Midpoint Derivative-Based Closed Newton-Cotes Quadrature

Abstract

A novel family of numerical integration of closed Newton-Cotes quadrature rules is presented which uses the derivative value at the midpoint. It is proved that these kinds of quadrature rules obtain an increase of two orders of precision over the classical closed Newton-Cotes formula, and the error terms are given. The computational cost for these methods is analyzed from the numerical point of view, and it has shown that the proposed formulas are superior computationally to the same order closed Newton-Cotes formula when they reduce the error below the same level. Finally, some numerical examples show the numerical superiority of the proposed approach with respect to closed Newton-Cotes formulas.

1. Introduction

Definite integration is one of the most important and basic concepts in mathematics. It has numerous applications in fields such as physics and engineering. In several practical problems, we need to calculate integrals. As is known to all, as for 𝐼=∫^𝑏_𝑎𝑓(𝑥)𝑑𝑥, once the primitive function 𝐹(𝑥) of integrand 𝑓(𝑥) is known, the definite integral of 𝑓(𝑥) over the interval [𝑎,𝑏] is given by Newton-Leibniz formula, that is, ^𝑏_𝑎𝑓(𝑥)𝑑𝑥=𝐹(𝑏)−𝐹(𝑎). (1)

However, the explicit primitive function 𝐹(𝑥) is not available or its primitive function is not easy to obtain, such as 𝑒^±𝑥2, sin𝑥², and sin𝑥/𝑥. Moreover, some of the integrand 𝑓(𝑥) is only available at certain points 𝑥_𝑖, 𝑖=0,1,…,𝑛. It is often the case that the values of 𝑓(𝑥_𝑖) come from experimental data, such as sampling [1]. But the need often arises for calculating the definite integral. And how to get high-precision numerical integration formulas becomes one of the challenges in fields of mathematics [2].

The methods of quadrature are usually based on the interpolation polynomials and can be written in the following form: ^𝑏_𝑎𝑓(𝑥)𝑑𝑥≈^𝑛_𝑖=0𝑤_𝑖𝑓𝑥_𝑖, (2) where there are 𝑛+1 distinct integration points at 𝑥₀,𝑥₁,…,𝑥_𝑛 within the interval [𝑎,𝑏] and 𝑛+1 weights 𝑤_𝑖,  𝑖=0,1,…,𝑛. If the integration points are uniformly distributed over the interval, so 𝑥_𝑖=𝑥₀+𝑖ℎ in which ℎ=(𝑏−𝑎)/𝑛.

These 𝑤_𝑖 can be derived in several different ways [3–5]. One method is to interpolate 𝑓(𝑥) at the 𝑛+1 points 𝑥₀,𝑥₁,…,𝑥_𝑛, using the Lagrange polynomials and then integrating the foresaid polynomials to obtain (2).

The other method is based on the precision of a quadrature formula. Select the 𝑤_𝑖, 𝑖=0,1,…,𝑛, so that the error 𝑅_𝑛𝑓=^𝑏_𝑎𝑓(𝑥)𝑑𝑥−^𝑛_𝑖=0𝑤_𝑖𝑓𝑥_𝑖, (3) is exactly zero for 𝑓(𝑥)=𝑥^𝑗, 𝑗=0,1,…,𝑛. Using the method of undetermined coefficients, this approach generates a system of 𝑛+1 linear equations for weights 𝑤_𝑖. Since the monomials 1,𝑥,…,𝑥^𝑛 are linearly independent, the linear system of equations has a unique solution.

