Higher-Order Newton-Cotes Formulas

Transcription

1 Journal of Mathematics and Statistics 6 (): 19-0, 010 ISSN 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, , 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, , Belo Horizonte, Minas Gerais, Brazil Tel:

2 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 hy hy hy ( 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 = hy1 + 1.hy + 0.5hy 1.hy hy 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 = hy 1.hy hy (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 = hy hy hy hy + 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

3 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 = 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 = Error»O (h ) ci = ci = cr = cr = Boole s rule num = Points (N) = 5 den = 5 Segments (N-1) = ci 1 = ci 5 = 7 cr 1 = cr 5 = 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 = Error»O (h 6 ) ci = ci 5 = 75 cr = cr 5 = ci = ci = 50 cr = cr = num = 1 Points (N) = 7 den = 10 Segments (N-1) = 6 ci 1 = ci 7 = 1 cr 1 = cr 7 = Error»O (h 8 ) ci = ci 6 = 16 cr = cr 6 = ci = ci 5 = 7 cr = cr 5 = ci = 7 cr = num = 7 Points (N) = 8 den = 1780 Segments (N-1) = 7 ci 1 = ci 8 = 751 cr 1 = cr 8 = Error»O (h 8 ) ci = ci 7 = 577 cr = cr 7 = ci = ci 6 = 1 cr = cr 6 = ci = ci 5 = 989 cr = cr 5 = num = Points (N) = 9 den = 1175 Segments (N-1) = 8 ci 1 = ci 9 = 989 cr 1 = cr 9 = Error»O (h 10 ) ci = ci 8 = 5888 cr = cr 8 = ci = ci 7 = -98 cr = cr 7 = ci = ci 6 = 1096 cr = cr 6 = ci 5 = -50 cr 5 = num = 9 Points (N) = 10 den = Segments (N-1) = 9 ci 1 = ci 10 = 857 cr 1 = cr 10 = Error»O (h 10 ) ci = ci 9 = 1571 cr = cr 9 = ci = ci 8 = 1080 cr = cr 8 = ci = ci 7 = 19 cr = cr 7 = ci 5 = ci 6 = 5778 cr 5 = cr 6 = Points (N) = 11 den = 9976 Segments (N-1) = 10 ci 1 = ci 11 = cr 1 = cr 11 = Error»O (h 1 ) ci = ci 10 = cr = cr 10 = ci = ci 9 = -855 cr = cr 9 = ci = ci 8 = 700 cr = cr 8 = ci 5 = ci 7 = cr 5 = cr 7 = ci 6 = 768 cr 6 = num = 11 Points (N) = 1 den = Segments (N-1) = 11 ci 1 = ci 1 = cr 1 = cr 1 = Error»O (h 1 ) ci = ci 11 = cr = cr 11 = ci = ci 10 = -711 cr = cr 10 = ci = ci 9 = cr = cr 9 = ci 5 = ci 8 = cr 5 = cr 8 = ci 6 = ci 7 = cr 6 = cr 7 =

