Check for Equality between Two MultiIndexSet Instances#

import minterpy as mp
import numpy as np

Two instances of MultiIndexSet may be checked for equality in value via the == operator as demonstrated in this guide.

Two instances of MultiIndexSet are equal in value if and only if:

  • The multi-indices elements stored in the exponents property of the two instances are equal and

  • The underlying \(l_p\)-degrees of the two instances are equal.

Equality in value#

Consider the following three-dimensional multi-index sets defined for \(l_p\)-degree \(1.0\):

\[ A = \left\{ (0, 0, 0), (1, 0, 0), (2, 0, 0), (0, 1, 0), (0, 0, 1) \right\} \]

and create two instances of MultiIndexSet with identical exponents and \(l_p\)-degree.

mi_1 = mp.MultiIndexSet(
    np.array([[0, 0, 0], [1, 0, 0], [2, 0, 0], [0, 1, 0], [0, 0, 1]]),
    lp_degree=1.0,
)
mi_2 = mp.MultiIndexSet(
    np.array([[0, 0, 0], [1, 0, 0], [2, 0, 0], [0, 1, 0], [0, 0, 1]]),
    lp_degree=1.0,
)

To check the equality between these two instances, use the == operator as follows:

mi_1 == mi_2

True

Note that the two instances are not identical as shown by the following check:

mi_1 is mi_2

False

as well as the id’s of the instances:

id(mi_1), id(mi_2)

(139836170119408, 139836170120656)

Inequality in value#

As explained in the beginning of the guide, for two instances of MultiIndexSet to be equal both the exponents and \(l_p\)-degree must be equal.

If two instances of MultiIndexSet has the same exponents but a different \(l_p\)-degree, the instances are not equal. Consider a multi-index set with the same exponents as the set above but with \(l_p\)-degree of \(2.0\) (instead of \(1.0\)):

mi_3 = mp.MultiIndexSet(mi_1.exponents, lp_degree=2.0)

mi_1 == mi_3

False

The inequality between two instances may also be checked via the != operator as follows:

mi_1 != mi_3

True

Accordingly, and perhaps the more obvious case, if two instances has the same \(l_p\)-degree but different exponents, the instances are not equal as well. Consider another multi-index set defined also for \(l_p\)-degree \(2.0\):

mi_2 = mp.MultiIndexSet(
    np.array([[0, 0], [1, 0], [2, 0], [0, 1]]),
    lp_degree=mi_1.lp_degree,
)

This multi-index set is of dimension \(2\) instead of \(1\). As expected, the instance is not equal with the previous one:

mi_1 != mi_2

True

Finally, and as expected, two instances having different set of exponents and \(l_p\)-degree are not equal in value:

mi_4 = mp.MultiIndexSet(
    np.array([[0], [1], [2]]),
    lp_degree=np.inf
)
mi_5 = mp.MultiIndexSet(
    np.array([[0, 0], [1, 0], [2, 0]]),
    lp_degree=1.0
)

mi_4 != mi_5

True