LegendreFactory¶
(Source code, png, hires.png, pdf)

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.
Returns: 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.
Returns: coefficients :
NumericalPoint
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.
Returns: recurrence_coefficients : list of
NumericalPoint
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.
Returns: nodes :
NumericalPoint
The nodes of the th order quadrature scheme.
weights :
NumericalPoint
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.
Returns: recurrence_coefficients :
NumericalPoint
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.
Returns: roots :
NumericalPoint
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.