4 J. Math. & Stat., 6 (): 19-0, 010 Table 1: Continued num = 1 Points (N) = 1 den = Segments (N-1) = 1 ci 1 = ci 1 = cr 1 = cr 1 = Error»O (h 1 ) ci = ci 1 = cr = cr 1 = ci = ci 11 = cr = cr 11 = ci = ci 10 = cr = cr 10 = ci 5 = ci 9 = cr 5 = cr 9 = ci 6 = ci 8 = cr 6 = cr 8 = ci 7 = cr 7 = num = 1 Points (N) = 1 den = Segments (N-1) = 1 ci 1 = ci 1 = cr 1 = cr 1 = Error»O (h 1 ) ci = ci 1 = cr = cr 1 = ci = ci 1 = cr = cr 1 = ci = ci 11 = cr = cr 11 = ci 5 = ci 10 = cr 5 = cr 10 = ci 6 = ci 9 = cr 6 = cr 9 = ci 7 = ci 8 = cr 7 = cr 8 = Table : Newton-cotes closed integration formulas with points = 15.1 Rule Integer coefficient Real coefficient num = 7 Points (N) = 15 den = Segments (N-1) = 1 ci 1 = ci 15 = cr 1 = cr 15 = Error»O (h 16 ) ci = ci 1 = cr = cr 1 = ci = ci 1 = cr = cr 1 = ci = ci 1 = cr = cr 1 = ci 5 = ci 11 = cr 5 = cr 11 = ci 6 = ci 10 = cr 6 = cr 10 = ci 7 = ci 9 = cr 7 = cr 9 = ci 8 = cr 8 = Points (N) = 16 den = Segments (N-1) = 15 ci 1 = ci 16 = 5100 cr 1 = cr 16 = Error»O (h 16 ) ci = ci 15 = cr = cr 15 = ci = ci 1 = cr = cr 1 = ci = ci 1 = cr = cr 1 = ci 5 = ci 1 = cr 5 = cr 1 = ci 6 = ci 11 = cr 6 = cr 11 = ci 7 = ci 10 = cr 7 = cr 10 = ci 8 = ci 9 = cr 8 = cr 9 = num = 8 Points (N) = 17 den = Segments (N-1) = 16 ci 1 = ci 17 = cr 1 = cr 17 = Error»O (h 18 ) ci = ci 16 = cr = cr 16 = ci = ci 15 = cr = cr 15 = ci = ci 1 = cr = cr 1 = ci 5 = ci 1 = cr 5 = cr 1 = ci 6 = ci 1 = cr 6 = cr 1 = ci 7 = ci 11 = cr 7 = cr 11 = ci 8 = ci 10 = cr 8 = cr 10 = ci 9 = cr 9 = num = 17 Points (N) = 18 den = Segments (N-1) = 17 ci 1 = ci 18 = cr 1 = cr 18 = Error»O (h 18 ) ci = ci 17 = cr = cr 17 = ci = ci 16 = cr = cr 16 = ci = ci 15 = cr = cr 15 = ci 5 = ci 1 = cr 5 = cr 1 = ci 6 = ci 1 = cr 6 = cr 1 = ci 7 = ci 1 = cr 7 = cr 1 = ci 8 = ci 11 = cr 8 = cr 11 = ci 9 = ci 10 = cr 9 = cr 10 = num = Points (N) = 19 den =

