# 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) |

**How to Cite This Entry:**

Newton–Cotes quadrature formula.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Newton%E2%80%93Cotes_quadrature_formula&oldid=22842