# OrthogonalProductPolynomialFactory¶

class `OrthogonalProductPolynomialFactory`(*args)

Base class for orthogonal multivariate polynomials.

Available constructors:

OrthogonalProductPolynomialFactory(polynomials)

OrthogonalProductPolynomialFactory(polynomials, enumerateFunction)

OrthogonalProductPolynomialFactory(marginals)

Parameters
polynomials

List of orthogonal univariate polynomials factories with the same dimension as the orthogonal basis.

enumerateFunction`EnumerateFunction`

Associates to an integer its multi-index image in the dimension, which is the dimension of the basis. This multi-index represents the collection of degrees of the univariate polynomials.

marginalssequence of `Distribution`

List of physical space marginals.

Notes

OrthogonalProductPolynomialFactory is a particular case of implementation of the `OrthogonalBasis` in the case of polynomial chaos expansion. It provides to the OrthogonalBasis the persistent types of the univariate orthogonal polynomials (e.g. Hermite, Legendre, Laguerre and Jacobi) needed to determine the distribution measure of projection of the input variable. Let’s note that the exact hessian and gradient have been implemented for the product of polynomials. To facilitate the construction of the basis it is recommanded to use the class `StandardDistributionPolynomialFactory`.

Examples

```>>> import openturns as ot
>>> # Define the model
>>> myModel = ot.SymbolicFunction(['x1','x2','x3'], ['1+x1*x2 + 2*x3^2'])
>>> # Create a distribution of dimension 3
>>> Xdist = ot.ComposedDistribution([ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0)])
>>> # Construct the multivariate orthonormal basis
>>> polyColl = [ot.HermiteFactory(), ot.LegendreFactory(), ot.LaguerreFactory(2.75)]
>>> enumerateFunction = ot.LinearEnumerateFunction(3)
>>> productBasis = ot.OrthogonalProductPolynomialFactory(polyColl, enumerateFunction)
```
```>>> # Easier way to construct the same multivariate orthonormal basis
>>> marginals = [Xdist.getMarginal(i) for i in range(Xdist.getDimension())]
>>> productBasis = ot.OrthogonalProductPolynomialFactory(marginals)
```

Methods

 `build`(*args) Get the term of the basis collection at a given index or multi-indices. Accessor to the object’s name. Get the dimension of the Basis. Return the enumerate function. Accessor to the object’s id. Get the measure upon which the basis is orthogonal. Accessor to the object’s name. `getNodesAndWeights`(degrees) Get the nodes and the weights. Get the collection of univariate orthogonal polynomial families. Accessor to the object’s shadowed id. Get the size of the Basis. `getSubBasis`(indices) Get a sub-basis of the Basis. Accessor to the object’s visibility state. Test if the object is named. Test if the object has a distinguishable name. Tell whether the basis is finite. Tell whether the basis is orthogonal. `setName`(name) Accessor to the object’s name. Accessor to the object’s shadowed id. `setVisibility`(visible) Accessor to the object’s visibility state.
`__init__`(*args)

Initialize self. See help(type(self)) for accurate signature.

`build`(*args)

Get the term of the basis collection at a given index or multi-indices.

Parameters
indexint

Indicates the term of the basis which must be constructed. In other words, index is used by a bijection from to (with the dimension of the basis). The bijection is detailed in `EnumerateFunction`.

indicessequence of int

Indicates the term of the basis which must be constructed. In other words, indices is used by a bijection from to (with the dimension of the basis). The bijection is the inverse of `EnumerateFunction`.

Returns
function`Function`

The term of the basis collection at the index index or the inverse of indices.

Examples

```>>> import openturns as ot
>>> # Create an orthogonal basis
>>> polynomialCollection = [ot.LegendreFactory(), ot.LaguerreFactory(), ot.HermiteFactory()]
>>> productBasis = ot.OrthogonalBasis(ot.OrthogonalProductPolynomialFactory(polynomialCollection))
>>> termBasis = productBasis.build(4)
>>> print(termBasis.getEvaluation())
-1.11803 + 3.3541 * x0^2
>>> termBasis = productBasis.build(5)
>>> print(termBasis.getEvaluation())
(1.73205 * x0) * (-1 + x1)
>>> termBasis2 = productBasis.build([1,1,0])
>>> print(termBasis2.getEvaluation())
(1.73205 * x0) * (-1 + x1)
```
`getClassName`()

Accessor to the object’s name.