The Newton-Cotes formulas are the most well-known numerical integration rules of this type. There are several subclasses of Newton-Cotes formulas that depend on the integer value of 𝑛. We list some of them as follows. Trapezoidal rule (𝑛=1)^𝑏_𝑎𝑓(𝑥)𝑑𝑥=𝑏−𝑎2𝑓(𝑎)(𝑏)−()+𝑓𝑏−𝑎³𝑓12^′′(𝜉), (4)where 𝜉∈(𝑎,𝑏). Simpson’s rule (𝑛=2)^𝑏_𝑎𝑓(𝑥)𝑑𝑥=𝑏−𝑎6𝑓(𝑎)+4𝑓𝑎+𝑏2(𝑏−()+𝑓)𝑏−𝑎⁵𝑓2880⁽⁴⁾(𝜉), (5)where 𝜉∈(𝑎,𝑏). Simpson’s 3/8 rule (𝑛=3)^𝑏_𝑎𝑓(𝑥)=()𝑑𝑥𝑏−𝑎8𝑓(𝑎)+3𝑓2𝑎+𝑏3+3𝑓𝑎+2𝑏3(𝑏−()+𝑓)𝑏−𝑎⁵𝑓6480⁽⁴⁾(𝜉), (6)where 𝜉∈(𝑎,𝑏). Bool’s rule (𝑛=4)^𝑏_𝑎𝑓(𝑥)()𝑑𝑥=𝑏−𝑎(𝑎)907𝑓+32𝑓3𝑎+𝑏4+12𝑓𝑎+𝑏2+32𝑓𝑎+3𝑏4(𝑏−()+7𝑓)𝑏−𝑎⁷𝑓1935360⁽⁶⁾(𝜉), (7)where 𝜉∈(𝑎,𝑏).

Note that when 𝑛 is an even integer, the degree of precision is 𝑛+1. When 𝑛 is odd, the degree of precision is only 𝑛 [1, 2].

In spite of the many accurate and efficient methods for numerical integration being available in [3–5], recently Dehghan et al. [6] improved the precision degree of closed Newton-Cotes quadrature by including the location of boundaries of the interval as two additional variables and rescaling the original integral to fit the optimal boundary locations. In their following work, they have applied this method to Gauss-Legendre quadrature [7], Gauss-Chebyshev quadrature [8], and open Newton-Cotes quadrature [9]. These formulas increase the order of accuracy of standard numerical integration by two orders. They use the method of undermined coefficients to set up nonlinear equations for parameters, which are solved approximately by using a computer algebra system. Burg has proposed derivative-based closed Newton-Cotes numerical quadrature [10], which uses the function values on uniformly spaced intervals and 2 derivative values at the endpoints. The precision of the method in [10] is higher than the standard closed Newton-Cotes quadrature.

The motivation for this research lies in construction of midpoint derivative-based closed Newton-Cotes numerical quadrature rule for Newton-Cotes quadrature which uses the derivative value at the midpoint only. These new schemes are given in Section 2. In Section 3, the error terms are presented. In Section 4, compared with the Newton-Cotes quadrature, computational costs of these methods and run time on a given processor are presented, where the minimum number of subinterval to achieve the same level is calculated along with the number of function and derivative evaluations. The numerical experiments results are shown in Section 5. Finally, conclusions are drawn in Section 6.

2. Midpoint Derivative-Based Closed Newton-Cotes Quadrature

In this section, by adding the high derivative at the midpoint, schemes with higher precision than the Newton-Cotes quadrature rules are presented.

Theorem 1. Midpoint derivative-based closed Trapezoidal rule (𝑛=1) is ^𝑏_𝑎𝑓(𝑥)𝑑𝑥≈𝑇=𝑏−𝑎2𝑓(𝑎)(𝑏)−()+𝑓𝑏−𝑎³𝑓12^′′𝑎+𝑏2. (8) The precision of this method is 3.

Proof. Since the Trapezoidal rule has degree of precision 1, the formula (8) at least has 1 precision degree. Now, we just need to verify that the quadrature formula (8) is exact for 𝑓(𝑥)=𝑥²,𝑥³.
When 𝑓(𝑥)=𝑥², ∫^𝑏_𝑎𝑥²𝑑𝑥=(1/3)(𝑏³−𝑎³); 𝑇=((𝑏−𝑎)/2)(𝑎²+𝑏²)−2(𝑏−𝑎)³/12=(1/3)(𝑏³−𝑎³).
When 𝑓(𝑥)=𝑥³, ∫^𝑏_𝑎𝑥³𝑑𝑥=(1/4)(𝑏⁴−𝑎⁴); 𝑇=𝑏−𝑎2𝑎³+𝑏³−6(𝑏−𝑎)³12𝑎+𝑏2=14𝑏²−𝑎²2𝑏²+2𝑎²−2𝑎𝑏−𝑏²−𝑎²=1+2𝑎𝑏4𝑏²−𝑎²𝑏²+𝑎²=14𝑏⁴−𝑎⁴. (9) So the precision of midpoint derivative-based closed Trapezoidal rule is 3.

