci 7 = -95 cr = cr 7 = -8.076190761907619076190761907619076 ci = ci 6 = 196 cr = cr 6 = +18.59076190761907619076190761907619 ci 5 = -59 cr 5 = -0.8169116911691169116911691 197

J. Math. & Stat., 6 (): 19-0, 010 Table : Continued Points (N) = 10 num = 9 den = 80 Segments (N-1) = 9 ci = ci 9 = 1787 cr = cr 9 = +.5899555718571857185718571857 Error»O (h 8 ) ci = ci 8 = -80 cr = cr 8 = -5.610678571857185718571857186 ci = ci 7 = 967 cr = cr 7 = +9.9788185718571857185718571 ci 5 = ci 6 = -1711 cr 5 = cr 6 = -.77678571857185718571857186 Points (N) = 11 den = 56 Segments (N-1) = 10 ci = ci 10 = 05 cr = cr 10 = +.5877500917107587750091710758 Error»O (h 10 ) ci = ci 9 = -11690 cr = cr 9 = -1.88580691580691580691580691 ci = ci 8 = 0 cr = cr 8 = +6.7500917107587750091710758775 ci 5 = ci 7 = -55070 cr 5 = cr 7 = -60.7067865961199956786596119995 ci 6 = 678 cr 6 = +7.7597001766680509700176668051 num = 11 Points (N) = 1 den = 757600 Segments (N-1) = 11 ci = ci 11 = 7577 cr = cr 11 = +.1717988001567901567901567901 Error»O (h 10 ) ci = ci 10 = -660199 cr = cr 10 = -10.008155508775009171075877501 ci = ci 9 = 1567880 cr = cr 9 = +.7561560511687971781051168797 ci 5 = ci 8 = -17085616 cr 5 = cr 8 = -5.895857587750091710758775009 ci 6 = ci 7 = 889158 cr 6 = cr 7 = +1.76057958500881815167585008818 num = 1 Points (N) = 1 den = 195 Segments (N-1) = 1 ci = ci 1 = 966 cr = cr 1 = +5.000519805198051980519805198051981 Error»O (h 1 ) ci = ci 11 = -5771 cr = cr 11 = -18.58766766766766766766 ci = ci 10 = 1058 cr = cr 10 = +6.9676676676676676677 ci 5 = ci 9 = -6698 cr 5 = cr 9 = -18.66766766766766766 ci 6 = ci 8 = 7956 cr 6 = cr 8 = +.18051980519805198051980519805 ci 7 = -90 cr 7 = -56.651980519805198051980519805198 num = 1 Points (N) = 1 den = 9580000 Segments (N-1) = 1 ci = ci 1 = 8897 cr = cr 1 = +.76590878816010685905816770 Error»O (h 1 ) ci = ci 1 = -11607 cr = cr 1 = -15.8571556159680696996159680 ci = ci 11 = 7167557 cr = cr 11 = +5.75757656787976676811165566 ci 5 = ci 10 = -57189865 cr 5 = cr 10 = -77.5966011198198198198 ci 6 = ci 9 = 67787908 cr 6 = cr 9 = +85.1901079911157881995866115788 ci 7 = ci 8 = -67100 cr 7 = cr 8 = -6.879855758711580790758711 num = 7 Points (N) = 15 den = 16988000 Segments (N-1) = 1 ci = ci 1 = 9067 cr = cr 1 = +5.598589975005161775101780 Error»O (h 1 ) ci = ci 1 = -19711 cr = cr 1 = -5.1711876600765896578567156 ci = ci 1 = 6058888 cr = cr 1 = +101.70015006187809506167858950761 ci 5 = ci 11 = -1615958710 cr 5 = cr 11 = -71.710609798009550657086091066 ci 6 = ci 10 = 1575 cr 6 = cr 10 = +50.807779897798977989779897798978 ci 7 = ci 9 = -796678 cr 7 = cr 9 = -805.151707675010087170767501008 ci 8 = 5871088 cr 8 = +91.1755196018590717961068519957 Points (N) = 16 den = 1700 Segments (N-1) = 15 ci = ci 15 = 1811601 cr = cr 15 = +5.5961568167081777960151099 Error»O (h 1 ) ci = ci 1 = -77951959 cr = cr 1 = -1.66781105178861615759009019 ci = ci 1 = 6718566 cr = cr 1 = +77.578118796159197991066516080 ci 5 = ci 1 = -60181 cr 5 = cr 1 = -17.6156005799679967996799670 ci 6 = ci 11 = 95180 cr 6 = cr 11 = +7.61115655675169591955066 ci 7 = ci 10 = -961657 cr 7 = cr 