Returns
class_namestr

The object class name (object.__class__.__name__).

`getDimension`()

Get the dimension of the Basis.

Returns
dimensionint

Dimension of the Basis.

`getEnumerateFunction`()

Return the enumerate function.

Returns
enumerateFunction`EnumerateFunction`

Enumerate function that translates unidimensional indices into multidimensional indices.

`getId`()

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

`getMeasure`()

Get the measure upon which the basis is orthogonal.

Returns
measure`Distribution`

Measure upon which the basis is orthogonal.

Examples

```>>> import openturns as ot
>>> # Create an orthogonal basis
>>> polynomialCollection = [ot.LegendreFactory(), ot.LaguerreFactory(), ot.HermiteFactory()]
>>> productBasis = ot.OrthogonalBasis(ot.OrthogonalProductPolynomialFactory(polynomialCollection))
>>> measure = productBasis.getMeasure()
>>> print(measure.getMarginal(0))
Uniform(a = -1, b = 1)
>>> print(measure.getMarginal(1))
Gamma(k = 1, lambda = 1, gamma = 0)
>>> print(measure.getMarginal(2))
Normal(mu = 0, sigma = 1)
```
`getName`()

Accessor to the object’s name.

Returns
namestr

The name of the object.

`getNodesAndWeights`(degrees)

Get the nodes and the weights.

Parameters
degreeslist of positiv int ( )

List of polynomial orders associated with the univariate polynomials of the basis.

Returns
nodes`Sample`
weights`Point`

Nodes and weights of the multivariate polynomial associated with the marginal degrees ( ) as the tensor product of the marginal orthogonal univariate polynomials, to build multivariate quadrature rules.

Examples

```>>> import openturns as ot
>>> # Define the model
>>> myModel = ot.SymbolicFunction(['x1','x2','x3'], ['1+x1*x2 + 2*x3^2'])
>>> # Create a distribution of dimension 3
>>> Xdist = ot.ComposedDistribution([ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0)])
>>> # Construct the multivariate orthonormal basis
>>> polyColl = [ot.HermiteFactory(), ot.LegendreFactory(), ot.LaguerreFactory(2.75)]
>>> enumerateFunction = ot.LinearEnumerateFunction(3)
>>> productBasis = ot.OrthogonalProductPolynomialFactory(polyColl, enumerateFunction)
>>> nodes, weights = productBasis.getNodesAndWeights([2, 3, 1])
>>> print(nodes[:2])
[ v0        v1        v2        ]
0 : [ -1        -0.774597  3.75     ]
1 : [  1        -0.774597  3.75     ]
>>> print(weights[:2])
[0.138889,0.138889]
```
`getPolynomialFamilyCollection`()

Get the collection of univariate orthogonal polynomial families.

Returns
polynomialFamily

List of orthogonal univariate polynomials families.

`getShadowedId`()

Accessor to the object’s shadowed id.

Returns
idint

Internal unique identifier.

`getSize`()

Get the size of the Basis.

Returns
sizeint

Size of the Basis.

`getSubBasis`(indices)

Get a sub-basis of the Basis.

Parameters
indiceslist of int

Indices of the terms of the Basis put in the sub-basis.

Returns
subBasislist of `Function`

Functions defining a sub-basis.

Examples

```>>> import openturns as ot
>>> dimension = 3
>>> input = ['x0', 'x1', 'x2']
>>> functions = []
>>> for i in range(dimension):
...     functions.append(ot.SymbolicFunction(input, [input[i]]))
>>> basis = ot.Basis(functions)
>>> subbasis = basis.getSubBasis()
>>> print(subbasis.getEvaluation())
[x0,x1,x2]->[x1]
```
`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.

`isFinite`()

Tell whether the basis is finite.

Returns
isFinitebool

True if the basis is finite.

`isOrthogonal`()

Tell whether the basis is orthogonal.

Returns
isOrthogonalbool

True if the basis is orthogonal.

`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.