FejerAlgorithm¶

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class FejerAlgorithm(*args)¶

Fejer Integration algorithm

Available constructors:

FejerAlgorithm(dimension=1, method=FejerAlgorithm.FEJERTYPE1)

FejerAlgorithm(discretization, method=FejerAlgorithmFEJERTYPE1)

Parameters

dimensionint, $dimension>0$

The dimension of the functions to integrate. The default discretization is FejerAlgorithm-DefaultMarginalIntegrationPointsNumber in each dimension, see ResourceMap.

discretizationsequence of int

The number of nodes in each dimension. The sequence must be non-empty and must contain only positive values.

methodint, optional

Integer used to select the method of integration. (Amongst ot.FejerAlgorithm.FEJERTYPE1, ot.FejerAlgorithm.FEJERTYPE2 and ot.FejerAlgorithm.CLENSHAWCURTIS).

Default is ot.FejerAlgorithm.FEJERTYPE1

Notes

The FejerAlgorithm algorithm enables to approximate the definite integral:

$\int_{\vect{a}}^\vect{b} f(\vect{t})\di{\vect{t}}$

with $f: \Rset^d \mapsto \Rset^p$ , $\vect{a}, \vect{b}\in\Rset^d$ using a the approximation:

$\int_{\vect{a}}^\vect{b} f(\vect{t})\di{\vect{t}} = \sum_{\vect{n}\in \cN}\left(\prod_{i=1}^d w^{N_i}_{n_i}\right)f(\xi^{N_1}_{n_1},\dots,\xi^{N_d}_{n_d})$

where $\xi^{N_i}_{n_i}$ is the $n_i$ -points and $w^{N_i}_{n_i}$ the associated weight.

Let us note $\theta_k=\dfrac{k\pi}{n}, k=0,1,...,n-1$ , $x_k = cos(\theta_k)$ . The Clenshaw-Curtis nodes are $x_k$ and its associated weights are given as follows:

$w_k =\dfrac{c_k}{n}\left(1-\sum_{j=1}^{\lfloor n/2\rfloor}\dfrac{b_j}{4j^2-1}\cos\left(2j\theta_k\right)\right)$

with $b_j= 2$ if $j < n/2$ else $b_j=1$ . Also, $c_k = 1$ if $if k = 0[n]$ else $c_k = 2$

The type-1 Fejer quadrature rule relies on $x_k = cos(\theta_{k+1/2})$ nodes and associated weights are the following:

$w_k=\dfrac{2}{n}\left(1-2\sum_{j=1}^{\lfloor n/2\rfloor}\dfrac{1}{4j^2-1}\cos\left(j\theta_{2k+1}\right)\right)$

Finally, the type-2 Fejer quadrature rule is very close to the Clenshaw-Curtis one. The two method do share the same nodes (except the endpoints that are set to 0 within the Fejer method). Its associated weights are given as follows:

$w_k=\dfrac{4}{n+1}\sin\theta_k\sum_{j=1}^{\lfloor n/2\rfloor}\dfrac{\sin\left((2j-1)\theta_k\right)}{2j-1}$

Examples

Create a FejerAlgorithm algorithm:

>>> import openturns as ot
>>> algo = ot.FejerAlgorithm(2)
>>> algo = ot.FejerAlgorithm([2, 4, 5])

Methods

`getClassName`()	Accessor to the object's name.
`getDiscretization`()	Accessor to the discretization of the tensorized rule.
`getId`()	Accessor to the object's id.
`getName`()	Accessor to the object's name.
`getNodes`()	Accessor to the integration nodes.
`getShadowedId`()	Accessor to the object's shadowed id.
`getVisibility`()	Accessor to the object's visibility state.
`getWeights`()	Accessor to the integration weights.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`integrate`(*args)	Evaluation of the integral of $f$ on an interval.
`setName`(name)	Accessor to the object's name.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setVisibility`(visible)	Accessor to the object's visibility state.

integrateWithNodes

__init__(*args)¶

getClassName()¶

Accessor to the object’s name.

Returns

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

getDiscretization()¶

Accessor to the discretization of the tensorized rule.

Returns

discretizationIndices: The number of integration point in each dimension.

getId()¶

Accessor to the object’s id.

Returns

idint: Internal unique identifier.

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getNodes()¶

Accessor to the integration nodes.

Returns

nodesSample: The tensorized FejerAlgorithm integration nodes on $[0,1]^d$ where $d>0$ is the dimension of the integration algorithm.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns

idint: Internal unique identifier.

getVisibility()¶

Accessor to the object’s visibility state.

Returns

visiblebool: Visibility flag.

getWeights()¶

Accessor to the integration weights.

Returns

weightsPoint: The tensorized FejerAlgorithm integration weights on $[0,1]^d$ where $d>0$ is the dimension of the integration algorithm.

hasName()¶

Test if the object is named.

Returns

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns

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

integrate(*args)¶

Evaluation of the integral of $f$ on an interval.

Available usages:

integrate(f, interval)

integrate(f, interval, xi)

Parameters

fFunction, $f: \Rset^d \mapsto \Rset^p$: The integrand function.
intervalInterval, $interval \in \Rset^d$: The integration domain.
xiSample: The integration nodes.

Returns

valuePoint: Approximation of the integral.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x'], ['sin(x)'])
>>> a = -2.5
>>> b = 4.5
>>> algoF1 = ot.FejerAlgorithm([10])
>>> value = algoF1.integrate(f, ot.Interval(a, b))[0]
>>> print(value)
-0.590...

setName(name)¶

Accessor to the object’s name.

Parameters

namestr: The name of the object.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters

idint: Internal unique identifier.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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