OrthogonalUniVariatePolynomialFamily¶
- class OrthogonalUniVariatePolynomialFamily(*args)¶
Base class for orthogonal univariate polynomial factories.
Methods
build
(degree)Build the -th order orthogonal univariate polynomial.
Accessor to the object's name.
getId
()Accessor to the object's id.
Accessor to the underlying implementation.
Accessor to the associated probability measure.
getName
()Accessor to the object's name.
Build the -th order quadrature scheme.
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:
- polynomial
OrthogonalUniVariatePolynomial
Requested orthogonal univariate polynomial.
- 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:
- measure
Distribution
The associated probability measure (according to which the polynomials are orthogonal).
- measure
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:
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_coefficients
Point
The recurrence coefficients of the -th order orthogonal univariate polynomial.
- recurrence_coefficients
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:
- roots
Point
The roots of the -th order orthogonal univariate polynomial.
- roots
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.