10 = -71.08588076807666619095190576 ci 8 = ci 9 = 01701868 cr 8 = cr 9 = +117.516589057197100888956 The Table and shows of coefficients for higher order open integration formulas. Figure shows the generation of rules in Maple 1.0. does not appear in equation (Zhang et al., 009). The rule for a double interval is obtained by fitting a zerodegree polynomial to three discrete points. When N is odd, the semi-closed or semi-open integration formulas are same as open rules. The upper limit of integration is x = x 1 +h, then the Integral (I) have the formula: Newton-cotes semi-closed or semi-open integration equation: In the semi-open integration formulas, the last (y N ) point does not appear in equation. In the semi-closed integration formulas, the first (y 1 ) point I = h( y) or I = hy Error O( h ) (1) 198

J. Math. & Stat., 6 (): 19-0, 010 Fig. : Verification Newton-cotes open integration formulas with maple 1.0. with number of points N =..105 For three intervals, the semi-open rule is obtained by fitting a first-degree polynomial to four discrete points. The upper limit of integration is x = x 1 +h, then the Integral (I) have the formula: h I = ( y1+ y) or I = 0.75hy1+.5hy Error O( h ) (1) For N = 5, the rule is obtained by fitting a seconddegree polynomial to four equally spaced discrete points, same of the open formula. The upper limit of integration is x 5, then: h I = ( y y+ y) or Error O( h ) I =.666666666666666666666666666666666666667hy 1.hy +.666666666666666666666666666666666666667hy (15) Fig. : Verification Newton-cotes semi-open integration formulas with maple 1.0. with N =..105 The Table 5 and 6 shows of coefficients for higher order open integration formulas. Figure shows the generation of rules in Maple 1.0. The semi-closed integration formulas are same as semi-open rules. For examples, N = : ( ) ( ) I = h y or I = hy Error O h (18) For N = : h I = ( y + y) or I = 0.75hy +.5hy Error O( h ) (19) Generally, have the semi-closed formulas, where N is number of points, ci are integer coefficients and cr are real coefficients: For N = 6, the rule is obtained by fitting a thirddegree polynomial to five equally spaced discrete points. The upper limit of integration is x 6, then: 5h 5 I = (19y1-10 y + 10 y-70 y +85 y 5) or Error O( h ) 1 I = 0.6597hy1-0.7hy (16) +.166666666666666666666666666666666666667hy. 0555555555555555555555555555555555556hy +.95188888888888888888888888888888888889hy Generally, have the semi-open formulas, where N is number of points: num I= h(ciy 1 1+ciy +ciy +ci y +... den +cin y N +cin 1y N 1) cr 1 y1+ cr y + cr y +... or I = h + cr N yn + cr N 1yN 1 5 (17) 199 num I= h(ciy +ciy +ci y +... den +cin y N +cin 1y N 1+ciNy N) or cr y + cr y +... I= h + cr N yn + cr N 1yN 1+ cr Ny N (0) The semi-open or semi-closed rules can be used on type of improper integral-that is, one with a lower limit of - or an upper limit of +. Such integrals usually can be evaluated by making a change of variable that transforms the infinite range to one that is finite (Choi et al., 00). The following identity serves this purpose and works for any function that decreases toward zero at least as fast 1/x as x approaches infinity: 1 1 f(x)dx f dw b 1/a = a 1/bw w (1)

