# LegendreFactory¶

class LegendreFactory(*args)

Legendre specific orthonormal univariate polynomial family.

For the Uniform distribution.

Available constructor:
LegendreFactory()

Notes

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

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

Examples

>>> import openturns as ot
>>> polynomial_factory = ot.LegendreFactory()
>>> for i in range(3):
...     print(polynomial_factory.build(i))
1
1.73205 * X
-1.11803 + 3.3541 * 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. getId() Accessor to the object’s id. 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. getShadowedId() Accessor to the object’s shadowed id. getVisibility() Accessor to the object’s visibility state. hasName() Test if the object is named. hasVisibleName() Test if the object has a distinguishable name. 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.
__init__(*args)
build(degree)

Build the -th order orthogonal univariate polynomial.

Parameters: k : int, Polynomial order. polynomial : OrthogonalUniVariatePolynomial 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: k : int, Polynomial order. coefficients : Point 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: k : int, Polynomial order. recurrence_coefficients : list 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))
[[1,0,0],[0.707107,0,-0.707107]]

getClassName()

Accessor to the object’s name.

Returns: class_name : str The object class name (object.__class__.__name__).
getId()

Accessor to the object’s id.

Returns: id : int Internal unique identifier.
getMeasure()

Accessor to the associated probability measure.

Returns: measure : Distribution 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: name : str The name of the object.
getNodesAndWeights(n)

Build the -th order quadrature scheme.

Associated with the orthogonal univariate polynomials family.

Parameters: k : int, Polynomial order. nodes : Point The nodes of the -th order quadrature scheme. weights : Point 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: k : int, Polynomial order. recurrence_coefficients : Point 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: k : int, Polynomial order. roots : Point 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]

getShadowedId()

Accessor to the object’s shadowed id.

Returns: id : int Internal unique identifier.
getVisibility()

Accessor to the object’s visibility state.

Returns: visible : bool Visibility flag.
hasName()

Test if the object is named.

Returns: hasName : bool True if the name is not empty.
hasVisibleName()

Test if the object has a distinguishable name.

Returns: hasVisibleName : bool True if the name is not empty and not the default one.
setName(name)

Accessor to the object’s name.

Parameters: name : str The name of the object.
setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters: id : int Internal unique identifier.
setVisibility(visible)

Accessor to the object’s visibility state.

Parameters: visible : bool Visibility flag.