224 lines
7.7 KiB
Python
224 lines
7.7 KiB
Python
"""Fast Hankel transforms using the FFTLog algorithm.
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The implementation closely follows the Fortran code of Hamilton (2000).
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added: 14/11/2020 Nicolas Tessore <n.tessore@ucl.ac.uk>
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"""
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from ._basic import _dispatch
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from scipy._lib.uarray import Dispatchable
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from ._fftlog_backend import fhtoffset
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import numpy as np
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__all__ = ['fht', 'ifht', 'fhtoffset']
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@_dispatch
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def fht(a, dln, mu, offset=0.0, bias=0.0):
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r'''Compute the fast Hankel transform.
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Computes the discrete Hankel transform of a logarithmically spaced periodic
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sequence using the FFTLog algorithm [1]_, [2]_.
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Parameters
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----------
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a : array_like (..., n)
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Real periodic input array, uniformly logarithmically spaced. For
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multidimensional input, the transform is performed over the last axis.
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dln : float
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Uniform logarithmic spacing of the input array.
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mu : float
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Order of the Hankel transform, any positive or negative real number.
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offset : float, optional
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Offset of the uniform logarithmic spacing of the output array.
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bias : float, optional
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Exponent of power law bias, any positive or negative real number.
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Returns
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-------
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A : array_like (..., n)
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The transformed output array, which is real, periodic, uniformly
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logarithmically spaced, and of the same shape as the input array.
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See Also
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--------
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ifht : The inverse of `fht`.
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fhtoffset : Return an optimal offset for `fht`.
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Notes
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-----
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This function computes a discrete version of the Hankel transform
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.. math::
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A(k) = \int_{0}^{\infty} \! a(r) \, J_\mu(kr) \, k \, dr \;,
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where :math:`J_\mu` is the Bessel function of order :math:`\mu`. The index
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:math:`\mu` may be any real number, positive or negative. Note that the
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numerical Hankel transform uses an integrand of :math:`k \, dr`, while the
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mathematical Hankel transform is commonly defined using :math:`r \, dr`.
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The input array `a` is a periodic sequence of length :math:`n`, uniformly
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logarithmically spaced with spacing `dln`,
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.. math::
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a_j = a(r_j) \;, \quad
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r_j = r_c \exp[(j-j_c) \, \mathtt{dln}]
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centred about the point :math:`r_c`. Note that the central index
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:math:`j_c = (n-1)/2` is half-integral if :math:`n` is even, so that
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:math:`r_c` falls between two input elements. Similarly, the output
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array `A` is a periodic sequence of length :math:`n`, also uniformly
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logarithmically spaced with spacing `dln`
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.. math::
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A_j = A(k_j) \;, \quad
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k_j = k_c \exp[(j-j_c) \, \mathtt{dln}]
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centred about the point :math:`k_c`.
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The centre points :math:`r_c` and :math:`k_c` of the periodic intervals may
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be chosen arbitrarily, but it would be usual to choose the product
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:math:`k_c r_c = k_j r_{n-1-j} = k_{n-1-j} r_j` to be unity. This can be
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changed using the `offset` parameter, which controls the logarithmic offset
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:math:`\log(k_c) = \mathtt{offset} - \log(r_c)` of the output array.
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Choosing an optimal value for `offset` may reduce ringing of the discrete
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Hankel transform.
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If the `bias` parameter is nonzero, this function computes a discrete
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version of the biased Hankel transform
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.. math::
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A(k) = \int_{0}^{\infty} \! a_q(r) \, (kr)^q \, J_\mu(kr) \, k \, dr
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where :math:`q` is the value of `bias`, and a power law bias
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:math:`a_q(r) = a(r) \, (kr)^{-q}` is applied to the input sequence.
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Biasing the transform can help approximate the continuous transform of
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:math:`a(r)` if there is a value :math:`q` such that :math:`a_q(r)` is
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close to a periodic sequence, in which case the resulting :math:`A(k)` will
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be close to the continuous transform.
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References
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----------
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.. [1] Talman J. D., 1978, J. Comp. Phys., 29, 35
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.. [2] Hamilton A. J. S., 2000, MNRAS, 312, 257 (astro-ph/9905191)
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Examples
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--------
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This example is the adapted version of ``fftlogtest.f`` which is provided
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in [2]_. It evaluates the integral
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.. math::
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\int^\infty_0 r^{\mu+1} \exp(-r^2/2) J_\mu(k, r) k dr
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= k^{\mu+1} \exp(-k^2/2) .