Theorem 2. Midpoint derivative-based closed Simpson’s rule (𝑛=2) is ^𝑏_𝑎𝑓(𝑥)𝑑𝑥≈𝑆=𝑏−𝑎6𝑓(𝑎)+4𝑓𝑎+𝑏2(𝑏−()+𝑓)𝑏−𝑎⁵𝑓2880⁽⁴⁾𝑎+𝑏2. (10) The precision of this method is 5.

Proof. Since the Simpson’s rule has degree of precision 3, the formula (10) at least has 3 precision degree. Now, we just need to verify that the quadrature formula (10) is exact for 𝑓(𝑥)=𝑥⁴,𝑥⁵.
When 𝑓(𝑥)=𝑥⁴, ∫^𝑏_𝑎𝑥⁴𝑑𝑥=(1/5)(𝑏⁵−𝑎⁵); 𝑆=𝑏−𝑎6𝑎⁴+4𝑎+𝑏2⁴+𝑏⁴−24(𝑏−𝑎)⁵=2880𝑏−𝑎12020𝑎⁴+5(𝑎+𝑏)⁴+20𝑏⁴−(𝑏−𝑎)⁴=15(𝑏−𝑎)𝑏⁴+𝑏³𝑎+𝑏²𝑎²+𝑏𝑎³+𝑎⁴=15𝑏⁵−𝑎⁵. (11)
When 𝑓(𝑥)=𝑥⁵, ∫^𝑏_𝑎𝑥⁵𝑑𝑥=(1/6)(𝑏⁶−𝑎⁶); 𝑆=𝑏−𝑎6𝑎⁵+4𝑎+𝑏2⁵+𝑏⁵−60(𝑏−𝑎)⁵(𝑎+𝑏)=2880𝑏−𝑎488𝑎⁵+(𝑎+𝑏)⁵+8𝑏⁵−(𝑎+𝑏)(𝑏−𝑎)⁴=16(𝑏−𝑎)𝑏⁵+𝑏⁴𝑎+𝑏³𝑎²+𝑏²𝑎³+𝑏𝑎⁴+𝑎⁵=16𝑏⁶−𝑎⁶. (12) So the precision of midpoint derivative-based closed Simpson’s rule is 5.

Similarly, we obtain the midpoint derivative-based closed Simpson’s 3/8 rule and Bool’s rule.

Theorem 3. Midpoint derivative-based closed Simpson’s 3/8 rule (𝑛=3) is ^𝑏_𝑎𝑓(𝑥)𝑑𝑥≈𝑆_3/8=(𝑏−𝑎)8𝑓(𝑎)+3𝑓2𝑎+𝑏3+3𝑓𝑎+2𝑏3(𝑏−()+𝑓)𝑏−𝑎⁵𝑓6480⁽⁴⁾𝑎+𝑏2. (13) The precision of this method is 5.

Proof. Since the Simpson’s 3/8 rule has degree of precision 3, the formula (13) at least has 3 precision degree. Similarly, we just need to verify that the quadrature formula (13) is exact for 𝑓(𝑥)=𝑥⁴,𝑥⁵.
When 𝑓(𝑥)=𝑥⁴, ∫^𝑏_𝑎𝑥⁴𝑑𝑥=(1/5)(𝑏⁵−𝑎⁵); 𝑆=𝑏−𝑎8𝑎⁴+32𝑎+𝑏3⁴+3𝑎+2𝑏3⁴+𝑏⁴−24(𝑏−𝑎)⁵=6480𝑏−𝑎21627𝑎⁴+(2𝑎+𝑏)⁴+(𝑎+2𝑏)⁴+27𝑏⁴−(𝑏−𝑎)⁵=270𝑏−𝑎1080135𝑎⁴+5(2𝑎+𝑏)⁴+5(𝑎+2𝑏)⁴+135𝑏⁴−4(𝑏−𝑎)⁴=216()𝑏1080𝑏−𝑎⁴+𝑏³𝑎+𝑏²𝑎²+𝑏𝑎³+𝑎⁴=15𝑏⁵−𝑎⁵. (14)
When 𝑓(𝑥)=𝑥⁵, ∫^𝑏_𝑎𝑥⁵𝑑𝑥=(1/6)(𝑏⁶−𝑎⁶); 𝑆=𝑏−𝑎8𝑎⁴+32𝑎+𝑏3⁴+3𝑎+2𝑏3⁴+𝑏⁴−60(𝑏−𝑎)⁵(𝑎+𝑏)=6480𝑏−𝑎21627𝑎⁵+(2𝑎+𝑏)⁴+(𝑎+2𝑏)⁴+27𝑏⁵−(𝑏−𝑎)⁵(𝑎+𝑏)=108𝑏−𝑎21627𝑎⁵+(2𝑎+𝑏)⁴+(𝑎+2𝑏)⁴+27𝑏⁵−2(𝑎+𝑏)(𝑏−𝑎)⁴=36(𝑏−𝑎)𝑏216⁵+𝑏⁴𝑎+𝑏³𝑎²+𝑏²𝑎³+𝑏𝑎⁴+𝑎⁵=16𝑏⁶−𝑎⁶. (15)