5 J. Math. & Stat., 6 (): 19-0, 010 Table : Continued Segments (N-1) = 18 ci 1 = ci 19 = cr 1 = cr 19 = Error»O (h 0 ) ci = ci 18 = cr = cr 18 = ci = ci 17 = cr = cr 17 = ci = ci 16 = cr = cr 16 = ci 5 = ci 15 = cr 5 = cr 15 = ci 6 = ci 1 = cr 6 = cr 1 = ci 7 = ci 1 = cr 7 = cr 1 = ci 8 = ci 1 = cr 8 = cr 1 = ci 9 = ci 11 = cr 9 = cr 11 = ci 10 = cr 10 = num = 19 Points (N) = 0 den = Segments (N-1) = 19 ci 1 = ci 0 = cr 1 = cr 0 = Error»O (h 0 ) ci = ci 19 = cr = cr 19 = ci = ci 18 = cr = cr 18 = ci = ci 17 = cr = cr 17 = ci 5 = ci 16 = cr 5 = cr 16 = ci 6 = ci 15 = cr 6 = cr 15 = ci 7 = ci 1 = cr 7 = cr 1 = ci 8 = ci 11 = cr 8 = cr 1 = ci 9 = ci 1 = cr 9 = cr 1 = ci 10 = ci 11 = cr 10 = cr 11 = num = 1 Points (N) = 1 den = Segments (N-1) = 0 ci 1 = ci 1 = cr 1 = cr 1 = Error»O (h ) ci = ci 0 = cr = cr 0 = ci = ci 19 = cr = cr 19 = ci = ci 18 = cr = cr 18 = ci 5 = ci 17 = cr 5 = cr 17 = ci 6 = ci 16 = cr 6 = cr 16 = ci 7 = ci 15 = cr 7 = cr 15 = ci 8 = ci 1 = cr 8 = cr 1 = ci 9 = ci 1 = cr 9 = cr 1 = ci 10 = ci 1 = cr 10 = cr 1 = ci 11 = cr 11 = 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 = Error»O (h ) Points (N) = num = Segments (N-1) = den = Error»O (h ) ci = ci = 1 cr = cr = Points (N) = 5 num = den = Segments (N-1) = ci = ci = cr = cr = Error»O (h ) ci = -1 cr = -1. Points (N) = 6 den = Segments (N-1) = 5 ci = ci 5 = 11 cr = cr 5 = Error»O (h ) ci = ci = 1 cr = cr = Points (N) = 7 num = den = 10 Segments (N-1) = 6 ci = ci 6 = 11 cr = cr 6 = Error»O (h 6 ) ci = ci 5 = -1 cr = cr 5 = ci = 6 cr = Points (N) = 8 num = 7 den = 10 Segments (N-1) = 7 ci = ci 7 = 611 cr = cr 7 = Error»O (h 6 ) ci = ci 6 = -5 cr = cr 6 = ci = ci 6 = 56 cr = cr 5 = Points (N) = 9 num = 8 den = 95 Segments (N-1) = 8 ci = ci 8 = 60 cr = cr 8 = Error»O (h 8 ) ci = ci 7 = -95 cr = cr 7 = ci = ci 6 = 196 cr = cr 6 = ci 5 = -59 cr 5 =

6 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 = Error»O (h 8 ) ci = ci 8 = -80 cr = cr 8 = ci = ci 7 = 967 cr = cr 7 = ci 5 = ci 6 = cr 5 = cr 6 = Points (N) = 11 den = 56 Segments (N-1) = 10 ci = ci 10 = 05 cr = cr 10 = Error»O (h 10 ) ci = ci 9 = cr = cr 9 = ci = ci 8 = 0 cr = cr 8 = ci 5 = ci 7 = cr 5 = cr 7 = ci 6 = 678 cr 6 = num = 11 Points (N) = 1 den = Segments (N-1) = 11 ci = ci 11 = 7577 cr = cr 11 = Error»O (h 10 ) ci = ci 10 = cr = cr 10 = ci = ci 9 = cr = cr 9 = ci 5 = ci 8 = cr 5 = cr 8 = ci 6 = ci 7 = cr 6 = cr 7 = num = 1 Points (N) = 1 den = 195 Segments (N-1) = 1 ci = ci 1 = 966 cr = cr 1 = Error»O (h 1 ) ci = ci 11 = cr = cr 11 = ci = ci 10 = 1058 cr = cr 10 = ci 5 = ci 9 = cr 5 = cr 9 = ci 6 = ci 8 = 7956 cr 6 = cr 8 = ci 7 = -90 cr 7 = num = 1 Points (N) = 1 den = Segments (N-1) = 1 ci = ci 1 = 8897 cr = cr 1 = Error»O (h 1 ) ci = ci 1 = cr = cr 1 = ci = ci 11 = cr = cr 11 = ci 5 = ci 10 = cr 5 = cr 10 = ci 6 = ci 9 = cr 6 = cr 9 = ci 7 = ci 8 = cr 7 = cr 8 = num = 7 Points (N) = 15 den = Segments (N-1) = 1 ci = ci 1 = 9067 cr = cr 1 = Error»O (h 1 ) ci = ci 1 = cr = cr 1 = ci = ci 1 = cr = cr 1 = ci 5 = ci 11 = cr 5 = cr 11 = ci 6 = ci 10 = 1575 cr 6 = cr 10 = ci 7 = ci 9 = cr 7 = cr 9 = ci 8 = cr 8 = Points (N) = 16 den = 1700 Segments (N-1) = 15 ci = ci 15 = cr = cr 15 = Error»O (h 1 ) ci = ci 1 = cr = cr 1 = ci = ci 1 = cr = cr 1 = ci 5 = ci 1 = cr 5 = cr 1 = ci 6 = ci 11 = cr 6 = cr 11 = ci 7 = ci 10 = cr 7 = cr 10 = ci 8 = ci 9 = cr 8 = cr 9 = 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

