OrthogonalUniVariatePolynomialFamily¶

class
OrthogonalUniVariatePolynomialFamily
(*args)¶ Base class for orthogonal univariate polynomial factories.
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
(*args)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)¶ x.__init__(…) initializes x; see help(type(x)) for signature

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

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.

getImplementation
(*args)¶ Accessor to the underlying implementation.
Returns: impl : Implementation
The implementation class.

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 :
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.
Returns: 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.
Returns: 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]

setName
(name)¶ Accessor to the object’s name.
Parameters: name : str
The name of the object.
