Note

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Estimate an integralΒΆ

In this example we are going to evaluate an integral of the form.

I_f = \int_{a}^{b}\, \int_{l_1(x_0)}^{u_1(x_0)}\, \int_{l_2(x_0, x_1)}^{u_2(x_0,x_1)}\, \int_{l_{n-1}(x_0, \dots, x_{n-2})}^{u_{n-1}(x_0, \dots, x_{n-2})} \, f(x_0, \dots, x_{n-1})\mathrm{d}{x_{n-1}}\dots\mathrm{d}{x_0}

with the iterated quadrature algorithm.

import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
import math as m
ot.Log.Show(ot.Log.NONE)

define the integrand and the bounds

a = -m.pi
b = m.pi
f = ot.SymbolicFunction(['x', 'y'], ['1+cos(x)*sin(y)'])
l = [ot.SymbolicFunction(['x'], [' 2+cos(x)'])]
u = [ot.SymbolicFunction(['x'], ['-2-cos(x)'])]

Draw the graph of the integrand and the bounds

g = ot.Graph('Integration nodes', 'x', 'y', True, 'topright')
g.add(f.draw([a, a], [b, b]))
curve = l[0].draw(a, b).getDrawable(0)
curve.setLineWidth(2)
curve.setColor('red')
g.add(curve)
curve = u[0].draw(a, b).getDrawable(0)
curve.setLineWidth(2)
curve.setColor('red')
g.add(curve)
view = viewer.View(g)
Integration nodes

compute the integral value

I2 = ot.IteratedQuadrature().integrate(f, a, b, l, u)
print(I2)
plt.show()

Out:

[-25.1327]

Total running time of the script: ( 0 minutes 0.110 seconds)

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