J. Math. & Stat., 6 (): 19-0, 010 Table : Newton-cotes open integration formulas with points = 17. Rule Integer coefficient Real coefficient num = 16 Points (N) = 17 den = 19155865 Segments (N-1) = 16 ci = ci 16 = 70696 cr = cr 16 = +6.089590516086179989896117 Error»O (h 16 ) ci = ci 15 = -890878 cr = cr 15 = -.509601610566068661606101070919 ci = ci 1 = 1815066 cr = cr 1 = +151.606998765165999700196119661909 ci 5 = ci 1 = -57687658 cr 5 = cr 1 = -80.0185611060766658810766691871 ci 6 = ci 1 = 170561016 cr 6 = cr 1 = +11.6185605905508057091115 ci 7 = ci 11 = -9560801 cr 7 = cr 11 = -08.511005965097675856058109778081 ci 8 = ci 10 = 5581906 cr 8 = cr 10 = +97.0660179015586905061856818706 ci 9 = -99697109 cr 9 = -8.566797519997168108188787 num = 17 Points (N) = 18 den = 67686966000 Segments (N-1) = 17 ci = ci 17 = 5100 cr = cr 17 = +5.7760800801696597819896975786 Error»O (h 16 ) ci = ci 16 = 6555865 cr = cr 16 = -8.087656101199911757790188159 ci = ci 15 = -96065 cr = cr 15 = +10.78571195187765906566850791888 ci 5 = ci 1 = 107777585 cr 5 = cr 1 = -6.0196956567106706005810115988 ci 6 = ci 1 = -15680685 cr 6 = cr 1 = +675.977817595819859597660179 ci 7 = ci 1 = 6188669 cr 7 = cr 1 = -957.76175966990650655016601968 ci 8 = ci 11 = -000805 cr 8 = cr 11 = +90.0788609808018971879695195 ci 9 = ci 10 = 1018807605 cr 9 = cr 10 = -7.89618888511091976590695809 num = 9 Points (N) = 19 den = 95950000 Segments (N-1) = 18 ci = ci 18 = 69117119 cr = cr 18 = +6.5808775655975069991518609555 Error»O (h 18 ) ci = ci 17 = -087617 cr = cr 17 = -0.695119789087786888650155 ci = ci 16 = 7788759000 cr = cr 16 = +15.1187686767019810695081799769 ci 5 = ci 15 = -88510 cr 5 = cr 15 = -787.966787987516689956765915778 ci 6 = ci 1 = 1767615100 cr 6 = cr 1 = +188.600111655911550718189765155856 ci 7 = ci 1 = -98890768 cr 7 = cr 1 = -711.01706801601759589586678665 ci 8 = ci 1 = 855791775 cr 8 = cr 1 = +8050.89997500069678619716098805 ci 9 = ci 11 = -11696807750 cr 9 = cr 11 = -1106.8558166568608700515667 ci 10 = 19909700970 cr 10 = +169.110867099010601976600186 num = 19 Points (N) = 0 den = 607705780000 Segments (N-1) = 19 ci = ci 19 = 115611186619 cr = cr 19 = +6.78898556881859811061797789995 Error»O (h 18 ) ci = ci 18 = -11978990609715 cr = cr 18 = -6.19779175887806577616790698 ci = ci 17 = 5960706791516 cr = cr 17 = +176.888001919671898911017695 ci 5 = ci 16 = -1967199708100 cr 5 = cr 16 = -58.7998518055078115586777051071767 ci 6 = ci 15 = 79789610000 cr 6 = cr 15 = +1.161716997955575198187591601 ci 7 = ci 1 = -865005806 cr 7 = cr 1 = -56.57706101768069500718668866 ci 8 = ci 1 = 11195961688150 cr 8 = cr 1 = +88.9187011558801891111881080087 ci 9 = ci 1 = -108511851908 cr 9 = cr 1 = -01.087191086775691907809 ci 10 = ci 11 = 1757879000789850 cr 10 = cr 11 = +19.7169150687069898159096606 Points (N) = 1 den = 177556 Segments (N-1) = 0 ci = ci 0 = 1798150066 cr = cr 0 = +7.018665507807776190960780089 Error»O (h 0 ) ci = ci 19 = -18995060697 cr = cr 19 = -9.66560575658617769956679886866 