venv/lib/python3.10/site-packages/scipy/fft/_fftlog.py

"""Fast Hankel transforms using the FFTLog algorithm.

The implementation closely follows the Fortran code of Hamilton (2000).

added: 14/11/2020 Nicolas Tessore <[email protected]>
"""

from ._basic import _dispatch
from scipy._lib.uarray import Dispatchable
from ._fftlog_backend import fhtoffset
import numpy as np

__all__ = ['fht', 'ifht', 'fhtoffset']


@_dispatch
def fht(a, dln, mu, offset=0.0, bias=0.0):
    r'''Compute the fast Hankel transform.

    Computes the discrete Hankel transform of a logarithmically spaced periodic
    sequence using the FFTLog algorithm [1]_, [2]_.

    Parameters
    ----------
    a : array_like (..., n)
        Real periodic input array, uniformly logarithmically spaced.  For
        multidimensional input, the transform is performed over the last axis.
    dln : float
        Uniform logarithmic spacing of the input array.
    mu : float
        Order of the Hankel transform, any positive or negative real number.
    offset : float, optional
        Offset of the uniform logarithmic spacing of the output array.
    bias : float, optional
        Exponent of power law bias, any positive or negative real number.

    Returns
    -------
    A : array_like (..., n)
        The transformed output array, which is real, periodic, uniformly
        logarithmically spaced, and of the same shape as the input array.

    See Also
    --------
    ifht : The inverse of `fht`.
    fhtoffset : Return an optimal offset for `fht`.

    Notes
    -----
    This function computes a discrete version of the Hankel transform

    .. math::

        A(k) = \int_{0}^{\infty} \! a(r) \, J_\mu(kr) \, k \, dr \;,

    where :math:`J_\mu` is the Bessel function of order :math:`\mu`.  The index
    :math:`\mu` may be any real number, positive or negative.  Note that the
    numerical Hankel transform uses an integrand of :math:`k \, dr`, while the
    mathematical Hankel transform is commonly defined using :math:`r \, dr`.

    The input array `a` is a periodic sequence of length :math:`n`, uniformly
    logarithmically spaced with spacing `dln`,

    .. math::

        a_j = a(r_j) \;, \quad
        r_j = r_c \exp[(j-j_c) \, \mathtt{dln}]

    centred about the point :math:`r_c`.  Note that the central index
    :math:`j_c = (n-1)/2` is half-integral if :math:`n` is even, so that
    :math:`r_c` falls between two input elements.  Similarly, the output
    array `A` is a periodic sequence of length :math:`n`, also uniformly
    logarithmically spaced with spacing `dln`

    .. math::

       A_j = A(k_j) \;, \quad
       k_j = k_c \exp[(j-j_c) \, \mathtt{dln}]

    centred about the point :math:`k_c`.

    The centre points :math:`r_c` and :math:`k_c` of the periodic intervals may
    be chosen arbitrarily, but it would be usual to choose the product
    :math:`k_c r_c = k_j r_{n-1-j} = k_{n-1-j} r_j` to be unity.  This can be
    changed using the `offset` parameter, which controls the logarithmic offset
    :math:`\log(k_c) = \mathtt{offset} - \log(r_c)` of the output array.
    Choosing an optimal value for `offset` may reduce ringing of the discrete
    Hankel transform.

    If the `bias` parameter is nonzero, this function computes a discrete
    version of the biased Hankel transform

    .. math::

        A(k) = \int_{0}^{\infty} \! a_q(r) \, (kr)^q \, J_\mu(kr) \, k \, dr

    where :math:`q` is the value of `bias`, and a power law bias
    :math:`a_q(r) = a(r) \, (kr)^{-q}` is applied to the input sequence.
    Biasing the transform can help approximate the continuous transform of
    :math:`a(r)` if there is a value :math:`q` such that :math:`a_q(r)` is
    close to a periodic sequence, in which case the resulting :math:`A(k)` will
    be close to the continuous transform.

    References
    ----------
    .. [1] Talman J. D., 1978, J. Comp. Phys., 29, 35
    .. [2] Hamilton A. J. S., 2000, MNRAS, 312, 257 (astro-ph/9905191)

    Examples
    --------

    This example is the adapted version of ``fftlogtest.f`` which is provided
    in [2]_. It evaluates the integral

    .. math::

        \int^\infty_0 r^{\mu+1} \exp(-r^2/2) J_\mu(k, r) k dr
        = k^{\mu+1} \exp(-k^2/2) .

