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 $r > 0$ 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`(degree)	Build the $k$ -th order orthogonal univariate polynomial.
`buildCoefficients`(degree)	Build the $k$ -th order orthogonal univariate polynomial coefficients.
`buildRecurrenceCoefficientsCollection`(degree)	Build the recurrence coefficients.
`getAlpha`()	Accessor to the alternative shape parameter $\alpha$ .
`getBeta`()	Accessor to the alternative shape parameter $\beta$ .
`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 $k$ -th order quadrature scheme.
`getRecurrenceCoefficients`(n)	Accessor to the recurrence coefficients of the $k$ -th order.
`getRoots`(n)	Accessor to the recurrence coefficients of the $k$ -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 $k$ -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(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(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))
0 : [  1         0         0        ]
1 : [  0.707107  0        -0.707107 ]

getAlpha()¶

Accessor to the alternative shape parameter $\alpha$ .

Of the Beta distribution.

Returns:

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

getBeta()¶

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()¶

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.

getMeasure()¶

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()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getNodesAndWeights(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(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(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()¶

Accessor to the object’s shadowed id.

Returns:

idint: Internal unique identifier.

getVisibility()¶

Accessor to the object’s visibility state.

Returns:

visiblebool: Visibility flag.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns:

hasVisibleNamebool: True if the name is not empty and not the default one.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters:

idint: Internal unique identifier.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters:

visiblebool: Visibility flag.

Examples using the class¶

Polynomial chaos graphs

Advanced polynomial chaos construction

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

Table of Contents

Previous topic

Next topic

This Page

JacobiFactory¶

Examples using the class¶