nlft_qsp.solvers.layer_stripping

Module for inverse NLFT based on the layer stripping algorithm.

Functions:

Name	Description
inlft	Computes the inverse nonlinear Fourier transform using the plain layer stripping algorithm.

inlft(a: Polynomial, b: Polynomial) -> NonLinearFourierSequence

Computes the inverse nonlinear Fourier transform using the plain layer stripping algorithm.

Parameters:

Name	Type	Description	Default
a	Polynomial	The pair \((a, b)\) is the NLFT we want to compute the sequence for.	required
b	Polynomial	The pair \((a, b)\) is the NLFT we want to compute the sequence for.	required

Note

This algorithm is guaranteed to be numerically stable only if \(a\) is outer. To generate an outer complementary polynomial, you can use weiss.complete.

Returns:

Type	Description
NonLinearFourierSequence	A sequence whose NLFT is equal to \((a, b)\).

Source code in nlft_qsp/solvers/layer_stripping.py

def inlft(a: Polynomial, b: Polynomial) -> NonLinearFourierSequence:
    """Computes the inverse nonlinear Fourier transform using the plain layer stripping algorithm.

    Args:
        a (Polynomial): The pair $(a, b)$ is the NLFT we want to compute the sequence for.
        b (Polynomial): The pair $(a, b)$ is the NLFT we want to compute the sequence for.

    Note:
        This algorithm is guaranteed to be numerically stable only if $a$ is outer.
        To generate an outer complementary polynomial, you can use `weiss.complete`.

    Returns:
        A sequence whose NLFT is equal to $(a, b)$.
    """
    if len(a.support()) != len(b.support()) or a.support().stop != 1:
        return ValueError("(a, b) must be in the image of the NLFT.")

    n = a.effective_degree()

    a_star = a.conjugate()
    b = b.duplicate()

    F = []
    for k in range(n+1):
        Fk = b[0]/a_star[0]
        F.append(Fk)

        s = np.sqrt(1 + Fk * np.conj(Fk))
        a_star, b = (a_star + np.conj(Fk) * b).truncate(0, n-k)/s, \
                    (b - Fk * a_star).truncate(b.support_start+1, b.support_start+n-k)/s

        b.support_start -= 1 # divide by z

    return NonLinearFourierSequence(F, support_start=b.support_start)