newton_cotes(rn, equal=0)
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 $B_N (\Delta x)^{N+3} f^{N+2}(\xi)$ .
Normally, the Newton-Cotes rules are used on smaller integration regions and a composite rule is used to return the total integral.
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 :None:None:`rn`. N is the order of the Newton-Cotes integration.
Set to 1 to enforce equally spaced data.
1-D array of weights to apply to the function at the provided sample positions.
Error coefficient.
Return weights and error coefficient for Newton-Cotes integration.
Compute the integral of sin(x) in [0, $\pi$ ]:
>>> from scipy.integrate import newton_cotesSee :
... 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
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.integrate._quadrature.newton_cotes
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