ci = ci 18 = 778578168 cr = cr 18 = +9.79967975580661661115587995997771 ci 5 = ci 17 = -06705570086 cr 5 = cr 17 = -11.896165689589185581759961751 ci 6 = ci 16 = 96616911570 cr 6 = cr 16 = +87.51669585155177810510505950 ci 7 = ci 15 = -0015869858188 cr 7 = cr 15 = -961.1819588890696501855581 ci 8 = ci 1 = 7807757976 cr 8 = cr 1 = +19170.67976868509008188777171 ci 9 = ci 1 = -7751977518986 cr 9 = cr 1 = -107.09901101181689888699658169999 ci 10 = ci 1 = 1081599009718 cr 10 = cr 1 = +179.05675961885960876787778068 ci 11 = -11975077889170 cr 11 = -595.9110066651589857865577917516596 num = 7 Points (N) = den = 161690680000 Segments (N-1) = 1 ci = ci 1 = 1171567611 cr = cr 1 = +6.768990801960056710815069761 Error»O (h 0 ) ci = ci 0 = -8715095959975 cr = cr 0 = -.7859105977986959568009057180 ci = ci 19 = 819866509865 cr = cr 19 = +7.88117501769597787855859085 ci 5 = ci 18 = -18551896985 cr 5 = cr 18 = -9.7677006650068767060595957190 ci 6 = ci 17 = 5817596 cr 6 = cr 17 = +688.770755851919167189879618670 ci 7 = ci 16 = -118158507718 cr 7 = cr 16 = -586.51977866890861667995966188 ci 8 = ci 15 = 188589860806070 cr 8 = cr 15 = +967.16189057191591050660691 ci 9 = ci 1 = -65896587570 cr 9 = cr 1 = -1058.805989158879006106655985919519 00

J. Math. & Stat., 6 (): 19-0, 010 Table : Continued ci 10 = ci 1 = 0086558879170 cr 10 = cr 1 = +106.657718690789967090078 ci 11 = ci 1 = -81168710895 cr 11 = cr 1 = -170.056075898817669986589 num = 11 Points (N) = den = 160717698800000 Segments (N-1) = ci = ci = 960057000 cr = cr = +7.881078867819107767905196161867 Error»O (h ) ci = ci 1 = -798776165690 cr = cr 1 = -59.116078000780568599188910 ci = ci 0 = 8106117676170 cr = cr 0 = +88.86161806598108858188988156 ci 5 = ci 19 = -77009079870 cr 5 = cr 19 = -1808.915698161016769715700 ci 6 = ci 18 = 79806119169057115 cr 6 = cr 18 = +651.598179560101977151676006 ci 7 = ci 17 = -650177506907 cr 7 = cr 17 = -18160.6101997771116688500000787909 ci 8 = ci 16 = 511098718609610 cr 8 = cr 16 = +11.718861559117010899595765 ci 9 = ci 15 = -9566081075088960 cr 9 = cr 15 = -7701.0595658656198799977957759 ci 10 = ci 1 = 1891688518898910 cr 10 = cr 1 = +1008.8051505978007751055966 ci 11 = ci 1 = -19878579659900 cr 11 = cr 1 = -155895.99979168011917916791797700 ci 1 = 1016000191870708 cr 1 = +170017.0916778661691869807868518 Table 5: Newton-cotes semi-open integration formulas with points =, 6, 8, 10, 1, 1, 16 Rule Integer coefficient Real coefficient num = den = Points (N) = ci 1 = 1 cr 1 = +0.7500000000000000000000000000000000000000 Segments (N-1) = ci = 0 cr = +0.0000000000000000000000000000000000000000 Error»O (h ) ci = cr = +.500000000000000000000000000000000000000 den = 1 Points (N) = 6 ci 1 = 19 cr 1 = +0.6597 Segments (N-1) = 5 ci = -10 cr = -0.7 Error»O (h 5 ) ci = 10 cr = +.1666666666666666666666666666666666666667 ci = -70 cr = -.05555555555555555555555555555555555556 ci 5 = 85 cr 5 = +.951888888888888888888888888888888888889 num = 7 den = 860 Points (N) = 8 ci 1 = 751 cr 1 = +0.608907070707070707070707071 Segments (N-1) = 7 ci = -80 cr = -0.6805555555555555555555555555555555555556 