choreo.segm.quad.IntegrateOnSegment#

IntegrateOnSegment(fun, int ndim, (double, double) x_span, QuadTable quad, Py_ssize_t nint=1, bool DoEFT=True) → ndarray#

Computes an approximation of the integral of a function on a segment.

Denoting \(a \eqdef \text{x_span}[0]\), \(b \eqdef \text{x_span}[1]\) and \(\Delta \eqdef \text{x_span}[1]-\text{x_span}[0]\), the integral is first decomposed into nint smaller integrals as:

\[\int_a^b \operatorname{fun}(x) \dd x = \sum_{i = 0}^{\text{nint}-1} \int_{a + \frac{i}{\text{nint}} * \Delta}^{a + \frac{i+1}{\text{nint}} * \Delta} \operatorname{fun}(x) \dd x\]

Each of the smaller integrals is then approximated using the QuadTable quad (cf formula (1)).

The integrand can either be:

See also

Example

Let us compare the approximation and exact value of the Wallis integral of order 7.

\[\int_0^{\frac{\pi}{2}} \operatorname{sin}^7(x) \dd x = \frac{16}{35}\]
>>> import numpy as np
>>> import choreo
>>> Gauss = choreo.segm.multiprec_tables.ComputeQuadrature(10, method="Gauss")
>>> def fun(x):
...     return np.array([np.sin(x)**7])
...
>>> np.abs(choreo.segm.quad.IntegrateOnSegment(fun, ndim=1, x_span=(0.,np.pi/2), quad=Gauss) - 16/35)
array([4.65549821e-12])
Parameters:

  • fun (callable or scipy.LowLevelCallable) – Function to be integrated.

  • ndim (int) – Number of output dimensions of the integrand.

  • x_span (tuple (numpy.float64, numpy.float64)) – Lower and upper bound of the integration interval.

  • quad (QuadTable) – Weights and nodes of the integration.

  • nint (int, optional) – Number of sub-intervals for the integration, by default 1.

  • DoEFT (bool, optional) – Whether to use an error-free transformation for summation, by default True.

Returns:

The approximated value of the integral.

Return type:

numpy.ndarray(shape = (nsteps, ndim), dtype = np.float64)

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