7 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 = hy 1.hy hy (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 = hy1-0.7hy (16) hy hy hy 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)

8 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 = Segments (N-1) = 16 ci = ci 16 = cr = cr 16 = Error»O (h 16 ) ci = ci 15 = cr = cr 15 = ci = ci 1 = cr = cr 1 = ci 5 = ci 1 = cr 5 = cr 1 = ci 6 = ci 1 = cr 6 = cr 1 = ci 7 = ci 11 = cr 7 = cr 11 = ci 8 = ci 10 = cr 8 = cr 10 = ci 9 = cr 9 = num = 17 Points (N) = 18 den = Segments (N-1) = 17 ci = ci 17 = 5100 cr = cr 17 = Error»O (h 16 ) ci = ci 16 = cr = cr 16 = ci = ci 15 = cr = cr 15 = ci 5 = ci 1 = cr 5 = cr 1 = ci 6 = ci 1 = cr 6 = cr 1 = ci 7 = ci 1 = cr 7 = cr 1 = ci 8 = ci 11 = cr 8 = cr 11 = ci 9 = ci 10 = cr 9 = cr 10 = num = 9 Points (N) = 19 den = Segments (N-1) = 18 ci = ci 18 = cr = cr 18 = Error»O (h 18 ) ci = ci 17 = cr = cr 17 = ci = ci 16 = cr = cr 16 = ci 5 = ci 15 = cr 5 = cr 15 = ci 6 = ci 1 = cr 6 = cr 1 = ci 7 = ci 1 = cr 7 = cr 1 = ci 8 = ci 1 = cr 8 = cr 1 = ci 9 = ci 11 = cr 9 = cr 11 = ci 10 = cr 10 = num = 19 Points (N) = 0 den = Segments (N-1) = 19 ci = ci 19 = cr = cr 19 = Error»O (h 18 ) ci = ci 18 = cr = cr 18 = ci = ci 17 = cr = cr 17 = ci 5 = ci 16 = cr 5 = cr 16 = ci 6 = ci 15 = cr 6 = cr 15 = ci 7 = ci 1 = cr 7 = cr 1 = ci 8 = ci 1 = cr 8 = cr 1 = ci 9 = ci 1 = cr 9 = cr 1 = ci 10 = ci 11 = cr 10 = cr 11 = Points (N) = 1 den = Segments (N-1) = 0 ci = ci 0 = cr = cr 0 = Error»O (h 0 ) ci = ci 19 = cr = cr 19 = ci = ci 18 = cr = cr 18 = ci 5 = ci 17 = cr 5 = cr 17 = ci 6 = ci 16 = cr 6 = cr 16 = ci 7 = ci 15 = cr 7 = cr 15 = ci 8 = ci 1 = cr 8 = cr 1 = ci 9 = ci 1 = cr 9 = cr 1 = ci 10 = ci 1 = cr 10 = cr 1 = ci 11 = cr 11 = num = 7 Points (N) = den = Segments (N-1) = 1 ci = ci 1 = cr = cr 1 = Error»O (h 0 ) ci = ci 0 = cr = cr 0 = ci = ci 19 = cr = cr 19 = ci 5 = ci 18 = cr 5 = cr 18 = ci 6 = ci 17 = cr 6 = cr 17 = ci 7 = ci 16 = cr 7 = cr 16 = ci 8 = ci 15 = cr 8 = cr 15 = ci 9 = ci 1 = cr 9 = cr 1 =