Error»O (h 7 ) ci = 857 cr = +6.965777777777777777777777777777777778 ci = -1168 cr = -9.7070707070707070707070707 ci 5 = 167 cr 5 = +11.85868055555555555555555555555555555556 ci 6 = -7 cr 6 = -5.85777777777777777777777777777777777778 ci 7 = 17 cr 7 = +.5785879696969696969696969696 num = 9 den = 800 Points (N) = 10 ci 1 = 857 cr 1 = +0.5795089857185718571857185719 Segments (N-1) = 9 ci = -986 cr = -1.00165178571857185718571857186 Error»O (h 9 ) ci = 51966 cr = +10.9598185718571857185718571 ci = -110 cr = -.169017857185718571857185719 ci 5 = 18880 cr 5 = +6.79857185718571857185718571 ci 6 = -17710 cr 6 = -5.578567857185718571857185719 ci 7 = 19666 cr 7 = +6.0897185718571857185718571 ci 8 = -50886 cr 8 = -10.6985718571857185718571 ci 9 = 077 cr 9 = +.1690650000000000000000000000000000000 num = 11 den = 55600 Points (N) = 1 ci 1 = 17165 cr 1 = +0.5851080061981187567099867850 Segments (N-1) = 11 ci = -5199788 cr = -1.1511996619655085971860175191 Error»O (h 11 ) ci = 5809671 cr = +1.67571500617895061789506178951 ci = -166550 cr = -.0675706915806915806915807 ci 5 = 5995 cr 5 = +89.956699975975975975975 ci 6 = -98616 cr 6 = -1.757750091710758775009171076 ci 7 = 50955198 cr 7 = +18.6675878756617566175661756617566 ci 8 = -608996 cr 8 = -91.7195871915806915806915806916 ci 9 = 19175905 cr 9 = +8.00590795855791887150585579189 ci 1 = -618 cr 10 = -15.96651697691959116816905861 ci 11 = 186867 cr 11 = +.709880755797766019988106 num = 1 den = 0118067000 Points (N) = 1 ci 1 = 818190909 cr 1 = +0.587069671019588680960007157 Segments (N-1) = 1 ci = -50017078 cr = -1.6181780106777960896051917818 Error»O (h 1 ) ci = 06016516 cr = +19.609150858859005011779808 ci = -109197519010 cr = -70.56185676789898718605871058816179 ci 5 = 89011805 cr 5 = +18.1110698061510776900079859 01

J. Math. & Stat., 6 (): 19-0, 010 Table 5: Continued ci 6 = -51617108998 cr 6 = -.5950159558515787511966 ci 7 = 699851815616 cr 7 = +5.9969691718581580607707191 ci 8 = -70160088 cr 8 = -55.01980667517086070551180716569 ci 9 = 568975795 cr 9 = +6.897975680559797517086070551181 ci 10 = -00086079160 cr 10 = -19.911691186989196769910018 ci 11 = 180961096 cr 11 = +80.6715680199861517671018056590089 ci 1 = -781778 cr 1 = -1.696598669197866801010160079 ci 1 = 817679 cr 1 = +5.595667109796969068881086 Points (N) = 16 den = 0806 Segments (N-1) = 15 ci 1 = 5100 cr 1 = +0.5161899187780508050805080508 Error»O (h 15 ) ci = -1080 cr = -1.917017685759561880976166690507 ci = 17708175 cr = +5.165561886106895961181675679 ci = -7509110 cr = -109.015180181971105085907996765196 ci 5 = 1767055 cr 5 = +8.51806610179089660966096609660 ci 6 = -5178705700 cr 6 = -751.8701970886991970751711601 ci 7 = 876166155 cr 7 = +168.08951879876195809080976 ci 8 = -1110059500 cr 8 = -161.956580768659911517057197101 ci 9 = 111190805 cr 9 = +1656.769688701797197100888956 ci 10 = -89698960 cr 10 = -197.91085761907185761857190 ci 11 = 5891869 cr 11 = +787.567855961006850956578788699 ci 1 = -88051100 cr 1 = -61.0685956709670967096709670 ci 1 = 855691905 cr 1 = +1.6615650017770879880791656 ci 1 = -19700 cr 1 = -8.6007160917751010869656058 