JacobiFactory¶

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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 $r > 0$ of the Beta distribution.

If parameters_set == ot.JacobiFactory.ANALYSIS: alternative shape parameter $\alpha = t - r - 1 > -1$ of the Beta distribution.

arg2float

If parameters_set == ot.JacobiFactory.PROBABILITY: default shape parameter $t > r$ of the Beta distribution.

If parameters_set == ot.JacobiFactory.ANALYSIS: alternative shape parameter $\beta = r - 1$ 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).

See also

StandardDistributionPolynomialFactory

Notes

Any sequence of orthogonal polynomials has a recurrence formula relating any three consecutive polynomials as follows:

$P_{i + 1} = (a_i x + b_i) P_i + c_i P_{i - 1}, \quad 1 < i$

The recurrence coefficients for the Jacobi polynomials come analytically and read:

$\begin{array}{rcl} a_i & = & \displaystyle K_{2,i} (2 i + \alpha + \beta + 2) \\ b_i & = & \displaystyle K_{2,i} \frac{(\alpha - \beta)(\alpha + \beta)} {2 i + \alpha + \beta} \\ c_i & = & \displaystyle - \frac{2 i + \alpha + \beta + 2} {2 i + \alpha + \beta} \left[(i + \alpha) (i + \beta) (i + \alpha + \beta) i \frac{K_{1,i}} {2 i + \alpha + \beta - 1} \right]^{1/2} \end{array}, \quad 1 < i$

where $\alpha$ and $\beta$ are the alternative shape parameters of the Beta distribution, and:

$\begin{array}{rcl} K_{1,i} & = & \displaystyle \frac{2 i + \alpha + \beta + 3} {(i + 1) (i + \alpha + 1) (i + \beta + 1) (i + \alpha + \beta + 1)} \\ K_{2,i} & = & \displaystyle \frac{1}{2} \sqrt{(2 i + \alpha + \beta + 1) K_{1,i}} \end{array}, \quad i > 1$

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 $k$ -th order orthogonal univariate polynomial.
`buildCoefficients`(self, degree)	Build the $k$ -th order orthogonal univariate polynomial coefficients.
`buildRecurrenceCoefficientsCollection`(self, …)	Build the recurrence coefficients.
`getAlpha`(self)	Accessor to the alternative shape parameter $\alpha$ .
`getBeta`(self)	Accessor to the alternative shape parameter $\beta$ .
`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 $k$ -th order quadrature scheme.
`getRecurrenceCoefficients`(self, n)	Accessor to the recurrence coefficients of the $k$ -th order.
`getRoots`(self, n)	Accessor to the recurrence coefficients of the $k$ -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 $k$ -th order orthogonal univariate polynomial.

Parameters

kint, $0 \leq k$: 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

buildCoefficients(self, degree)¶

Build the $k$ -th order orthogonal univariate polynomial coefficients.

Parameters

kint, $0 \leq k$: Polynomial order.

Returns

coefficientsPoint: 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(self, degree)¶

Build the recurrence coefficients.

Build the recurrence coefficients of the orthogonal univariate polynomial family up to the $k$ -th order.

Parameters

kint, $0 \leq k$: Polynomial order.

Returns

recurrence_coefficientslist of Point: 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]]

getAlpha(self)¶

Accessor to the alternative shape parameter $\alpha$ .

Of the Beta distribution.

Returns

alphafloat: Alternative shape parameter $\alpha = r - 1$ of the Beta distribution.

getBeta(self)¶

Accessor to the alternative shape parameter $\beta$ .

Of the Beta distribution.

Returns

betafloat: Alternative shape parameter $\beta = t - r - 1$ 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

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 $w(x) \di{x}$ if and only if their dot product:

$\langle P, Q \rangle = \int_{\alpha}^{\beta} P(x) Q(x) w(x)\di{x} = 0$

where $\alpha \in \Rset \cup \{-\infty\}$ and $\beta \in \Rset \cup \{+\infty\}$ .

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 $k$ -th order quadrature scheme.

Associated with the orthogonal univariate polynomials family.

Parameters

kint, $0 < k$: Polynomial order.

Returns

nodesPoint: The nodes of the $k$ -th order quadrature scheme.
weightsPoint: The weights of the $k$ -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(self, n)¶

Accessor to the recurrence coefficients of the $k$ -th order.

Of the orthogonal univariate polynomial.

Parameters

kint, $0 \leq k$: Polynomial order.

Returns

recurrence_coefficientsPoint: The recurrence coefficients of the $k$ -th order orthogonal univariate polynomial.

Notes

Any sequence of orthogonal polynomials has a recurrence formula relating any three consecutive polynomials as follows:

$P_{-1}=0, P_0=1, P_{n + 1} = (a_n x + b_n) P_n + c_n P_{n - 1}, \quad n > 1$

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 $k$ -th order.

Of the orthogonal univariate polynomial.

Parameters

kint, $k > 0$: Polynomial order.

Returns

rootsPoint: The roots of the $k$ -th order orthogonal univariate polynomial.

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.

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