9 J. Math. & Stat., 6 (): 19-0, 010 Table : Continued ci 10 = ci 1 = cr 10 = cr 1 = ci 11 = ci 1 = cr 11 = cr 1 = num = 11 Points (N) = den = Segments (N-1) = ci = ci = cr = cr = Error»O (h ) ci = ci 1 = cr = cr 1 = ci = ci 0 = cr = cr 0 = ci 5 = ci 19 = cr 5 = cr 19 = ci 6 = ci 18 = cr 6 = cr 18 = ci 7 = ci 17 = cr 7 = cr 17 = ci 8 = ci 16 = cr 8 = cr 16 = ci 9 = ci 15 = cr 9 = cr 15 = ci 10 = ci 1 = cr 10 = cr 1 = ci 11 = ci 1 = cr 11 = cr 1 = ci 1 = cr 1 = 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 = Segments (N-1) = ci = 0 cr = Error»O (h ) ci = cr = den = 1 Points (N) = 6 ci 1 = 19 cr 1 = Segments (N-1) = 5 ci = -10 cr = -0.7 Error»O (h 5 ) ci = 10 cr = ci = -70 cr = ci 5 = 85 cr 5 = num = 7 den = 860 Points (N) = 8 ci 1 = 751 cr 1 = Segments (N-1) = 7 ci = -80 cr = Error»O (h 7 ) ci = 857 cr = ci = cr = ci 5 = 167 cr 5 = ci 6 = -7 cr 6 = ci 7 = 17 cr 7 = num = 9 den = 800 Points (N) = 10 ci 1 = 857 cr 1 = Segments (N-1) = 9 ci = -986 cr = Error»O (h 9 ) ci = cr = ci = -110 cr = ci 5 = cr 5 = ci 6 = cr 6 = ci 7 = cr 7 = ci 8 = cr 8 = ci 9 = 077 cr 9 = num = 11 den = Points (N) = 1 ci 1 = cr 1 = Segments (N-1) = 11 ci = cr = Error»O (h 11 ) ci = cr = ci = cr = ci 5 = 5995 cr 5 = ci 6 = cr 6 = ci 7 = cr 7 = ci 8 = cr 8 = ci 9 = cr 9 = ci 1 = -618 cr 10 = ci 11 = cr 11 = num = 1 den = Points (N) = 1 ci 1 = cr 1 = Segments (N-1) = 1 ci = cr = Error»O (h 1 ) ci = cr = ci = cr = ci 5 = cr 5 =

10 J. Math. & Stat., 6 (): 19-0, 010 Table 5: Continued ci 6 = cr 6 = ci 7 = cr 7 = ci 8 = cr 8 = ci 9 = cr 9 = ci 10 = cr 10 = ci 11 = cr 11 = ci 1 = cr 1 = ci 1 = cr 1 = Points (N) = 16 den = 0806 Segments (N-1) = 15 ci 1 = 5100 cr 1 = Error»O (h 15 ) ci = cr = ci = cr = ci = cr = ci 5 = cr 5 = ci 6 = cr 6 = ci 7 = cr 7 = ci 8 = cr 8 = ci 9 = cr 9 = ci 10 = cr 10 = ci 11 = cr 11 = ci 1 = cr 1 = ci 1 = cr 1 = ci 1 = cr 1 = ci 15 = cr 15 = Table 6: Newton-cotes semi-open integration formulas with points = 18, 0, Rule Integer coefficient Real coefficient num = 17 Points (N) = 18 den = Segments (N-1) = 17 ci 1 = cr Error»O (h 17 ) ci = cr = ci = cr = ci = cr = ci 5 = cr 5 = ci 6 = cr 6 = ci 7 = cr 7 = ci 8 = cr 8 = ci 9 = cr 9 = ci 10 = cr 10 = ci 11 = cr 11 = ci 1 = cr 1 = ci 1 = cr 1 = ci 1 = cr 1 = ci 15 = cr 15 = ci 16 = cr 16 = ci 17 = cr 17 = num = 19 Points (N) = 0 den = Segments (N-1) = 19 ci 1 = cr 1 = Error»O (h 19 ) ci = cr = ci = cr = ci = cr = ci 5 = cr 5 = ci 6 = cr 6 = ci 7 = cr 7 = ci 8 = cr 8 = ci 9 = cr 9 = ci 10 = cr 10 = ci 11 = cr 11 = ci 1 = cr 1 = ci 1 = cr 1 = ci 1 = cr 1 = ci 15 = cr 15 =

