OrthogonalUniVariatePolynomialFamily

class OrthogonalUniVariatePolynomialFamily(*args)

Base class for orthogonal univariate polynomial factories.

See also

StandardDistributionPolynomialFactory

Methods

build(degree)	Build the -th order orthogonal univariate polynomial.
getClassName()	Accessor to the object's name.
getId()	Accessor to the object's id.
getImplementation()	Accessor to the underlying implementation.
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.
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

getClassName()

Accessor to the object’s name.

Returns:

class_namestr

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

getId()

Accessor to the object’s id.

Returns:

idint

Internal unique identifier.

getImplementation()

Accessor to the underlying implementation.

Returns:

implImplementation

A copy of the underlying implementation object.

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]

setName(name)

Accessor to the object’s name.

Parameters:

namestr

The name of the object.