ci 15 = 9760105 cr 15 = +5.775187769150967780790679951 Table 6: Newton-cotes semi-open integration formulas with points = 18, 0, Rule Integer coefficient Real coefficient num = 17 Points (N) = 18 den = 18805108990000 Segments (N-1) = 17 ci 1 = 5597087657 cr 1 +0.991950059518171709010519105 Error»O (h 17 ) ci = -9169711 cr = -.1107811967715008617118795 ci = 8909009750 cr = +1.986105887805806676177065716918 ci = -17585886850150 cr = -158.7665680578085518969785171851 ci 5 = 616590780110 cr 5 = +57.51575187155560551957761006 ci 6 = -166651580618666 cr 6 = -150.57851967790011771959775 ci 7 = 671106879186 cr 7 = +09.7907901887176089687990 ci 8 = -565105791009850 cr 8 = -808.71577878600601689990895 ci 9 = 670798800 cr 9 = +6050.77897797161686050579968857 ci 10 = -679875188870 cr 10 = -608.6906769519580800160819100858 ci 11 = 570676601086 cr 11 = +899.68850116596157810891 ci 1 = -761618707586 cr 1 = -18.68671797698507156179118510715 ci 1 = 175509577105150 cr 1 = +158.8175109198555179007859097 ci 1 = -68185176090550 cr 1 = -615.569065800001601060595006 ci 15 = 00156065510 cr 15 = +180.68918069567161505665811768 ci 16 = -01057798110 cr 16 = -6.91975751997706890687611 ci 17 = 69509788515777 cr 17 = +6.757589555868769560169910599775 num = 19 Points (N) = 0 den = 68899695605760000 Segments (N-1) = 19 ci 1 = 69087615560 cr 1 = +0.87756565618899971185557107 Error»O (h 19 ) ci = -5970881156 cr = -.50091168668719907607550996191 ci = 5858991957589 cr = +8.780109519179897605776 ci = -1969796879096 cr = -1.11569518897559600086951 ci 5 = 1860589687851650180 cr 5 = +908.7076608078658915805576186750 ci 6 = -901650188676906 cr 6 = -756.806757165157879758006 ci 7 = 9187785675890958 cr 7 = +691.9558166789756758019518196 ci 8 = -17171507758019767 cr 8 = -11.098860617161019955779606189789 ci 9 = 590118085798 cr 9 = +1801.680670159168698801619 ci 10 = -18079577080898560 cr 10 = -7.968519179105755605998800 ci 11 = 195971009006581 cr 11 = +58.00051555088876501190058958 ci 1 = -671017556198888 cr 1 = -1856.9895501687086517158109701 ci 1 = 1761071018578085 cr 1 = +1.16551869890101798770109991 ci 1 = -9511015180006808 cr 1 = -671.5817189818000666010600889 ci 15 = 1501915115080 cr 15 = +91.665759019069066509970788855 0

J. Math. & Stat., 6 (): 19-0, 010 Table 6: Continued ci 16 = -18995891175971968 cr 16 = -981.7917761855678096859561188171 ci 17 = 559565181779191 cr 17 = +51.50988075068695080108879971780 ci 18 = -66579507657675 cr 18 = -.97670059770815166160579808898571 ci 19 = 9575999111081 cr 19 = +6.7665995198165091809505009 num = 7 Points (N) = den = 1988590800000 Segments (N-1) = 1 ci 1 = 107795767 cr 1 +0.7781157185567991079916955558 Error»O (h 1 ) ci = -59661807890 cr = -.7875169511758179019516756 ci = 9866786108780 cr = +5.99877591187808960065606697811 ci = -6650585186070 cr = -97.56018686105781657686856755910 ci 5 = 97689875665 cr 5 = +17.8759087157769806959159898 ci 6 = -1010177707778 cr 6 = -719.15176968177050877956780 ci 7 = 7170960559660 cr 7 = +169.00919088177981988059999 ci 8 = -5857789795569600 cr 8 = -765.767599515658088019555 ci 9 = 10070508855916590 cr 9 = +811.081157966700595979971190770 ci 10 = -196585769701860 cr 10 = -69816.869676988909567685580587 ci 11 = 180085001517756 cr 11 = +8108.709150891770767917586897 ci 1 = -1807101906578870 cr 1 = -8.060670551891058959619601 ci 1 = 15117906691155956690 cr 1 = +7066.180655716890759677179760 ci 1 = -10509857677110 cr 1 = -9098.7106689788711781900111590755 ci 15 = 605097167958500 cr 15 = +819.1576158558957751500615061809 ci 16 = -8917809509168 cr 16 = -1.5098906690060175950795666 ci 17 = 107108180859655 cr 17 = +500.766005690689709667099011969 ci 18 = -1896651788770 cr 18 = -187.51600556801085988171155690 ci 19 = 779090710880 cr 19 = +8.195016907656015977158 ci 0 = -11610600880590 cr 0 = -5.161919616919788675168 ci 1 = 155115196071877 cr 1 = +7.6800990056066877188061961986 Table 7: Set of points x 0.0 0.1 0. 0. 0. 0.5 y = f(x) 1.0000 1.105 1.1 1.99 1.918 1.687 For ab>0. Therefore, it can be used only when a is positive and b is or when a is - and b is negative. For cases where the limits are from - to, the integral can be implemented in three steps. For example: A A f (x)dx = f (x)dx + f (x)dx + f (x)dx () A Where, A is a positive number. One problem with using Eq. 1 to evaluate an integral is that the transformed function will be singular at one of the limits (Sun and Wu, 005). The semi-open or semi-closed integration formulas can be used to circumvent this dilemma as they allow evaluation of integral without employing data at the end points of integration interval. Analysis of error: Let s solve the example problem presented in Eq. using Simpson s 1/ rule (Sun and Wu, 008). The exact solution is I exact = e 0.8 -e 0.0 = 1.5509: A 0. I(h = 0.) = 1+ (1.91870) +.5509 = 1.571196 Error = 1.571196-1.5509 = 0.0001710 [ ] () Breaking the total range of integration into four increments of h = 0. and two intervals and applying the composite rule yields: 0. I(h = 0.) = [1 + (1.1076) + (1.91870) + (1.811880) +,55098] = 1.555177 Error = 1.555177-1.5509 = 0.0000108 () The global error of Simpson s 1/ rule is O (h ). Thus, for successive increment havings: 0.0001710 Ratio = = 15.775 0.0000108 O (h) Error(h) Ratio = = = = 16 15.775 Error(h / ) O (h/) () 0.8 x e dx 0 I = () Solving the problems for two increments of h = 0., the minimum permissible number of increments for Simpson s 1/ rule and one interval yields: Mixing rules: To estimate the error of integration in discrete function, we can apply different rules or mix several integration formulas of Newton-cotes (Berriochoa et al., 007). The difference in the result of each formula provides an approximation of the error. For example, calculate the integral of the points in Table 7. 0

