Norms¶

Functional norms are useful to quantify metrics such as distance between polynomials. The Polynomial class provides mainly two norms that can be used.

$L^2$ norm¶

The $L^2$ norm of a function is defined as

\[ \lVert P \rVert_{L^2}^2 := \frac{1}{2\pi} \int_0^{2\pi} | P(e^{i\theta}) |^2 \ d\theta = \sum_k |p_k|^2 \]

The package easily computes the $L^2$ norm using the rightmost expression, exposed by the methods l2_squared_norm() and l2_norm().

from nlft_qsp import *

P = Polynomial([1, 2, 3])
print(P.l2_squared_norm())
print(P.l2_norm())

(14+0j)
(3.7416573867739413+0j)

Supremum norm¶

We define the supremum norm, also known as the $L^{\infty}$ norm, as follows $$ \lVert P \rVert_{\infty} := \sup_{|z| = 1} |P(z)| $$

Unlike the $L^2$ norm, we cannot compute the supremum norm exactly. The sup_norm() method takes an argument $N$ and computes the maximum absolute value over the polynomial computed in the $N$-th roots of unity $e^{2\pi i k/N}$. We recommend $N$ to be at least twice the highest frequency appearing in the polynomial.

Norms¶

\(L^2\) norm¶

Supremum norm¶