numpoly.monomial

numpoly.monomial(start: ArrayLike, stop: ArrayLike | None = None, dimensions: int = 1, cross_truncation: ArrayLike = 1.0, graded: bool = False, reverse: bool = False, allocation: int | None = None) → ndpoly

Create an polynomial monomial expansion.

Args:

start:

The minimum polynomial to include. If int is provided, set as lowest total order. If array of int, set as lower order along each indeterminant.

stop:

The maximum shape included. If omitted: stop <- start; start <- 0 If int is provided, set as largest total order. If array of int, set as largest order along each indeterminant.

dimensions:

The indeterminants names used to create the monomials expansion. If int provided, set the number of names to be used.

cross_truncation:

The hyperbolic cross truncation scheme to reduce the number of terms in expansion. In other words, it sets the \(q\) in the \(L_q\)-norm.

graded:

Graded sorting, meaning the indices are always sorted by the index sum. E.g. q0**2*q1**2*q2**2 has an exponent sum of 6, and will therefore be consider larger than both q0**3*q1*q2, q0*q1**2*q2 and q0*q1*q2**2, which all have exponent sum of 5.

reverse:

Reverse lexicographical sorting meaning that q0*q1**3 is considered bigger than q0**3*q1, instead of the opposite.

allocation:

The maximum number of polynomial exponents. If omitted, use length of exponents for allocation.

Return:

Monomial expansion.

Example:

>>> numpoly.monomial(4)
polynomial([1, q0, q0**2, q0**3])
>>> numpoly.monomial(start=4, stop=5, dimensions=2,
...                  graded=True, reverse=True)
polynomial([q1**4, q0*q1**3, q0**2*q1**2, q0**3*q1, q0**4])
>>> numpoly.monomial(2, [3, 4], graded=True)
polynomial([q0**2, q0*q1, q1**2, q1**3])
>>> numpoly.monomial(
...     start=0,
...     stop=4,
...     dimensions=("q2", "q4"),
...     cross_truncation=0.5,
...     graded=True,
...     reverse=True,
... )
polynomial([1, q4, q2, q4**2, q2**2, q4**3, q2**3])