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>>> import numpy as np
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>>> from scipy import fft
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>>> import matplotlib.pyplot as plt
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Parameters for the transform.
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>>> mu = 0.0 # Order mu of Bessel function
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>>> r = np.logspace(-7, 1, 128) # Input evaluation points
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>>> dln = np.log(r[1]/r[0]) # Step size
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>>> offset = fft.fhtoffset(dln, initial=-6*np.log(10), mu=mu)
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>>> k = np.exp(offset)/r[::-1] # Output evaluation points
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Define the analytical function.
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>>> def f(x, mu):
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... """Analytical function: x^(mu+1) exp(-x^2/2)."""
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... return x**(mu + 1)*np.exp(-x**2/2)
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Evaluate the function at ``r`` and compute the corresponding values at
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``k`` using FFTLog.
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>>> a_r = f(r, mu)
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>>> fht = fft.fht(a_r, dln, mu=mu, offset=offset)
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For this example we can actually compute the analytical response (which in
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this case is the same as the input function) for comparison and compute the
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relative error.
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>>> a_k = f(k, mu)
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>>> rel_err = abs((fht-a_k)/a_k)
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Plot the result.
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>>> figargs = {'sharex': True, 'sharey': True, 'constrained_layout': True}
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>>> fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4), **figargs)
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>>> ax1.set_title(r'$r^{\mu+1}\ \exp(-r^2/2)$')
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>>> ax1.loglog(r, a_r, 'k', lw=2)
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>>> ax1.set_xlabel('r')
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>>> ax2.set_title(r'$k^{\mu+1} \exp(-k^2/2)$')
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>>> ax2.loglog(k, a_k, 'k', lw=2, label='Analytical')
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>>> ax2.loglog(k, fht, 'C3--', lw=2, label='FFTLog')
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>>> ax2.set_xlabel('k')
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>>> ax2.legend(loc=3, framealpha=1)
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>>> ax2.set_ylim([1e-10, 1e1])
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>>> ax2b = ax2.twinx()
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>>> ax2b.loglog(k, rel_err, 'C0', label='Rel. Error (-)')
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>>> ax2b.set_ylabel('Rel. Error (-)', color='C0')
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>>> ax2b.tick_params(axis='y', labelcolor='C0')
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>>> ax2b.legend(loc=4, framealpha=1)
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>>> ax2b.set_ylim([1e-9, 1e-3])
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>>> plt.show()
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'''
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return (Dispatchable(a, np.ndarray),)
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@_dispatch
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def ifht(A, dln, mu, offset=0.0, bias=0.0):
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r"""Compute the inverse fast Hankel transform.
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Computes the discrete inverse Hankel transform of a logarithmically spaced
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periodic sequence. This is the inverse operation to `fht`.
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Parameters
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----------
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A : array_like (..., n)
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Real periodic input array, uniformly logarithmically spaced. For
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multidimensional input, the transform is performed over the last axis.
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dln : float
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Uniform logarithmic spacing of the input array.
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mu : float
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Order of the Hankel transform, any positive or negative real number.
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offset : float, optional
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Offset of the uniform logarithmic spacing of the output array.
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bias : float, optional
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Exponent of power law bias, any positive or negative real number.
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Returns
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-------
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a : array_like (..., n)
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The transformed output array, which is real, periodic, uniformly
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logarithmically spaced, and of the same shape as the input array.
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See Also
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--------
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fht : Definition of the fast Hankel transform.
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fhtoffset : Return an optimal offset for `ifht`.
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Notes
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-----
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This function computes a discrete version of the Hankel transform
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.. math::
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a(r) = \int_{0}^{\infty} \! A(k) \, J_\mu(kr) \, r \, dk \;,
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where :math:`J_\mu` is the Bessel function of order :math:`\mu`. The index
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:math:`\mu` may be any real number, positive or negative. Note that the
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numerical inverse Hankel transform uses an integrand of :math:`r \, dk`, while the
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mathematical inverse Hankel transform is commonly defined using :math:`k \, dk`.
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See `fht` for further details.
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"""
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return (Dispatchable(A, np.ndarray),)
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