11 J. Math. & Stat., 6 (): 19-0, 010 Table 6: Continued ci 16 = cr 16 = ci 17 = cr 17 = ci 18 = cr 18 = ci 19 = cr 19 = num = 7 Points (N) = den = Segments (N-1) = 1 ci 1 = cr Error»O (h 1 ) ci = cr = ci = cr = ci = cr = ci 5 = cr 5 = ci 6 = cr 6 = ci 7 = cr 7 = ci 8 = cr 8 = ci 9 = cr 9 = ci 10 = cr 10 = ci 11 = cr 11 = ci 1 = cr 1 = ci 1 = cr 1 = ci 1 = cr 1 = ci 15 = cr 15 = ci 16 = cr 16 = ci 17 = cr 17 = ci 18 = cr 18 = ci 19 = cr 19 = ci 0 = cr 0 = ci 1 = cr 1 = Table 7: Set of points x y = f(x) 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 = : A 0. I(h = 0.) = 1+ ( ) = Error = = [ ] () 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) + ( ) + ( ) +,55098] = Error = = () The global error of Simpson s 1/ rule is O (h ). Thus, for successive increment havings: Ratio = = O (h) Error(h) Ratio = = = = 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

12 J. Math. & Stat., 6 (): 19-0, 010 Using trapezoidal rule: REFERENCES 0.1 I = [ (1.105) + (1.1) + (1.99) + (1.918) ] = 0.69 Using Simpson s 1/ and /8 rule: 0.1 I = [ (1.105) + 1.1] + (0.1) [1.1 + (1.99) + (1.918) ] 8 = Using closed rule with N = 6: 5(0.1) I = [19(1.0000) + 75(1.105) + 50(1.1) (1.99) + 75(1.918) + 19(1.687)] = Using open rule with N = 6: 5(0.1) I = [0(1.0000) + 11(1.105) (1.918) + 0(1.687)] = (5) (6) (7) (8) Then, estimation of the error is = and the best result is 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, : DOI: /s 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: DOI: /BF Espelid, T.O., 00. Doubly adaptive quadrature routines based on Newton cotes rules. BIT Numer. Mathe., : DOI: 10.10/A: Kalogiratou, Z. and T.E. Simos, 00. Newton-cotes formulae for long-time integration. J. Comput. Applied Math., 158: DOI: /S077-07(0) Sermutlu, E. and H.T. Eyyubog lu, 007. A new quadrature routine for improper and oscillatory integrals. Applied Math. Comput., 189: DOI: /j.amc Sermutlu, E., 005. Comparison of Newton-cotes and Gaussian methods of quadrature. Applied Math. Compu., 171: DOI: /j.amc Simos, T.E., 008. High-order closed Newton-cotes trigonometrically-fitted formulae for long-time integration of orbital problems. Comput. Phys Commun., 178: DOI: /j.cpc Simos, T.E., 009. High order closed Newton-cotes trigonometrically-fitted formulae for the numerical solution of the schrodinger equation. Applied Math. Comput., 09: DOI: /j.amc Sun, W. and J. Wu, 005. Newton-cotes formulae for the numerical evaluation of certain hypersingular integrals. Computing, 75: DOI: /s Sun, W. and J. Wu, 008. The superconvergence of Newton-cotes rules for the hadamard finite-part integral on an interval. Numer. Math., 109: DOI: /s 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: DOI: /j.compfluid 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: /s