Theorem 4. Midpoint derivative-based closed Bool’s rule (𝑛=4) is ^𝑏_𝑎𝑓(𝑥)()𝑑𝑥≈𝐵=𝑏−𝑎(𝑎)907𝑓+32𝑓3𝑎+𝑏4+12𝑓𝑎+𝑏2+32𝑓𝑎+3𝑏4(𝑏−()+7𝑓)𝑏−𝑎⁷𝑓1935360⁽⁶⁾𝑎+𝑏2. (16) The precision of this method is 7.

Proof. Since the Bool’s rule has degree of precision 5, the formula (16) at least has 5 precision degree. We just need to verify that the quadrature formula (16) is exact for 𝑓(𝑥)=𝑥⁶,𝑥⁷ in like manner.
When 𝑓(𝑥)=𝑥⁶, ∫^𝑏_𝑎𝑥⁶𝑑𝑥=(1/7)(𝑏⁷−𝑎⁷); 𝑆=𝑏−𝑎907𝑎⁶+323𝑎+𝑏4⁶+12𝑎+𝑏2⁶+32𝑎+3𝑏4⁶+7𝑏⁶−720(𝑏−𝑎)⁷=1935360𝑏−𝑎11520896𝑎⁶+(3𝑎+𝑏)⁶+24(𝑎+𝑏)⁶+(𝑎+3𝑏)⁶+896𝑏⁶−(𝑏−𝑎)⁷=2688𝑏−𝑎806406272𝑎⁶+7(3𝑎+𝑏)⁶+168(𝑎+𝑏)⁶+7(𝑎+3𝑏)⁶+6272𝑏⁶−30(𝑏−𝑎)⁶=11520()×𝑏80640𝑏−𝑎₆+𝑏⁵𝑎+𝑏⁴𝑎²+𝑏³𝑎³+𝑏²𝑎⁴+𝑏𝑎⁵+𝑎⁶=17𝑏⁷−𝑎⁷. (17)
When 𝑓(𝑥)=𝑥⁷, ∫^𝑏_𝑎𝑥⁷𝑑𝑥=(1/8)(𝑏⁸−𝑎⁸); 𝑆=𝑏−𝑎907𝑎⁷+323𝑎+𝑏4⁷+12𝑎+𝑏2⁷+32𝑎+3𝑏4⁷+7𝑏⁷−5040(𝑏−𝑎)⁷(𝑎+𝑏)=1935360𝑏−𝑎460803584𝑎⁷+(3𝑎+𝑏)⁷+48(𝑎+𝑏)⁷+(𝑎+3𝑏)⁷+3584𝑏⁷−(𝑏−𝑎)⁷(𝑎+𝑏)=768𝑏−𝑎460803584𝑎⁷+(3𝑎+𝑏)⁷+48(𝑎+𝑏)⁷+(𝑎+3𝑏)⁷+3584𝑏⁷−60(𝑏−𝑎)⁶(=()𝑎+𝑏)5760𝑏−𝑎×𝑏46080₇+𝑏⁶𝑎+𝑏⁵𝑎²+𝑏⁴𝑎³+𝑏³𝑎⁴+𝑏²𝑎⁵+𝑏𝑎⁶+𝑎⁷=18𝑏⁸−𝑎⁸. (18)

