JacobiFactory¶
(Source code, png, hires.png, pdf)
-
class
JacobiFactory
(*args)¶ Jacobi specific orthonormal univariate polynomial family.
For the
Beta
distribution.- Available constructors:
Jacobi(arg1=1.0, arg2=1.0, parameters_set=ot.JacobiFactory.ANALYSIS)
- Parameters
- arg1float
If parameters_set == ot.JacobiFactory.PROBABILITY: default shape parameter of the
Beta
distribution.If parameters_set == ot.JacobiFactory.ANALYSIS: alternative shape parameter of the
Beta
distribution.- arg2float
If parameters_set == ot.JacobiFactory.PROBABILITY: default shape parameter of the
Beta
distribution.If parameters_set == ot.JacobiFactory.ANALYSIS: alternative shape parameter of the
Beta
distribution.- parameters_setint, optional
Integer telling which parameters set is used for defining the distribution (amongst ot.JacobiFactory.ANALYSIS, ot.JacobiFactory.PROBABILITY).
Notes
Any sequence of orthogonal polynomials has a recurrence formula relating any three consecutive polynomials as follows:
The recurrence coefficients for the Jacobi polynomials come analytically and read:
where and are the alternative shape parameters of the
Beta
distribution, and:Examples
>>> import openturns as ot >>> polynomial_factory = ot.JacobiFactory() >>> for i in range(3): ... print(polynomial_factory.build(i)) 1 2.23607 * X -0.935414 + 4.67707 * X^2
Methods
build
(self, degree)Build the -th order orthogonal univariate polynomial.
buildCoefficients
(self, degree)Build the -th order orthogonal univariate polynomial coefficients.
Build the recurrence coefficients.
getAlpha
(self)Accessor to the alternative shape parameter .
getBeta
(self)Accessor to the alternative shape parameter .
getClassName
(self)Accessor to the object’s name.
getId
(self)Accessor to the object’s id.
getMeasure
(self)Accessor to the associated probability measure.
getName
(self)Accessor to the object’s name.
getNodesAndWeights
(self, n)Build the -th order quadrature scheme.
getRecurrenceCoefficients
(self, n)Accessor to the recurrence coefficients of the -th order.
getRoots
(self, n)Accessor to the recurrence coefficients of the -th order.
getShadowedId
(self)Accessor to the object’s shadowed id.
getVisibility
(self)Accessor to the object’s visibility state.
hasName
(self)Test if the object is named.
hasVisibleName
(self)Test if the object has a distinguishable name.
setName
(self, name)Accessor to the object’s name.
setShadowedId
(self, id)Accessor to the object’s shadowed id.
setVisibility
(self, visible)Accessor to the object’s visibility state.
-
__init__
(self, \*args)¶ Initialize self. See help(type(self)) for accurate signature.
-
build
(self, 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
-
buildCoefficients
(self, degree)¶ Build the -th order orthogonal univariate polynomial coefficients.
- Parameters
- kint,
Polynomial order.
- Returns
- coefficients
Point
Coefficients of the requested orthogonal univariate polynomial.
- coefficients
Examples
>>> import openturns as ot >>> polynomial_factory = ot.HermiteFactory() >>> print(polynomial_factory.buildCoefficients(2)) [-0.707107,0,0.707107]
-
buildRecurrenceCoefficientsCollection
(self, degree)¶ Build the recurrence coefficients.
Build the recurrence coefficients of the orthogonal univariate polynomial family up to the -th order.
- Parameters
- kint,
Polynomial order.
- Returns
- recurrence_coefficientslist of
Point
All the tecurrence coefficients up to the requested order.
- recurrence_coefficientslist of
Examples
>>> import openturns as ot >>> polynomial_factory = ot.HermiteFactory() >>> print(polynomial_factory.buildRecurrenceCoefficientsCollection(2)) [[1,0,0],[0.707107,0,-0.707107]]
-
getAlpha
(self)¶ Accessor to the alternative shape parameter .
Of the
Beta
distribution.- Returns
- alphafloat
Alternative shape parameter of the
Beta
distribution.
-
getBeta
(self)¶ Accessor to the alternative shape parameter .
Of the
Beta
distribution.- Returns
- betafloat
Alternative shape parameter of the
Beta
distribution.
-
getClassName
(self)¶ Accessor to the object’s name.
- Returns
- class_namestr
The object class name (object.__class__.__name__).
-
getId
(self)¶ Accessor to the object’s id.
- Returns
- idint
Internal unique identifier.
-
getMeasure
(self)¶ 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
(self)¶ Accessor to the object’s name.
- Returns
- namestr
The name of the object.
-
getNodesAndWeights
(self, 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
(self, 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
(self, 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]
-
getShadowedId
(self)¶ Accessor to the object’s shadowed id.
- Returns
- idint
Internal unique identifier.
-
getVisibility
(self)¶ Accessor to the object’s visibility state.
- Returns
- visiblebool
Visibility flag.
-
hasName
(self)¶ Test if the object is named.
- Returns
- hasNamebool
True if the name is not empty.
-
hasVisibleName
(self)¶ Test if the object has a distinguishable name.
- Returns
- hasVisibleNamebool
True if the name is not empty and not the default one.
-
setName
(self, name)¶ Accessor to the object’s name.
- Parameters
- namestr
The name of the object.
-
setShadowedId
(self, id)¶ Accessor to the object’s shadowed id.
- Parameters
- idint
Internal unique identifier.
-
setVisibility
(self, visible)¶ Accessor to the object’s visibility state.
- Parameters
- visiblebool
Visibility flag.