J. Math. & Stat., 6 (): 19-0, 010 Using trapezoidal rule: REFERENCES 0.1 I = [1.0000 + (1.105) + (1.1) + (1.99) + (1.918) + 1.687] = 0.69 Using Simpson s 1/ and /8 rule: 0.1 I = [1.0000 + (1.105) + 1.1] + (0.1) [1.1 + (1.99) + (1.918) + 1.687] 8 = 0.687 Using closed rule with N = 6: 5(0.1) I = [19(1.0000) + 75(1.105) + 50(1.1) + 88 50.(1.99) + 75(1.918) + 19(1.687)] = 0.687 Using open rule with N = 6: 5(0.1) I = [0(1.0000) + 11(1.105) + 1.1 + 1.99 + 11(1.918) + 0(1.687)] = 0.687 (5) (6) (7) (8) Then, estimation of the error is 0.687-0.69 = 0.0006 and the best result is 0.687. CONCLUSION Newton-Cotes Integration is very important. To improve the error, formulas of high order should be applied. To estimate the error of integration for set of points proposed to apply different rules or mix various integration formulas. The difference in the result of each formula provides an approximation of the error of integration. And we can choose the best result as the average of all values or the value that appeared more times using the various formulas of integration. Computational routines to generate Newton-Cotes integration rules were presented with number of points until one hundred. The equations of Newton-Cotes shown in the article can be used in many scientific applications. ACKNOWLEDGMENT The researchers thank the generous support of the Pontificia Universidade Catolica de Minas Gerais- PUCMINAS. 0 Berriochoa, E., A. Cachafeiro and F. Marcellan, 007. A new numerical quadrature formula on the unit circle. Numer. Algorithms, : 91-01. DOI: 10.1007/s11075-007-911- Choi, S.H., J.H. Park and Y.Y. Park, 00. A study on the average case error of composite Newton-cotes quadratures. J. Applied Math. Comput., 1: 107-117. DOI: 10.1007/BF096185 Espelid, T.O., 00. Doubly adaptive quadrature routines based on Newton cotes rules. BIT Numer. Mathe., : 19-7. DOI: 10.10/A:10608770168 Kalogiratou, Z. and T.E. Simos, 00. Newton-cotes formulae for long-time integration. J. Comput. Applied Math., 158: 75-8. DOI: 10.1016/S077-07(0)0079-5 Sermutlu, E. and H.T. Eyyubog lu, 007. A new quadrature routine for improper and oscillatory integrals. Applied Math. Comput., 189: 5-61. DOI: 10.1016/j.amc.006.08.176 Sermutlu, E., 005. Comparison of Newton-cotes and Gaussian methods of quadrature. Applied Math. Compu., 171: 108-1057. DOI: 10.1016/j.amc.005.01.10 Simos, T.E., 008. High-order closed Newton-cotes trigonometrically-fitted formulae for long-time integration of orbital problems. Comput. Phys Commun., 178: 199-07. DOI: 10.1016/j.cpc.007.08.016 Simos, T.E., 009. High order closed Newton-cotes trigonometrically-fitted formulae for the numerical solution of the schrodinger equation. Applied Math. Comput., 09: 17-151. DOI: 10.1016/j.amc.008.06.00 Sun, W. and J. Wu, 005. Newton-cotes formulae for the numerical evaluation of certain hypersingular integrals. Computing, 75: 97-09. DOI: 10.1007/s00607-005-011-5 Sun, W. and J. Wu, 008. The superconvergence of Newton-cotes rules for the hadamard finite-part integral on an interval. Numer. Math., 109: 1-165. DOI: 10.1007/s0011-007-015-7 Witteveen, J.A.S., A. Loeven and H. Bijl, 009. An adaptive stochastic finite elements approach based on Newton-cotes quadrature in simplex elements. Comput. Fluids, 8: 170-188. DOI: 10.1016/j.compfluid.008.1.00 Zhang, X. J. Wu and D. Yu, 009. The superconvergence of composite Newton-cotes rules for hadamard finite-part integral on a circle. Computing, 85: 19-. DOI: 10.1007/s00607-009-008-5

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