3. The Error Terms of Midpoint Derivative-Based Closed Newton-Cotes Quadrature

In this section, the error terms of midpoint derivative-based closed Newton-Cotes quadrature are given. The error term can be given in mainly 3 different ways [5, 10]. Here, we use the concept of precision to calculate the error term, where the error term is related to the difference between the quadrature formula for the monomial 𝑥^𝑝+1/(𝑝+1)! and the exact value (1/(𝑝+1)!)∫^𝑏_𝑎𝑥^𝑝+1𝑑𝑥=(𝑏^𝑝+2−𝑎^𝑝+2)/(𝑝+2)!, where 𝑝 is the precision of the quadrature formula.

Theorem 5. Midpoint derivative-based closed Trapezoidal rule (𝑛=1) with the error term is ^𝑏_𝑎𝑓(𝑥)𝑑𝑥=𝑏−𝑎2𝑓(𝑎)(𝑏)−()+𝑓𝑏−𝑎³𝑓12^′′𝑎+𝑏2−(𝑏−𝑎)⁵𝑓480⁽⁴⁾(𝜉), (19) where 𝜉∈(𝑎,𝑏). Thus, this scheme is fifth order accurate with the error term 𝑅₁[𝑓]=−((𝑏−𝑎)⁵/480)𝑓⁽⁴⁾(𝜉), and the associate composite method is fourth order.

Proof. Let 𝑓(𝑥)=𝑥⁴/4!. So (1/4!)∫^𝑏_𝑎𝑥⁴𝑑𝑥=(1/120)(𝑏⁵−𝑎⁵), 𝑏−𝑎2𝑓(𝑎)(𝑏)−()+𝑓𝑏−𝑎³𝑓12^′′𝑎+𝑏2=𝑏−𝑎2𝑎⁴+𝑏4!⁴−()4!𝑏−𝑎³24𝑎+𝑏2²=𝑏−𝑎𝑏96²+𝑎²². (20) Therefore, 1𝑏120⁵−𝑎⁵−𝑏−𝑎𝑏96²+𝑎²²=(𝑏−𝑎)480−𝑏⁴+4𝑎𝑏³−6𝑎²𝑏²+4𝑎³𝑏−𝑎⁴()=−𝑏−𝑎()480𝑏−𝑎⁴()=−𝑏−𝑎⁵.480 (21) This implies that 𝑅₁𝑓()=−𝑏−𝑎⁵𝑓480⁽⁴⁾(𝜉). (22)

Theorem 6. Midpoint derivative-based closed Simpson’s rule (𝑛=2) with the error term is ^𝑏_𝑎𝑓(𝑥)𝑑𝑥=𝑏−𝑎6𝑓(𝑎)+4𝑓𝑎+𝑏2(𝑏−()+𝑓)𝑏−𝑎⁵𝑓2880⁽⁴⁾𝑎+𝑏2−(𝑏−𝑎)⁷𝑓241920⁽⁶⁾(𝜉), (23) where 𝜉∈(𝑎,𝑏). Thus, this scheme is seventh order accurate with the error term 𝑅₂[𝑓]=−((𝑏−𝑎)⁷/241920)𝑓⁽⁶⁾(𝜉), and the associate composite method is sixth order.

