LaguerreFactory

(Source code, png)

class LaguerreFactory(*args)

Laguerre specific orthonormal univariate polynomial family.

For the Gamma distribution.

Parameters:

kfloat

If parameters_set == ot.JacobiFactory.PROBABILITY: default shape parameter of the Gamma distribution.

If parameters_set == ot.JacobiFactory.ANALYSIS: alternative shape parameter of the Gamma distribution.

parameters_setint, optional

Integer telling which parameters set is used for defining the distribution (amongst ot.LaguerreFactory.ANALYSIS, ot.LaguerreFactory.PROBABILITY).

See also

StandardDistributionPolynomialFactory

Notes

Any sequence of orthogonal polynomials has a recurrence formula relating any three consecutive polynomials as follows:

The recurrence coefficients for the Laguerre polynomials come analytically and read:

where is the alternative shape parameter of the Gamma distribution, and:

Examples

>>> import openturns as ot
>>> polynomial_factory = ot.LaguerreFactory()
>>> for i in range(3):
...     print(polynomial_factory.build(i))
1
-1 + X
1 - 2 * X + 0.5 * X^2

Methods

build(degree)	Build the -th order orthogonal univariate polynomial.
buildCoefficients(degree)	Build the -th order orthogonal univariate polynomial coefficients.
buildRecurrenceCoefficientsCollection(degree)	Build the recurrence coefficients.
getClassName()	Accessor to the object's name.
getK()	Accessor to the alternative shape parameter .
getMeasure()	Accessor to the associated probability measure.
getName()	Accessor to the object's name.
getNodesAndWeights(n)	Build the -th order quadrature scheme.
getRecurrenceCoefficients(n)	Accessor to the recurrence coefficients of the -th order.
getRoots(n)	Accessor to the recurrence coefficients of the -th order.
hasName()	Test if the object is named.
setName(name)	Accessor to the object's name.

__init__(*args)

build(degree)

Build the -th order orthogonal univariate polynomial.

Parameters:

kint,

Polynomial order.

Returns:

polynomialOrthogonalUniVariatePolynomial

Requested orthogonal univariate polynomial.

Examples

>>> import openturns as ot
>>> polynomial_factory = ot.HermiteFactory()
>>> print(polynomial_factory.build(2))
-0.707107 + 0.707107 * X^2

buildCoefficients(degree)

Build the -th order orthogonal univariate polynomial coefficients.

Parameters:

kint,

Polynomial order.

Returns:

coefficientsPoint

Coefficients of the requested orthogonal univariate polynomial.

Examples

>>> import openturns as ot
>>> polynomial_factory = ot.HermiteFactory()
>>> print(polynomial_factory.buildCoefficients(2))
[-0.707107,0,0.707107]

buildRecurrenceCoefficientsCollection(degree)

Build the recurrence coefficients.

Build the recurrence coefficients of the orthogonal univariate polynomial family up to the -th order.

Parameters:

kint,

Polynomial order.

Returns:

recurrence_coefficientslist of Point

All the tecurrence coefficients up to the requested order.

Examples

>>> import openturns as ot
>>> polynomial_factory = ot.HermiteFactory()
>>> print(polynomial_factory.buildRecurrenceCoefficientsCollection(2))
0 : [  1         0         0        ]
1 : [  0.707107  0        -0.707107 ]

getClassName()

Accessor to the object’s name.

Returns:

class_namestr

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

getK()

Accessor to the alternative shape parameter .

Of the Gamma distribution.

Returns:

k_afloat

Alternative shape parameter of the Gamma distribution.

getMeasure()

Accessor to the associated probability measure.

Returns:

measureDistribution

The associated probability measure (according to which the polynomials are orthogonal).

Notes

Two polynomials P and Q are orthogonal with respect to the probability measure if and only if their dot product:

where and .

Examples

>>> import openturns as ot
>>> polynomial_factory = ot.HermiteFactory()
>>> print(polynomial_factory.getMeasure())
Normal(mu = 0, sigma = 1)

getName()

Accessor to the object’s name.

Returns:

namestr

The name of the object.

getNodesAndWeights(n)

Build the -th order quadrature scheme.

Associated with the orthogonal univariate polynomials family.

Parameters:

kint,

Polynomial order.

Returns:

nodesPoint

The nodes of the -th order quadrature scheme.

weightsPoint

The weights of the -th order quadrature scheme.

Examples

>>> import openturns as ot
>>> polynomial_factory = ot.HermiteFactory()
>>> nodes, weights = polynomial_factory.getNodesAndWeights(3)
>>> print(nodes)
[-1.73205,...,1.73205]
>>> print(weights)
[0.166667,0.666667,0.166667]

getRecurrenceCoefficients(n)

Accessor to the recurrence coefficients of the -th order.

Of the orthogonal univariate polynomial.

Parameters:

kint,

Polynomial order.

Returns:

recurrence_coefficientsPoint

The recurrence coefficients of the -th order orthogonal univariate polynomial.

Notes

Any sequence of orthogonal polynomials has a recurrence formula relating any three consecutive polynomials as follows:

Examples

>>> import openturns as ot
>>> polynomial_factory = ot.HermiteFactory()
>>> print(polynomial_factory.getRecurrenceCoefficients(3))
[0.5,0,-0.866025]

getRoots(n)

Accessor to the recurrence coefficients of the -th order.

Of the orthogonal univariate polynomial.

Parameters:

kint,

Polynomial order.

Returns:

rootsPoint

The roots of the -th order orthogonal univariate polynomial.

Examples

>>> import openturns as ot
>>> polynomial_factory = ot.HermiteFactory()
>>> print(polynomial_factory.getRoots(3))
[-1.73205,...,1.73205]

hasName()

Test if the object is named.

Returns:

hasNamebool

True if the name is not empty.

setName(name)

Accessor to the object’s name.

Parameters:

namestr

The name of the object.

Examples using the class

Polynomial chaos graphs

Polynomial chaos graphs

Advanced polynomial chaos construction

Advanced polynomial chaos construction