Newton-Cotes formula
In numerical analysis, Newton-Cotes formulas are a group of formulas for numerical integration (also called quadrature).
These formulas are named after Isaac Newton and Roger Cotes.
These formulas are also called Newton-Cotes rules.
It is assumed that the value of a function f is known at equally spaced points xi, for i = 0, ..., n.
(If the evaluation points are not assumed to be equally spaced,
another class of formulas, called Gaussian quadrature formulas,
can be derived.)
There are two types of Newton-Cotes formulas,
the "closed" type which uses the function value at all points,
and the "open" type which does not use the function values at the end points.
The closed Newton-Cotes formula of degree n is stated as
- <math>\int_a^b f(x) \,dx \approx \sum_{i=0}^n w_i\, f(x_i)<math>
where xi = h i + x0,
with h (called the step size) equal to (xn - x0)/n. The ωi are called weights.
As can be seen in the following derivation the weights are derived from the Langrange basis polynomials. This means they depend only on the xi and not on the function f. L(x) is the interpolation polynomial in the Lagrange form for the given data points (x0, f(x0) ),..,(xn, f(xn) )
- <math>\int_a^b f(x) \,dx \approx \int_a^b L(x)\,dx = \int_a^b \sum_{i=0}^n f(x_i)\, l_i(x)\, dx <math>
- <math>=\sum_{i=0}^n \int_{x_{i-1}}^{x_i} f(x_i)\, l_i(x)\, dx =
\sum_{i=0}^n f(x_i) \underbrace{\int_{x_{i-1}}^{x_i} l_i(x)\, dx}_{w_i}<math>
The open Newton-Cotes formula of degree n is stated as
- <math>\int_a^b f(x)\, dx \approx \sum_{i=1}^{n-1} w_i\, f(x_i)<math>
The weights are found in a manner similar to the closed formula.
A Newton-Cotes formula of any degree can be constructed.
Some of the formulas of low degree are known by conventional names.
This table lists some of the Newton-Cotes formulas of the closed type.
| Degree |
Common name |
Formula |
Error term |
| 1 |
Trapezoid rule |
<math> \frac{h}{2} (f_0 + f_1) <math> |
<math>-\frac{h^3}{12}\,f^{(2)}(\xi)<math> |
| 2 |
Simpson's rule |
<math> \frac{h}{3} (f_0 + 4 f_1 + f_2) <math> |
<math>-\frac{h^5}{90}\,f^{(4)}(\xi)<math> |
| 3 |
3/8 rule |
<math> \frac{3\, h}{8} (f_0 + 3 f_1 + 3 f_2 + f_3) <math> |
<math>-\frac{3\, h^5}{80}\,f^{(4)}(\xi)<math> |
| 4 |
Boole's rule |
<math> \frac{2\, h}{45} (7 f_0 + 32 f_1 + 12 f_2 + 32 f_3 + 7 f_4) <math> |
<math>-\frac{8\, h^7}{945}\,f^{(6)}(\xi)<math> |
The exponent of the step size h in the error term shows the rate at which the approximation error decreases.
The derivative of f in the error term shows which polynomials can be integrated exactly (i.e., with error equal to zero).
Note that the derivative of f in the error term increases by 2 for every other rule.
This table lists some of the Newton-Cotes formulas of the open type.
| Degree |
Common name |
Formula |
Error term |
| 0 |
Rectangle rule |
<math>2 h f_1 <math> |
<math>\frac{h^3}{24}\,f^{(2)}(\xi)<math> |
| 1 |
|
<math> \frac{3\, h}{2} (f_1 + f_2) <math> |
<math> \frac{h^3}{4}\,f^{(2)}(\xi) <math> |
| 2 |
|
<math> \frac{4 \,h}{3} (2 f_1 - f_2 + 2 f_3) <math> |
<math> \frac{28\, h^5}{90}\,f^{(4)}(\xi) <math> |
| 3 |
|
<math> \frac{5 \,h}{24} (11 f_1 + f_2 + f_3 + 11 f_4) <math> |
<math> \frac{95\, h^5}{144}\,f^{(4)}(\xi) <math> |
It is interesting to note that the rectangle rule is exact for the same class of functions as the trapezoidal rule.
What's more,
the error term of the trapezoidal rule is twice as great as that of the rectangle rule.
However,
the trapezoidal rule is important as a building block in the construction of extrapolation methods.
References
- M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Section 25.4.)
- George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler. Computer Methods for Mathematical Computations. Englewood Cliffs, NJ: Prentice-Hall, 1977. (See Section 5.1.)
- William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Section 4.1.)
- Josef Stoer and Roland Bulirsch. Introduction to Numerical Analysis. New York: Springer-Verlag, 1980. (See Section 3.1.)