Proof. Let 𝑓(𝑥)=𝑥⁶/6!. So (1/6!)∫^𝑏_𝑎𝑥⁶𝑑𝑥=(1/5040)(𝑏⁷−𝑎⁷), 𝑏−𝑎6𝑓(𝑎)+4𝑓𝑎+𝑏2(𝑏−()+𝑓)𝑏−𝑎⁵𝑓2880⁽⁴⁾𝑎+𝑏2=𝑏−𝑎6𝑎⁶()6!+4𝑎+𝑏⁶2⁶+𝑏⋅6!⁶−()6!𝑏−𝑎⁵5760𝑎+𝑏2²=(𝑏−𝑎)×345607𝑏⁶+6𝑎𝑏⁵+9𝑎²𝑏⁵+4𝑎³𝑏³+9𝑎⁴𝑏²+6𝑎⁵𝑏+7𝑎⁶. (24) Therefore,1𝑏5040⁷−𝑎⁷−(𝑏−𝑎)×345607𝑏⁶+6𝑎𝑏⁵+9𝑎²𝑏⁵+4𝑎³𝑏³+9𝑎⁴𝑏²+6𝑎⁵𝑏+7𝑎⁶=48(𝑏−𝑎)×𝑏241920₆+𝑏⁵𝑎+𝑏⁴𝑎²+𝑏³𝑎³+𝑏²𝑎⁴+𝑏𝑎⁵+𝑎⁶−7(𝑏−𝑎)×2419207𝑏⁶+6𝑎𝑏⁵+9𝑎²𝑏⁵+4𝑎³𝑏³+9𝑎⁴𝑏²+6𝑎⁵𝑏+7𝑎⁶=(𝑏−𝑎)×241920−𝑏⁶+6𝑏⁵𝑎−15𝑏⁴𝑎²+20𝑏³𝑎³−15𝑏²𝑎⁴+6𝑏𝑎⁵−𝑎⁶()=−𝑏−𝑎⁷.241920 (25)This implies that 𝑅₂𝑓()=−𝑏−𝑎⁷𝑓241920⁽⁶⁾(𝜉). (26)

Theorem 7. Midpoint derivative-based closed Simpson’s 3/8 rule (𝑛=3) with the error term is ^𝑏_𝑎𝑓(𝑥)=()𝑑𝑥𝑏−𝑎8𝑓(𝑎)+3𝑓2𝑎+𝑏3+3𝑓𝑎+2𝑏3(𝑏−()+𝑓)𝑏−𝑎⁵𝑓6480⁽⁴⁾𝑎+𝑏2−23(𝑏−𝑎)⁷𝑓9797760⁽⁶⁾(𝜉), (27) where 𝜉∈(𝑎,𝑏). It has a seventh order leading order error term and is sixth order accurate in its composite form. And the error term of this method is 𝑅₃[𝑓]=−(23(𝑏−𝑎)⁷/9797760)𝑓⁽⁶⁾(𝜉).

Proof. Let 𝑓(𝑥)=𝑥⁶/6!. So (1/6!)∫^𝑏_𝑎𝑥⁶𝑑𝑥=(1/5040)(𝑏⁷−𝑎⁷), 𝑏−𝑎8𝑓(𝑎)+3𝑓2𝑎+𝑏3+3𝑓𝑎+2𝑏3(𝑏−()+𝑓)𝑏−𝑎⁵𝑓6480⁽⁴⁾𝑎+𝑏2=𝑏−𝑎8𝑎⁶()6!+32𝑎+𝑏⁶3⁶()⋅6!+3𝑎+2𝑏⁶3⁶+𝑏⋅6!⁶−()6!𝑏−𝑎⁵12960𝑎+𝑏2²=(𝑏−𝑎)97977601967𝑏⁶+1806𝑎𝑏⁵+2289𝑎²𝑏⁵+1484𝑎³𝑏³+2289𝑎⁴𝑏²+1806𝑎⁵𝑏+1967𝑎⁶. (28) Therefore, 1𝑏5040⁷−𝑎⁷−(𝑏−𝑎)×97977601967𝑏⁶+1806𝑎𝑏⁵+2289𝑎²𝑏⁵+1484𝑎³𝑏³+2289𝑎⁴𝑏²+1806𝑎⁵𝑏+1967𝑎⁶=1944(𝑏−𝑎)×𝑏9797760₆+𝑏⁵𝑎+𝑏⁴𝑎²+𝑏³𝑎³+𝑏²𝑎⁴+𝑏𝑎⁵+𝑎⁶−(𝑏−𝑎)97977601967𝑏⁶+1806𝑎𝑏⁵+2289𝑎²𝑏⁵+1484𝑎³𝑏³+2289𝑎⁴𝑏²+1806𝑎⁵𝑏+1967𝑎⁶=23(𝑏−𝑎)9797760−𝑏⁶+6𝑏⁵𝑎−15𝑏⁴𝑎²+20𝑏³𝑎³−15𝑏²𝑎⁴+6𝑏𝑎⁵−𝑎⁶()=−23𝑏−𝑎⁷.9797760 (29) This implies that 𝑅₃𝑓()=−23𝑏−𝑎⁷𝑓9797760⁽⁶⁾(𝜉). (30)

