Newton-Cotes quadrature formula

The interpolation quadrature formula

$$ \int\limits _ { a } ^ { b } f ( x) dx \cong \ ( b - a) \sum _ {k = 0 } ^ { n } B _ {k} ^ {(} n) f ( x _ {k} ^ {(} n) ) $$

for the computation of an integral over a finite interval $ [ a, b] $, with nodes $ x _ {k} ^ {(} n) = a + kh $, $ k = 0 \dots n $, where $ n $ is a natural number, $ h = ( b - a)/n $, and the number of nodes is $ N = n + 1 $. The coefficients are determined by the fact that the quadrature formula is interpolational, that is,

$$ B _ {k} ^ {(} n) = \ \frac{(- 1) ^ {n - k } }{k! ( n - k)! n } \int\limits _ { 0 } ^ { n } \frac{t ( t - 1) \dots ( t - n) }{t - k } dt. $$

For $ n = 1 \dots 7 , 9 $ all coefficients are positive, for $ n = 8 $ and $ n \geq 10 $ there are both positive and negative ones among them. The algebraic degree of accuracy (the number $ d $ such that the formula is exact for all polynomials of degree at most $ d $ and not exact for $ x ^ {d + 1 } $) is $ n $ for odd $ n $ and $ n + 1 $ for even $ n $. The simplest special cases of the Newton–Cotes quadrature formula are: $ n = 1 $, $ h = b - a $, $ N = 2 $,

$$ \int\limits _ { a } ^ { b } f ( x) dx \cong \ { \frac{b - a }{2} } [ f ( a) + f ( b)], $$

the trapezium formula; $ n = 2 $, $ h = ( b - a)/2 $, $ N = 3 $,

$$ \int\limits _ { a } ^ { b } f ( x) dx \cong \ { \frac{b - a }{6} } \left [ f ( a) + 4f \left ( { \frac{a + b }{2} } \right ) + f ( b) \right ] , $$

the Simpson formula; $ n = 3 $, $ h = ( b - a)/3 $, $ N = 4 $,

$$ \int\limits _ { a } ^ { b } f ( x) dx \cong \ { \frac{b - a }{8} } \left [ f ( a) + 3f ( a + h) + 3f ( a + 2h) + f ( b) \right ] , $$

the "three-eighths" quadrature formula. For large $ n $ the Newton–Cotes formula is seldom used (because of the property of the coefficients for $ n \geq 10 $ mentioned above). One prefers to use for small $ n $ the compound Newton–Cotes quadrature formulas, namely, the trapezium formula and Simpson's formula.

The coefficients of the Newton–Cotes quadrature formula for $ n $ from 1 to 20 are listed in [3].

The formula first appeared in a letter from I. Newton to G. Leibniz in 1676 (see [1]) and later in the book [2] by R. Cotes, where the coefficients of the formula are given for $ n $ from 1 to 10.

References

[1]	I. Newton, "Mathematical principles of natural philosophy" A.N. Krylov (ed.) , Collected works , 7 , Moscow-Leningrad (1936) (In Russian; translated from Latin)
[2]	R. Cotes, "Harmonia Mensurarum" , 1–2 , London (1722) (Published by R. Smith after Cotes' death)
[3]	V.I. Krylov, L.T. Shul'gina, "Handbook on numerical integration" , Moscow (1966) (In Russian)

Comments

The formulas above are often referred to as closed Newton–Cotes formulas, in contrast to open Newton–Cotes formulas, which do not include the end points as nodes.

References

[a1]	H. Engels, "Numerical quadrature and cubature" , Acad. Press (1980)
[a2]	H. Brass, "Quadraturverfahren" , Vandenhoeck & Ruprecht (1977)
[a3]	P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984)
[a4]	A.H. Stroud, "Numerical quadrature and solution of ordinary differential equations" , Springer (1974)