    >>> import numpy as np
    >>> from scipy import fft
    >>> import matplotlib.pyplot as plt

    Parameters for the transform.

    >>> mu = 0.0                     # Order mu of Bessel function
    >>> r = np.logspace(-7, 1, 128)  # Input evaluation points
    >>> dln = np.log(r[1]/r[0])      # Step size
    >>> offset = fft.fhtoffset(dln, initial=-6*np.log(10), mu=mu)
    >>> k = np.exp(offset)/r[::-1]   # Output evaluation points

    Define the analytical function.

    >>> def f(x, mu):
    ...     """Analytical function: x^(mu+1) exp(-x^2/2)."""
    ...     return x**(mu + 1)*np.exp(-x**2/2)

    Evaluate the function at ``r`` and compute the corresponding values at
    ``k`` using FFTLog.

    >>> a_r = f(r, mu)
    >>> fht = fft.fht(a_r, dln, mu=mu, offset=offset)

    For this example we can actually compute the analytical response (which in
    this case is the same as the input function) for comparison and compute the
    relative error.

    >>> a_k = f(k, mu)
    >>> rel_err = abs((fht-a_k)/a_k)

    Plot the result.

    >>> figargs = {'sharex': True, 'sharey': True, 'constrained_layout': True}
    >>> fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4), **figargs)
    >>> ax1.set_title(r'$r^{\mu+1}\ \exp(-r^2/2)$')
    >>> ax1.loglog(r, a_r, 'k', lw=2)
    >>> ax1.set_xlabel('r')
    >>> ax2.set_title(r'$k^{\mu+1} \exp(-k^2/2)$')
    >>> ax2.loglog(k, a_k, 'k', lw=2, label='Analytical')
    >>> ax2.loglog(k, fht, 'C3--', lw=2, label='FFTLog')
    >>> ax2.set_xlabel('k')
    >>> ax2.legend(loc=3, framealpha=1)
    >>> ax2.set_ylim([1e-10, 1e1])
    >>> ax2b = ax2.twinx()
    >>> ax2b.loglog(k, rel_err, 'C0', label='Rel. Error (-)')
    >>> ax2b.set_ylabel('Rel. Error (-)', color='C0')
    >>> ax2b.tick_params(axis='y', labelcolor='C0')
    >>> ax2b.legend(loc=4, framealpha=1)
    >>> ax2b.set_ylim([1e-9, 1e-3])
    >>> plt.show()

    '''
    return (Dispatchable(a, np.ndarray),)


@_dispatch
def ifht(A, dln, mu, offset=0.0, bias=0.0):
    r"""Compute the inverse fast Hankel transform.

    Computes the discrete inverse Hankel transform of a logarithmically spaced
    periodic sequence. This is the inverse operation to `fht`.

    Parameters
    ----------
    A : array_like (..., n)
        Real periodic input array, uniformly logarithmically spaced.  For
        multidimensional input, the transform is performed over the last axis.
    dln : float
        Uniform logarithmic spacing of the input array.
    mu : float
        Order of the Hankel transform, any positive or negative real number.
    offset : float, optional
        Offset of the uniform logarithmic spacing of the output array.
    bias : float, optional
        Exponent of power law bias, any positive or negative real number.

    Returns
    -------
    a : array_like (..., n)
        The transformed output array, which is real, periodic, uniformly
        logarithmically spaced, and of the same shape as the input array.

    See Also
    --------
    fht : Definition of the fast Hankel transform.
    fhtoffset : Return an optimal offset for `ifht`.

    Notes
    -----
    This function computes a discrete version of the Hankel transform

    .. math::

        a(r) = \int_{0}^{\infty} \! A(k) \, J_\mu(kr) \, r \, dk \;,

    where :math:`J_\mu` is the Bessel function of order :math:`\mu`.  The index
    :math:`\mu` may be any real number, positive or negative. Note that the
    numerical inverse Hankel transform uses an integrand of :math:`r \, dk`, while the
    mathematical inverse Hankel transform is commonly defined using :math:`k \, dk`.

    See `fht` for further details.
    """
    return (Dispatchable(A, np.ndarray),)