Theorem 8. Midpoint derivative-based closed Bool’s rule (𝑛=4) with the error term is ^𝑏_𝑎𝑓(𝑥)=()𝑑𝑥𝑏−𝑎(𝑎)907𝑓+32𝑓3𝑎+𝑏4+12𝑓𝑎+𝑏2+32𝑓𝑎+3𝑏4(𝑏−()+7𝑓)𝑏−𝑎⁷𝑓1935360⁽⁶⁾𝑎+𝑏2−17(𝑏−𝑎)⁹45⋅2¹¹𝑓⋅8!⁽⁸⁾(𝜉), (31) where 𝜉∈(𝑎,𝑏). It has a ninth order leading order error term and is eighth order accurate in its composite form. And the error term of this method is 𝑅₄[𝑓]=−(17(𝑏−𝑎)⁹/45⋅2¹¹⋅8!)𝑓⁽⁸⁾(𝜉).

Proof. Similarly, let 𝑓(𝑥)=𝑥⁸/8!. So (1/8!)∫^𝑏_𝑎𝑥⁸𝑑𝑥=(1/362880)(𝑏⁹−𝑎⁹), (𝑏−𝑎)(𝑎)907𝑓+32𝑓3𝑎+𝑏4+12𝑓𝑎+𝑏2+32𝑓𝑎+3𝑏4(𝑏−()+7𝑓)𝑏−𝑎⁷𝑓1935360⁽⁶⁾𝑎+𝑏2=𝑏−𝑎907𝑎⁸()8!+323𝑎+𝑏⁸4⁸()⋅8!+12𝑎+𝑏⁸2⁸()⋅8!+32𝑎+3𝑏⁸4⁸+⋅8!7𝑏⁸−()8!𝑏−𝑎⁷3870720𝑎+𝑏2²=(𝑏−𝑎)45⋅2¹¹⋅8!10257𝑏⁸+10104𝑎𝑏⁷+10716𝑎²𝑏⁶+9288𝑎³𝑏⁵+11430𝑎⁴𝑏⁴+9288𝑎⁵𝑏³+10716𝑎⁶𝑏²+10104𝑎⁷𝑏+10257𝑎⁸. (32) Therefore, 1𝑏362880⁹−𝑎⁹−(𝑏−𝑎)45⋅2¹¹×⋅8!10257𝑏⁸+10104𝑎𝑏⁷+10716𝑎²𝑏⁶+9288𝑎³𝑏⁵+11430𝑎⁴𝑏⁴+9288𝑎⁵𝑏³+10716𝑎⁶𝑏²+10104𝑎⁷𝑏+10257𝑎⁸=10240(𝑏−𝑎)45⋅2¹¹𝑏⋅8!⁸+𝑏⁷𝑎+𝑏⁶𝑎²+𝑏⁵𝑎³+𝑏⁴𝑎⁴+𝑏³𝑎⁵+𝑏²𝑎⁶+𝑏𝑎⁷+𝑎⁸−(𝑏−𝑎)45⋅2¹¹⋅8!10257𝑏⁸+10104𝑎𝑏⁷+10716𝑎²𝑏⁶+9288𝑎³𝑏⁵+11430𝑎⁴𝑏⁴+9288𝑎⁵𝑏³+10716𝑎⁶𝑏²+10104𝑎⁷𝑏+10257𝑎⁸=17(𝑏−𝑎)45⋅2¹¹⋅8!−𝑏⁸+8𝑏⁷𝑎−28𝑏⁶𝑎²+56𝑏⁵𝑎³−70𝑏⁴𝑎⁴+56𝑏³𝑎⁵−28𝑏²𝑎⁶+8𝑏𝑎⁷−𝑎⁶()=−17𝑏−𝑎⁹45⋅2¹¹.⋅8! (33) This implies that 𝑅₄𝑓()=−17𝑏−𝑎⁹45⋅2¹¹𝑓⋅8!⁽⁸⁾(𝜉). (34)

Precision, the orders and the error terms for midpoint derivative-based closed Newton-Cotes quadrature are summarized in Table 1.

Table 1 

Precision, the orders and the error terms for midpoint derivative-based closed Newton-Cotes quadrature.

