Quadratures Newton-Cotes

Abstract An improvement is made to an automatic quadrature due to Ninomiya (J. Inf. Process. 3 162-170, 1980) of adaptive type based on the Newton-Cotes rule by incorporating a doubly-adaptive algorithm due to Favati, Lotti and Romani (ACM Trans. Math. Softw. 17 207-217,1991 ACM Trans. Math. Softw. 17 218-232, 1991). We compare the present method in performance with some others by using various test problems including Kahaner s ones (Computation of numerical quadrature formulas. In Rice, J.R. (ed.) Mathematical Software, 229-259. Academic, Orlando, FL, 1971). 

Ninomiya, L Improvement of adaptive Newton-Cotes quadrature methods. J. Inf. Process. 3, 162-170 (1980)... 

One of the easiest ways to obtain useful quadrature formulas for our purpose is to use Lagrangian interpolation on an equally spaced mesh and integrate the result. This class of quadrature formulas is called the Newton-Cotes... 

An alternative closed Newton-Cotes quadrature formula of second order can be obtained by a polynomial of degree 1 which passes through the end points. This quadrature formula is called the trapezoid rule. In 2D this surface integral approximation requires the integrand values at the GCV corners. 

It should be clear to the reader that other Newton-Cotes quadratures can also be employed to obtain alternative expressions for discretized DAFs (44). Now note that unlike, e.g., interpolation schemes or the popular discrete variable representation (DVR) (16),... 

A more accurate and systematic way of evaluating integrals is to perform the integration numerically. In the next two sections, we derive Newton-Cotes integration formulas for equally spaced intervals and Gauss quadrature for unequally spaced points. 

Numerical differentiation and integration, including differentiation by backward, forward, and central finite differences Newton-Cotes formulas and the Gauss Quadrature... 

This method has been applied to the behavior of fast reactor clad in the program PECT-1 (48). The problem is solved by taking 11 evenly spaced radii and performing the integrations with the Newton-Cotes quadrature formula. The time-dependent part of the solution is carried out using a... 

---