scipy.integrate.newton_cotes

scipy.integrate.newton_cotes(rn, equal=0)

Return weights and error coefficient for Newton-Cotes integration.

Suppose we have (N+1) samples of f at the positions x_0, x_1, …, x_N. Then an N-point Newton-Cotes formula for the integral between x_0 and x_N is:

\(\int_{x_0}^{x_N} f(x)dx = \Delta x \sum_{i=0}^{N} a_i f(x_i) + B_N (\Delta x)^{N+2} f^{N+1} (\xi)\)

where \(\xi \in [x_0,x_N]\) and \(\Delta x = \frac{x_N-x_0}{N}\) is the average samples spacing.

If the samples are equally-spaced and N is even, then the error term is \(B_N (\Delta x)^{N+3} f^{N+2}(\xi)\).

Parameters:

rn : int

The integer order for equally-spaced data or the relative positions of the samples with the first sample at 0 and the last at N, where N+1 is the length of rn. N is the order of the Newton-Cotes integration.

equal : int, optional

Set to 1 to enforce equally spaced data.

Returns:

an : ndarray

1-D array of weights to apply to the function at the provided sample positions.

B : float

Error coefficient.

Notes

Normally, the Newton-Cotes rules are used on smaller integration regions and a composite rule is used to return the total integral.

Examples

Compute the integral of sin(x) in [0, \(\pi\)]:

>>> from scipy.integrate import newton_cotes
>>> def f(x):
...     return np.sin(x)
>>> a = 0
>>> b = np.pi
>>> exact = 2
>>> for N in [2, 4, 6, 8, 10]:
...     x = np.linspace(a, b, N + 1)
...     an, B = newton_cotes(N, 1)
...     dx = (b - a) / N
...     quad = dx * np.sum(an * f(x))
...     error = abs(quad - exact)
...     print('{:2d}  {:10.9f}  {:.5e}'.format(N, quad, error))
...
 2   2.094395102   9.43951e-02
 4   1.998570732   1.42927e-03
 6   2.000017814   1.78136e-05
 8   1.999999835   1.64725e-07
10   2.000000001   1.14677e-09