EnumerateFunction¶

class EnumerateFunction(*args)¶

Enumerate function.

See also

LinearEnumerateFunction, HyperbolicAnisotropicEnumerateFunction, NormInfEnumerateFunction

Notes

EnumerateFunction represents a bijection $\tau$ from $\Nset$ to $\Nset^{dim}$ . This bijection is based on a particular enumeration rule of the set of multi-indices.

For every integer $i \in \Nset$ , we associate a multi-index

$\boldsymbol{\tau}(i) = (\alpha_1,\dots, \alpha_{n_X}) \in {\Nset}^{n_X}.$

The first multi-index is

$\boldsymbol{\alpha}(0) = (0, 0, \dots, 0).$

Let $i, j \in \Nset$ be any pair of indices. If $|i - j|\leq 1$ then

$\left|\sum_{k=1}^{n_X} \left[ \alpha_k(i) - \alpha_k(j) \right]\right| \leq 1$

where $\alpha_k(i)$ represents the k-th component of the multi-index $\boldsymbol{\alpha}(i)$ .

This provides a necessary but insufficient condition for the construction of the bijection: a supplementary hypothesis must be made.

For example, consider the dimension $\textrm{dim}=2$ . The following mapping is an enumerate function:

$\phi(0) &= (0, \; 0) \\ \phi(1) &= (1, \; 0) \\ \phi(2) &= (0, \; 1) \\ \phi(3) &= (2, \; 0) \\ \phi(4) &= (1, \; 1) \\ \phi(5) &= (0, \; 2) \\ \phi(6) &= (3, \; 0)$

For the functional expansion (respectively polynomial chaos expansion), the multi-index $\vect{i_p}$ represents the collection of degrees of the selected orthogonal functions (respectively orthogonal polynomials). More precisely, after the selection of the type of orthogonal functions (respectively orthogonal polynomials) for the construction of the orthogonal basis, the EnumerateFunction characterizes the term of the basis by providing the degrees of the univariate functions (respectively univariate polynomials).

The total degree of the $k^{th}$ polynomial of the multivariate basis is equal to the sum of all the integers given in the list.

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> for i in range(6):
...     print(enum_func(i))
[0,0]
[1,0]
[0,1]
[2,0]
[1,1]
[0,2]

Methods

`__call__`(index)	Call self as a function.
`getBasisSizeFromTotalDegree`(maximumDegree)	Get the basis size corresponding to a total degree.
`getClassName`()	Accessor to the object's name.
`getDimension`()	Return the dimension of the EnumerateFunction.
`getId`()	Accessor to the object's id.
`getImplementation`()	Accessor to the underlying implementation.
`getMaximumDegreeCardinal`(maximumDegree)	Get the number of terms of total degree below a threshold.
`getMaximumDegreeStrataIndex`(maximumDegree)	Get the largest index of the strata containing terms of maximum degree.
`getName`()	Accessor to the object's name.
`getStrataCardinal`(strataIndex)	Get the number of terms in the basis inside a given strata.
`getStrataCumulatedCardinal`(strataIndex)	Get the number of terms in the basis inside a range of stratas.
`inverse`(indices)	Get the antecedent of a indices list in the EnumerateFunction.
`setDimension`(dimension)	Set the dimension of the EnumerateFunction.
`setName`(name)	Accessor to the object's name.

getUpperBound
setUpperBound

__init__(*args)¶

getBasisSizeFromTotalDegree(maximumDegree)¶

Get the basis size corresponding to a total degree.

Parameters:

max_degint: Maximum total degree.

Returns:

sizeint: Number of terms in the basis of total degree $\leq max_deg$ .

Notes

In the specific context of a linear enumeration (LinearEnumerateFunction) this is also the cumulated cardinal of stratas up to max_deg.

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> enum_func.getBasisSizeFromTotalDegree(3)
10
>>> enum_func.getStrataCumulatedCardinal(3)
10

getClassName()¶

Accessor to the object’s name.

Returns:

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

getDimension()¶

Return the dimension of the EnumerateFunction.

Returns:

dimint, $dim \geq 0$: Dimension of the EnumerateFunction.

getId()¶

Accessor to the object’s id.

Returns:

idint: Internal unique identifier.

getImplementation()¶

Accessor to the underlying implementation.

Returns:

implImplementation: A copy of the underlying implementation object.

getMaximumDegreeCardinal(maximumDegree)¶

Get the number of terms of total degree below a threshold.

Parameters:

max_degint: Maximum total degree.

Returns:

cardinalint: Number of terms in the basis of total degree $\leq max_deg$ .

Notes

In the specific context of a linear enumeration (LinearEnumerateFunction) this is also the cumulated cardinal of stratas of index $\leq max_deg$ .

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> enum_func.getMaximumDegreeCardinal(2)
6

getMaximumDegreeStrataIndex(maximumDegree)¶

Get the largest index of the strata containing terms of maximum degree.

Parameters:

max_degint: Maximum total degree.

Returns:

indexint: Index of the last strata that contains terms of total degree $\leq max_deg$ .

Notes

In the specific context of a linear enumeration (LinearEnumerateFunction) this is the strata of index max_deg.

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> enum_func.getMaximumDegreeStrataIndex(2)
2

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getStrataCardinal(strataIndex)¶

Get the number of terms in the basis inside a given strata.

Parameters:

strataIndexint: Index of the strata of the tensorized basis.

Returns:

cardinalint: Number of terms in the basis inside the strata strataIndex.

Notes

In the specific context of a linear enumeration (LinearEnumerateFunction) the strata strataIndex consists of a hyperplane of all the terms of total degree strataIndex, and its cardinal is strataIndex + 1.

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> enum_func.getStrataCardinal(2)
3

getStrataCumulatedCardinal(strataIndex)¶

Get the number of terms in the basis inside a range of stratas.

Parameters:

strataIndexint: Index of the strata of the tensorized basis.

Returns:

cardinalint: Number of terms in the basis inside the stratas of index lower of equal to strataIndex.

Notes

The number of terms is the total of terms inside each consecutive stratas. In the specific context of a linear enumeration (LinearEnumerateFunction) this returns the number of terms of maximal total degree strataIndex.

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> enum_func.getStrataCumulatedCardinal(2)
6
>>> sum([enum_func.getStrataCardinal(i) for i in range(3)])
6

inverse(indices)¶

Get the antecedent of a indices list in the EnumerateFunction.

Parameters:

multiIndexsequence of int: List of indices.

Returns:

antecedentint: Represents the antecedent of the multiIndex in the EnumerateFunction.

Examples

>>> import openturns as ot
>>> dim = 2
>>> enum_func = ot.LinearEnumerateFunction(dim)
>>> for i in range(6):
...     print(str(i)+' '+str(enum_func(i)))
0 [0,0]
1 [1,0]
2 [0,1]
3 [2,0]
4 [1,1]
5 [0,2]
>>> print(enum_func.inverse([1,1]))
4