class IteratedQuadrature(*args)

Multivariate integration algorithm.

Parameters
univariateQuadratureIntegrationAlgorithm

By default, the integration algorithm is the Gauss-Kronrod algorithm (GaussKronrod) with the following parameters: maximumSubIntervals=32, maximumError= and GKRule = G3K7. Note that the default parametrisation of the GaussKronrod class leads to a more precise evaluation of the integral but the cost is greater. It is recommended to increase the order of the quadratic rule and the number of sub intervals if the integrand or one of the bound functions is smooth but with many oscillations.

Notes

This class enables to approximate the following integral:

with , and . For , there is no bound functions and .

Examples

>>> import openturns as ot
>>> import math as m
>>> 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')
>>> curve = l[0].draw(a, b).getDrawable(0)
>>> curve.setLineWidth(2)
>>> curve.setColor('red')
>>> curve = u[0].draw(a, b).getDrawable(0)
>>> curve.setLineWidth(2)
>>> curve.setColor('red')


Evaluate the integral with high precision:

>>> Iref = ot.IteratedQuadrature(ot.GaussKronrod(100000, 1e-13, ot.GaussKronrodRule(ot.GaussKronrodRule.G11K23))).integrate(f, a, b, l, u)


Evaluate the integral with the default GaussKronrod algorithm, and get evaluation points:

>>> f = ot.MemoizeFunction(f)
>>> I1 = ot.IteratedQuadrature(ot.GaussKronrod()).integrate(f, a, b, l, u)
>>> sample1 = f.getInputHistory()
>>> print(I1)
[-25.132...]
>>> n_evals = sample1.getSize()
>>> print(n_evals)
2116
>>> err = abs(100.0*(1.0-I1[0]/Iref[0]))
>>> print(err)
2.2...e-14
>>> cloud = ot.Cloud(sample1)
>>> cloud.setPointStyle('fcircle')
>>> cloud.setColor('green')
>>> f.clearHistory()


Evaluate the integral with the default IteratedQuadrature algorithm:

>>> I2 = ot.IteratedQuadrature().integrate(f, a, b, l, u)
>>> sample2 = f.getInputHistory()
>>> print(I2)
[-25.132...]
>>> n_evals = sample2.getSize()
>>> print(n_evals)
5236
>>> err = abs(100.0*(1.0-I2[0]/Iref[0]))
>>> print(err)
4.6...e-10
>>> cloud = ot.Cloud(sample2)
>>> cloud.setPointStyle('fcircle')
>>> cloud.setColor('gold')


Methods

 getClassName(self) Accessor to the object’s name. getId(self) Accessor to the object’s id. getName(self) Accessor to the object’s name. getShadowedId(self) Accessor to the object’s shadowed id. getVisibility(self) Accessor to the object’s visibility state. hasName(self) Test if the object is named. hasVisibleName(self) Test if the object has a distinguishable name. integrate(self, \*args) Evaluation of the integral of on a domain. setName(self, name) Accessor to the object’s name. setShadowedId(self, id) Accessor to the object’s shadowed id. setVisibility(self, visible) Accessor to the object’s visibility state.
__init__(self, \*args)

Initialize self. See help(type(self)) for accurate signature.

getClassName(self)

Accessor to the object’s name.

Returns
class_namestr

The object class name (object.__class__.__name__).

getId(self)

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getName(self)

Accessor to the object’s name.

Returns
namestr

The name of the object.

getShadowedId(self)

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

getVisibility(self)

Accessor to the object’s visibility state.

Returns
visiblebool

Visibility flag.

hasName(self)

Test if the object is named.

Returns
hasNamebool

True if the name is not empty.

hasVisibleName(self)

Test if the object has a distinguishable name.

Returns
hasVisibleNamebool

True if the name is not empty and not the default one.

integrate(self, \*args)

Evaluation of the integral of on a domain.

Available usages:

integrate(f, interval)

integrate(f, a, b, lowerBoundFunctions, upperBoundFunctions)

Parameters
f

The integrand function.

interval

The integration domain.

a,bfloat

Bounds of the integration interval of the first scalar input

lowerBoundFunctions, upperBoundFunctionslist of Function

List of functions and where defining the integration domain as defined above. The bound functions can cross each other.

Returns
valuePoint

Approximation of the integral.

setName(self, name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

setShadowedId(self, id)

Accessor to the object’s shadowed id.

Parameters
idint

Internal unique identifier.

setVisibility(self, visible)

Accessor to the object’s visibility state.

Parameters
visiblebool

Visibility flag.