JacobiFactory¶

(Source code, png)

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

Create multivariate functions

OpenTURNS

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

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JacobiFactory¶

Examples using the class¶