Nonlinear Dynamics in Applied Sciences Systems: Advances and Perspectives
View this Special IssueResearch Article  Open Access
Weijing Zhao, Hongxing Li, "Midpoint DerivativeBased Closed NewtonCotes Quadrature", Abstract and Applied Analysis, vol. 2013, Article ID 492507, 10 pages, 2013. https://doi.org/10.1155/2013/492507
Midpoint DerivativeBased Closed NewtonCotes Quadrature
Abstract
A novel family of numerical integration of closed NewtonCotes 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 NewtonCotes 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 NewtonCotes 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 NewtonCotes 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 , once the primitive function of integrand is known, the definite integral of over the interval is given by NewtonLeibniz formula, that is,
However, the explicit primitive function is not available or its primitive function is not easy to obtain, such as , , and . Moreover, some of the integrand is only available at certain points , . It is often the case that the values of come from experimental data, such as sampling [1]. But the need often arises for calculating the definite integral. And how to get highprecision 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: where there are distinct integration points at within the interval and weights , . If the integration points are uniformly distributed over the interval, so in which .
These can be derived in several different ways [3–5]. One method is to interpolate at the points , 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 , , so that the error is exactly zero for , . Using the method of undetermined coefficients, this approach generates a system of linear equations for weights . Since the monomials are linearly independent, the linear system of equations has a unique solution.
The NewtonCotes formulas are the most wellknown numerical integration rules of this type. There are several subclasses of NewtonCotes formulas that depend on the integer value of . We list some of them as follows. Trapezoidal rule where . Simpson’s rule where . Simpson’s rule where . Bool’s rule where .
Note that when is an even integer, the degree of precision is . When is odd, the degree of precision is only [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 NewtonCotes 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 GaussLegendre quadrature [7], GaussChebyshev quadrature [8], and open NewtonCotes 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 derivativebased closed NewtonCotes 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 NewtonCotes quadrature.
The motivation for this research lies in construction of midpoint derivativebased closed NewtonCotes numerical quadrature rule for NewtonCotes 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 NewtonCotes 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 DerivativeBased Closed NewtonCotes Quadrature
In this section, by adding the high derivative at the midpoint, schemes with higher precision than the NewtonCotes quadrature rules are presented.
Theorem 1. Midpoint derivativebased closed Trapezoidal rule is 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 .
When , ; .
When , ;
So the precision of midpoint derivativebased closed Trapezoidal rule is 3.
Theorem 2. Midpoint derivativebased closed Simpson’s rule is 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 .
When , ;
When , ;
So the precision of midpoint derivativebased closed Simpson’s rule is 5.
Similarly, we obtain the midpoint derivativebased closed Simpson’s rule and Bool’s rule.
Theorem 3. Midpoint derivativebased closed Simpson’s rule is The precision of this method is 5.
Proof. Since the Simpson’s 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 .
When , ;
When , ;
Theorem 4. Midpoint derivativebased closed Bool’s rule is 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 in like manner.
When , ;
When , ;
3. The Error Terms of Midpoint DerivativeBased Closed NewtonCotes Quadrature
In this section, the error terms of midpoint derivativebased closed NewtonCotes 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 and the exact value , where is the precision of the quadrature formula.
Theorem 5. Midpoint derivativebased closed Trapezoidal rule with the error term is where . Thus, this scheme is fifth order accurate with the error term , and the associate composite method is fourth order.
Proof. Let . So , Therefore, This implies that
Theorem 6. Midpoint derivativebased closed Simpson’s rule with the error term is where . Thus, this scheme is seventh order accurate with the error term , and the associate composite method is sixth order.
Proof. Let . So , Therefore,This implies that
Theorem 7. Midpoint derivativebased closed Simpson’s rule with the error term is where . 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 .
Proof. Let . So , Therefore, This implies that
Theorem 8. Midpoint derivativebased closed Bool’s rule with the error term is where . 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 .
Proof. Similarly, let . So , Therefore, This implies that
Precision, the orders and the error terms for midpoint derivativebased closed NewtonCotes quadrature are summarized in Table 1.

4. Computational Efficiency in Composite Form
In this section, in order to compare the computational efficiency of the closed NewtonCotes and the midpoint derivativebased quadrature formula, the number of calculations required by each quadrature formula to obtain a certain level of accuracy of and is calculated for the following integrals and , respectively.
In Tables 2 and 3, the number of function and derivative evaluations for the various quadrature formula presented for and are listed, respectively, using Matlab 6.5.


Take 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 timeconsuming 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 and as examples and compare the CPU time for when they reach the same level of accuracy of , , and . The comparative experimental results are shown in Tables 4 and 5.


5. Numerical Results
So far, we have proposed midpoint derivativebased closed NewtonCotes 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 NewtonCotes quadrature and the midpoint derivativebased closed NewtonCotes quadrature, we calculate the following integrals: , . The comparison results are shown in Tables 6, 7, 8, 9, 10, 11, 12, and 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 = .
It can be seen from Tables 6–13 that midpoint derivativebased closed NewtonCotes quadrature formulas have a much higher accuracy than classical closed NewtonCotes quadrature formulas.
6. Conclusion
We briefly summarize our main conclusions in this paper as follows.(1)A family of numerical integration formulas of closed NewtonCotes 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 NewtonCotes 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 NewtonCotes 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 derivativebased closed NewtonCotes 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).
References
 N. Petrovskaya and E. Venturino, “Numerical integration of sparsely sampled data,” Simulation Modelling Practice and Theory, vol. 19, no. 9, pp. 1860–1872, 2011. View at: Publisher Site  Google Scholar
 D. H. Bailey and J. M. Borwein, “Highprecision numerical integration: progress and challenges,” Journal of Symbolic Computation, vol. 46, no. 7, pp. 741–754, 2011. View at: Publisher Site  Google Scholar  MathSciNet
 K. E. Atkinson, An Introduction to Numerical Analysis, John Wiley & Sons, New York, NY, USA, 2nd edition, 1989. View at: MathSciNet
 R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole, Boston, Mass, USA, 9th edition, 2011.
 E. Isaacson and H. B. Keller, Analysis of Numerical Methods, John Wiley & Sons, New York, NY, USA, 1966. View at: MathSciNet
 M. Dehghan, M. MasjedJamei, and M. R. Eslahchi, “On numerical improvement of closed NewtonCotes quadrature rules,” Applied Mathematics and Computation, vol. 165, no. 2, pp. 251–260, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 E. Babolian, M. Masjedjamei, and M. R. Eslahchi, “On numerical improvement of GaussLegendre quadrature rules,” Applied Mathematics and Computation, vol. 160, no. 3, pp. 779–789, 2005. View at: Publisher Site  Google Scholar
 M. R. Eslahchi, M. Dehghan, and M. MasjedJamei, “On numerical improvement of the first kind GaussChebyshev quadrature rules,” Applied Mathematics and Computation, vol. 165, no. 1, pp. 5–21, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. Dehghan, M. MasjedJamei, and M. R. Eslahchi, “On numerical improvement of open NewtonCotes quadrature rules,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 618–627, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 C. O. E. Burg, “Derivativebased closed NewtonCotes numerical quadrature,” Applied Mathematics and Computation, vol. 218, no. 13, pp. 7052–7065, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
Copyright
Copyright © 2013 Weijing Zhao and Hongxing Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.