EnumerateFunction¶

class EnumerateFunction(*args)¶

Enumerate function.

See also

LinearEnumerateFunction, HyperbolicEnumerateFunction

Notes

EnumerateFunction represents a bijection from $\Nset$ to $\Nset^{dim}$ . This bijection is based on a particular procedure of enumerating the set of multi-indices. It begins from the multi-index $\{0,0,\dots,0\}$ .

We associate a multi-index $\{j_{p1}, j_{p2},\dots, j_{pdim}\}$ for every integer $i_p$ in $\Nset$ :

For more details, let us consider any $i_p, i_q \in \Nset$ :

if $|i_p-i_q|\leq 1$ then $|\sum_{k=1}^{dim} (j_{pk}- j_{qk})| \leq 1$ . This proposition provides a necessary but unsufficient condition for the construction of the bijection. Another assumption was done indicating the way of iteration. Below an example showing this assumption.

Example for $dim=2$ :

$phi(0) &= [0 ~ 0] \\ phi(1) &= [1 ~ 0] \\ phi(2) &= [0 ~ 1] \\ phi(3) &= [2 ~ 0] \\ phi(4) &= [1 ~ 1] \\ phi(5) &= [0 ~ 2] \\ phi(6) &= [3 ~ 0]$

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). In fact, 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).

In order to know the degree of the $k^{th}$ polynomial of the multivariate basis, it is enough to sum all the integers given in the list.

Examples

>>> import openturns as ot
>>> enumerateFunction = ot.LinearEnumerateFunction(2)
>>> for i in range(6):
...     print(enumerateFunction(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 cardinal of indices of degree inferior or equal to a given value.
`getMaximumDegreeStrataIndex`(maximumDegree)	Get the largest index of the strata of degree lower or equal to a given value.
`getName`()	Accessor to the object's name.
`getStrataCardinal`(strataIndex)	Get the number of members of the basis associated to a given strata.
`getStrataCumulatedCardinal`(strataIndex)	Get the cardinal of the cumulated strata above or equal to the given strata.
`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.

__init__(*args)¶

getBasisSizeFromTotalDegree(maximumDegree)¶

Get the basis size corresponding to a total degree.

Parameters

maximumDegreeint: Degree.

Returns

sizeint: Number of members of the basis of degree $max \leq maximumDegree$ .

Examples

>>> import openturns as ot
>>> ef = ot.LinearEnumerateFunction(2)
>>> size = ef.getBasisSizeFromTotalDegree(3)

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 cardinal of indices of degree inferior or equal to a given value.

Parameters

maximumDegreeint: Degree.

Returns

cardinalint: Cardinal of indices of degree $max \leq maximumDegree$ .

Examples

>>> import openturns as ot
>>> enumerateFunction = ot.LinearEnumerateFunction(2)
>>> for i in range(6):
...     indices = enumerateFunction(i)
...     degree = sum(indices)
...     print(str(int(degree))+' '+str(indices))
0 [0,0]
1 [1,0]
1 [0,1]
2 [2,0]
2 [1,1]
2 [0,2]
>>> print(enumerateFunction.getMaximumDegreeCardinal(2))
6

getMaximumDegreeStrataIndex(maximumDegree)¶

Get the largest index of the strata of degree lower or equal to a given value.

Parameters

maximumDegreeint: Degree.

Returns

indexint: Index of the last strata of degree $max \leq maximumDegree$ .

Examples

>>> import openturns as ot
>>> enumerateFunction = ot.LinearEnumerateFunction(2)
>>> for i in [1, 2]:
...     indices = enumerateFunction(i)
...     strataIndex = sum(indices) + 1
...     print(str(int(strataIndex))+' '+str(indices))
2 [1,0]
2 [0,1]
>>> print(enumerateFunction.getMaximumDegreeStrataIndex(2))
2

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getStrataCardinal(strataIndex)¶

Get the number of members of the basis associated to a given strata.

Parameters

strataIndexint: Index of the strata in the hierarchical basis. In the context of product of polynomial basis, this is the total polynom degree.

Returns

cardinalint: Number of members of the basis associated to the strata strataIndex. In the context of product of polynomial basis, this is the number of polynoms of the basis which total degree is strataIndex.

Examples

>>> import openturns as ot
>>> enumerateFunction = ot.LinearEnumerateFunction(2)
>>> for i in [3, 4, 5]:
...     indices = enumerateFunction(i)
...     degree = sum(indices)
...     print(str(int(degree))+' '+str(indices))
2 [2,0]
2 [1,1]
2 [0,2]
>>> print(enumerateFunction.getStrataCardinal(2))
3

getStrataCumulatedCardinal(strataIndex)¶

Get the cardinal of the cumulated strata above or equal to the given strata.

Parameters

strataIndexint: Index of the strata in the hierarchical basis. In the context of product of polynomial basis, this is the total polynomial degree.

Returns

cardinalint: Number of members of the basis associated to the strata of index inferior or equal to strataIndex. In the context of product of polynomial basis, this is the number of polynomials of the basis which total degree is inferior or equal to strataIndex.

Examples

>>> import openturns as ot
>>> enumerateFunction = ot.LinearEnumerateFunction(2)
>>> for i in range(6):
...     indices = enumerateFunction(i)
...     degree = sum(indices)
...     print(str(int(degree))+' '+str(indices))
0 [0,0]
1 [1,0]
1 [0,1]
2 [2,0]
2 [1,1]
2 [0,2]
>>> print(enumerateFunction.getStrataCumulatedCardinal(2))
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
>>> enumerateFunction = ot.LinearEnumerateFunction(2)
>>> for i in range(6):
...     print(str(i)+' '+str(enumerateFunction(i)))
0 [0,0]
1 [1,0]
2 [0,1]
3 [2,0]
4 [1,1]
5 [0,2]
>>> print(enumerateFunction.inverse([1,1]))
4

setDimension(dimension)¶

Set the dimension of the EnumerateFunction.

Parameters

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

setName(name)¶

Accessor to the object’s name.

Parameters

namestr: The name of the object.

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