# Integrate <`compmech.integrate`>

Brings together many convenient algorithms for numerical integration that are not available in SciPy, such as the integration of vector valued functions.

It would be nice to integrate this into SciPy one day.

## Examples

Below you can see how to use the available Cython functions.

Integration of scalar functions:

>     from numpy import isclose, pi, sin
>     import numpy as np

>     from compmech.integrate.integrate import (trapz2d_points, simps2d_points,
>                                               python_trapz_quad)

>     def test_integrate_sinsin():
>         ans = 4/pi**2
>         nx = ny = 40
>         xs, ys, alphas, betas = trapz2d_points(0, 1., nx, 0, 1., ny)
>         out = 0.
>         for x, y, alpha, beta in zip(xs, ys, alphas, betas):
>             out = beta*(out) + alpha*(sin(pi*x)*sin(pi*y))
>         assert isclose(out, ans, atol=0.001, rtol=0)

>         xs, ys, alphas, betas = simps2d_points(0, 1., nx, 0, 1., ny)
>         out = 0.
>         for x, y, alpha, beta in zip(xs, ys, alphas, betas):
>             out = beta*(out) + alpha*(sin(pi*x)*sin(pi*y))
>         assert isclose(out, ans, atol=0.00001, rtol=0)

>     def test_integrate_sinsin_using_trapz_quad():
>         ans = 4/pi**2
>         nx = ny = 40
>         xis = np.zeros(nx, dtype=np.float64)
>         weightsxi = np.zeros(nx, dtype=np.float64)
>         etas = np.zeros(ny, dtype=np.float64)
>         weightseta = np.zeros(ny, dtype=np.float64)

>         python_trapz_quad(nx, xis, weightsxi)
>         python_trapz_quad(ny, etas, weightseta)

>         out = 0.

>         for xi, weightxi in zip(xis, weightsxi):
>             for eta, weighteta in zip(etas, weightseta):
>                 out += weightxi*weighteta*(sin(pi*(xi+1)/2.)*sin(pi*(eta+1)/2.))
>         out *= 1/2.*1/2.

>         assert isclose(out, ans, atol=0.001, rtol=0)

>     if __name__ == '__main__':
>         test_integrate_sinsin()
>         test_integrate_sinsin_using_trapz_quad()

Integration of vector-valued functions:

>     import pyximport; pyximport.install()
>     import numpy as np

>     from compmech.integrate.integratev import _test_integratev

>     def test_integratev_trapz2d(nx=30, ny=30, method='trapz2d'):
>         out = _test_integratev(nx, ny, method)
>         assert np.allclose(out, (4/np.pi**2, 4/(9*np.pi**2), 4/(25*np.pi**2)),
>                 atol=1.e-3)

>     def test_integratev_simps2d(nx=20, ny=20, method='simps2d'):
>         out = _test_integratev(nx, ny, method)
>         assert np.allclose(out, (4/np.pi**2, 4/(9*np.pi**2), 4/(25*np.pi**2)),
>                 atol=1.e-4)

>     if __name__ == '__main__':
>         test_integratev_trapz2d()
>         test_integratev_simps2d()