4. Computational Efficiency in Composite Form

In this section, in order to compare the computational efficiency of the closed Newton-Cotes and the midpoint derivative-based quadrature formula, the number of calculations required by each quadrature formula to obtain a certain level of accuracy of 10⁻¹⁰ and 10⁻⁵ is calculated for the following integrals ∫¹₀(𝑑𝑥/(1+𝑥)) and ∫²₀𝑒^𝑥𝑑𝑥, respectively.

In Tables 2 and 3, the number of function and derivative evaluations for the various quadrature formula presented for ∫¹₀(𝑑𝑥/(1+𝑥)) and ∫²₀𝑒^𝑥𝑑𝑥 are listed, respectively, using Matlab 6.5.

Table 2 

Computational cost to estimate ∫¹₀(𝑑𝑥/(1+𝑥)).

Table 3 

Computational cost to estimate ∫²₀𝑒^𝑥𝑑𝑥.

Take ∫¹₀(𝑑𝑥/(1+𝑥)) as an example, for the composite Trapezoidal rule, 25002 function evaluations are required, and the computing time is 0.125 seconds; while for the composite midpoint derivative Trapezoidal rule, 106 function evaluations and 105 second derivative evaluations are required (total = 211), and the computing time is 0.031 seconds on the same processor. So the midpoint derivative Trapezoidal rule is less time-consuming than Trapezoidal rule when they obtain the same level of accuracy.

In order to compare the different methods with the same computational cost, the numerical experiments between Trapezoidal rule and Midpoint derivative Trapezoidal rule are performed. We choose the following two integrals ∫²₀𝑒^𝑥𝑑𝑥 and ∫¹₀(ln(1+𝑥)/(1+𝑥²))𝑑𝑥 as examples and compare the CPU time for when they reach the same level of accuracy of 10⁻³, 10⁻⁶, and 10⁻⁹. The comparative experimental results are shown in Tables 4 and 5.

Table 4 

CPU time for ∫²₀𝑒^𝑥𝑑𝑥.

Table 5 

CPU time for ∫¹₀(ln(1+𝑥)/(1+𝑥²))𝑑𝑥.

5. Numerical Results

So far, we have proposed midpoint derivative-based closed Newton-Cotes quadrature in Section 2 and demonstrate the results that the proposed methods use fewer evaluations in Section 4.

In this section, many numerical experiments are carried out to determine whether the novel methods are of high precision. In order to compare the precision of Newton-Cotes quadrature and the midpoint derivative-based closed Newton-Cotes quadrature, we calculate the following integrals: ∫¹₀(4𝑑𝑥/(1+𝑥²)), ∫²₀𝑒^𝑥𝑑𝑥. The comparison results are shown in Tables 6, 7, 8, 9, 10, 11, 12, and 13.

Table 6 

Exact value of ∫¹₀(4𝑑𝑥/(1+𝑥²))=𝜋≈3.1415926536.

Table 7 

Table 8 

Table 9 

Table 10 

Exact value of ∫²₀𝑒^𝑥𝑑𝑥=𝑒²−1≈6.3890560989.

Table 11 

Table 12 

Table 13 

In Tables 6, 7, 8, 9, 10, 11, 12, and 13, the item Int. stands for the number of composite intervals.

Let us define Error = |Exactvalue−Approximatevalue|.

It can be seen from Tables 6–13 that midpoint derivative-based closed Newton-Cotes quadrature formulas have a much higher accuracy than classical closed Newton-Cotes quadrature formulas.

6. Conclusion

We briefly summarize our main conclusions in this paper as follows.(1)A family of numerical integration formulas of closed Newton-Cotes quadrature rules is presented, which uses the derivative value at the midpoint. (2)It is proved that these kinds of quadrature rules obtain an increase of two orders of precision over the classical closed Newton-Cotes formula, and the error terms are given. (3)The computational cost for these methods is analyzed for several examples. And it has shown that the proposed formulas are superior computationally to the same order closed Newton-Cotes formulas when they reduce the error below the same level.(4)Finally, some numerical examples are given to show the efficiency of the proposed approach.

Dehghan’s technique may be applied for midpoint derivative-based closed Newton-Cotes quadrature and how to accelerate the convergence of the quadrature formulas by using Richardson extrapolation algorithm will be achieved by further research.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (nos. 61074004, 61104038, 11101066).