boundlab.utils.Polynomial

class boundlab.utils.Polynomial

Bases: object

Dense univariate polynomial \(P(x) = \sum_k c_k x^k\).

coeffs runs from the constant term up. Coefficients may be tensors, giving an independent polynomial per element. Besides ring arithmetic it supports composition (chain()), argument substitution (apply_add(), apply_ax()), differentiation, and sound interval evaluation (ibp()).

Methods

X	Polynomial representing x^n.
__init__
apply_add	P(x + shift)
apply_ax	P(a x)
apply_ax_reversed	P(x / a) * a^n
chain	P(Q(x))
derivative
ibp	Sound range of \(P\) over \([c - hw,\, c + hw]\).
mul

coeffs: list[Tensor | float]

__init__(coeffs)

property degree: int

static X(n=1)

Polynomial representing x^n.

__add__(other)

__mul__(other)

mul(other, mul=<built-in function mul>)

apply_add(shift, mul=<built-in function mul>)

P(x + shift)

apply_ax(a, mul=<built-in function mul>)

P(a x)

apply_ax_reversed(a, mul=<built-in function mul>)

P(x / a) * a^n

__call__(x)

Call self as a function.

chain(other, mul=<built-in function mul>)

P(Q(x))

derivative(order=1)

ibp(c, hw)

Sound range of \(P\) over \([c - hw,\, c + hw]\).

Substituting \(x = c + hw\,u\) reduces to bounding over \(u \in [-1, 1]\), where each monomial is elementary: odd powers of u range over \(\pm|a_k|\) and even powers over \([\min(a_k, 0), \max(a_k, 0)]\). Summing those per-monomial ranges is sound (though not tight — monomial correlations are ignored).