Abstract
A new family of numerical integration formula is presented, which uses the function evaluation at the midpoint of the interval and odd derivatives at the endpoints. Because the weights for the odd derivatives sum to zero, the derivative calculations cancel out for the interior points in the composite form, so that these derivatives must only be calculated at the endpoints of the overall interval of integration. When using N subintervals, the basic rule which uses the midpoint function evaluation and the first derivative at the endpoints achieves fourth order accuracy for the cost of N/2 function evaluations and 2 derivative evaluations, whereas the three point open Newton-Cotes method uses 3N/4 function evaluations to achieve the same order of accuracy. These derivative-based midpoint quadrature methods are shown to be more computationally efficient than both the open and closed Newton-Cotes quadrature rules of the same order. This family of derivative-based midpoint quadrature rules are derived using the concept of precision, along with the error term. A theorem concerning the order of accuracy of quadrature rule using the concept of precision is provided to justify its use to determine the leading order error term.
Highlights
Open Newton-Cotes quadrature formula rely on a weighted averaged of function evaluations of the form xn n 1f x dx wi f xi (1)x0 i 1 where there are n + 1 distinct uniformly distributed integration points at x0, x1, xn within the interval [a, b], where xi a ih, and n − 1 weights wi
To demonstrate the accuracy of the new numerical integration formula based on the inclusion of the derivative, the values of e x2 dx and e 2x sin 4x dx are estimated using the midpoint rule and the first three derivative-based midpoint quadrature rules, dealing with the first derivative, the first and third derivatives and the first, third and fifth derivatives
The observed order of accuracy of the derivative-based midpoint quadrature formula is converging to the appropriate theoretical order of accuracy
Summary
X0 i 1 where there are n + 1 distinct uniformly distributed integration points at x0 , x1, , xn within the interval [a, b], where xi a ih , and n − 1 weights wi. For Gauss-Legendre integration, both the locations and the weights need to be specified, for the generic interval [−1, 1], so there are twice as many parameters as evaluation locations for this type of quadrature In each of these methods, by increasing the number of parameters, the precision and the order of accuracy of these methods increases. Burg [9] took a different approach by including first and higher order derivatives at the evaluation locations within the closed Newton-Cotes quadrature framework, in order to increase the number of parameters and the precision and order of accuracy of the resulting formula. Because of the odd nature of the midpoint rule, the use of odd derivatives produces highly advantageous results, creating quadrature rules that are much more efficient than existing open or closed NewtonCotes quadrature rules A theorem concerning the leading order error term in a quadrature rule is proved in the appendix, along with an associated conjecture concerning the overall error in a quadrature rule
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
