Check for Membership (Containment)

import minterpy as mp
import numpy as np

This guide shows how to check the membership of a set of exponents in a MultiIndexSet instance.

Motivating example

As a motivating example, consider the following three multi-index sets of the same dimension and \(l_p\)-degree of \(2.0\):

\[\begin{align*} A & = \left\{ (0, 0, 0), (1, 0, 0), (2, 0, 0), (0, 1, 0), (1, 1, 0) \right\}\\ B & = \left\{ (0, 0, 0), (1, 0, 0) \right\}\\ C & = \left\{ (0, 0, 0), (2, 0, 0), (0, 1, 0), (1, 1, 0), (0, 0, 2) \right\} \end{align*}\]

mi_a = mp.MultiIndexSet(
    np.array([[0, 0, 0], [1, 0, 0], [2, 0, 0], [0, 1, 0], [1, 1, 0]]),
    lp_degree=2.0,
)
mi_b = mp.MultiIndexSet(
    np.array([[0, 0, 0], [1, 0, 0]]),
    lp_degree=2.0,
)
mi_c = mp.MultiIndexSet(
    np.array(
        [[0, 0, 0], [2, 0, 0], [0, 1, 0], [1, 1, 0], [0, 0, 2]],
    ),
    lp_degree=2.0,
)

Check for exponents membership

The method contains_these_exponents() is used to check whether a given set of exponents are contained in the set of exponents of a MultiIndexSet instance. The method takes as input the set of exponents as an integer NumPy array. It returns True if the set of exponents are contained in the MultiIndexSet instance and False otherwise.

For instance, to check if the exponents \(\{(0, 0, 0), (2, 0, 0)\}\) are contained in the multi-index set \(A\) (they are):

mi_a.contains_these_exponents(np.array([[0, 0, 0], [2, 0, 0]]))

True

Attention

Note that the dimension of the set exponents must be consistent with the dimension of the multi-index set; otherwise, an exception will be raised. This behavior can be changed via the parameter expand_dim (default is set to False); if set to True, the set whose the lower dimension is first expanded before the membership check is carried out.

As another example, the exponent \((2, 0, 0)\) is not contained in \(B\):

mi_b.contains_these_exponents(np.array([[2, 0, 0]]))

False

If a set of exponents are checked then it is either all of them are contained in a multi-index set or none of them; a set of exponents cannot be partially contained in a multi-index set.

For instance, the set of exponents \(\{ (0, 0, 0), (1, 0, 0), (2, 0, 0) \}\) is not contained in \(B\) because the element \((2, 0, 0)\) is not contained in \(B\):

mi_b.contains_these_exponents(
    np.array([[0, 0, 0], [1, 0, 0], [2, 0, 0]])
)

False

in Keyword

A single index element may be checked for containment in a MultiIndexSet instance via the in keyword. For instance, the element \((1, 1, 0)\) is in \(C\):

np.array([1, 1, 0]) in mi_c

True

One may also use list instead of an array:

[0, 0, 0] in mi_a

True

Note that the containment check via the in keyword is forgiving in the sense that if the operation does not make any sense the statement simply returns False instead of raising an exception. For example:

"a" in mi_a

False

Different in spatial dimension

The behavior of the contains_these_exponents() method and the in keyword differs with respect to how it handle item of different spatial dimension. In the method call, if there is any difference in the spatial dimension, the method raises an exception. This behavior can be changed via the expand_dim parameter.

For example, the index element \((1, 1)\) when compared to a set whose spatial dimension is \(3\) is expanded to \((1, 1, 0)\) (zero padding) when expand_dim is set to True:

mi_a.contains_these_exponents(np.array([[1, 1]]), expand_dim=True)

True

When using the in keyword, this dimension expansion happens automatically:

[1, 1] in mi_a

True