/
_fftlog.py
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/
_fftlog.py
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'''Fast Hankel transforms using the FFTLog algorithm.
The implementation closely follows the Fortran code of Hamilton (2000).
'''
import math
from warnings import warn
import cupy
from cupyx.scipy.fft import _fft
from cupyx.scipy.special import loggamma, poch
try:
# fht only exists in SciPy >= 1.7
from scipy.fft import fht as _fht
_scipy_fft = _fft._scipy_fft
del _fht
except ImportError:
class _DummyModule:
def __getattr__(self, name):
return None
_scipy_fft = _DummyModule()
# Note scipy also defines fhtoffset but this only operates on scalars
__all__ = ['fht', 'ifht']
# constants
LN_2 = math.log(2)
@_fft._implements(_scipy_fft.fht)
def fht(a, dln, mu, offset=0.0, bias=0.0):
"""Compute the fast Hankel transform.
Computes the discrete Hankel transform of a logarithmically spaced periodic
sequence using the FFTLog algorithm [1]_, [2]_.
Parameters
----------
a : cupy.ndarray (..., 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 : cupy.ndarray (..., n)
The transformed output array, which is real, periodic, uniformly
logarithmically spaced, and of the same shape as the input array.
See Also
--------
:func:`scipy.special.fht`
:func:`scipy.special.fhtoffset` : Return an optimal offset for `fht`.
References
----------
.. [1] Talman J. D., 1978, J. Comp. Phys., 29, 35
.. [2] Hamilton A. J. S., 2000, MNRAS, 312, 257 (astro-ph/9905191)
"""
# size of transform
n = a.shape[-1]
# bias input array
if bias != 0:
# a_q(r) = a(r) (r/r_c)^{-q}
j_c = (n-1)/2
j = cupy.arange(n)
a = a * cupy.exp(-bias*(j - j_c)*dln)
# compute FHT coefficients
u = fhtcoeff(n, dln, mu, offset=offset, bias=bias)
# transform
A = _fhtq(a, u)
# bias output array
if bias != 0:
# A(k) = A_q(k) (k/k_c)^{-q} (k_c r_c)^{-q}
A *= cupy.exp(-bias*((j - j_c)*dln + offset))
return A
@_fft._implements(_scipy_fft.ifht)
def ifht(A, dln, mu, offset=0.0, bias=0.0):
"""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 : cupy.ndarray (..., 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 : cupy.ndarray (..., n)
The transformed output array, which is real, periodic, uniformly
logarithmically spaced, and of the same shape as the input array.
See Also
--------
:func:`scipy.special.ifht`
:func:`scipy.special.fhtoffset` : Return an optimal offset for `fht`.
"""
# size of transform
n = A.shape[-1]
# bias input array
if bias != 0:
# A_q(k) = A(k) (k/k_c)^{q} (k_c r_c)^{q}
j_c = (n - 1) / 2
j = cupy.arange(n)
A = A * cupy.exp(bias * ((j - j_c) * dln + offset))
# compute FHT coefficients
u = fhtcoeff(n, dln, mu, offset=offset, bias=bias)
# transform
a = _fhtq(A, u, inverse=True)
# bias output array
if bias != 0:
# a(r) = a_q(r) (r/r_c)^{q}
a /= cupy.exp(-bias * (j - j_c) * dln)
return a
def fhtcoeff(n, dln, mu, offset=0.0, bias=0.0):
'''Compute the coefficient array for a fast Hankel transform.
'''
lnkr, q = offset, bias
# Hankel transform coefficients
# u_m = (kr)^{-i 2m pi/(n dlnr)} U_mu(q + i 2m pi/(n dlnr))
# with U_mu(x) = 2^x Gamma((mu+1+x)/2)/Gamma((mu+1-x)/2)
xp = (mu + 1 + q)/2
xm = (mu + 1 - q)/2
y = cupy.linspace(0, math.pi * (n // 2) / (n * dln), n // 2 + 1)
u = cupy.empty(n // 2 + 1, dtype=complex)
v = cupy.empty(n // 2 + 1, dtype=complex)
u.imag[:] = y
u.real[:] = xm
loggamma(u, out=v)
u.real[:] = xp
loggamma(u, out=u)
y *= 2 * (LN_2 - lnkr)
u.real -= v.real
u.real += LN_2 * q
u.imag += v.imag
u.imag += y
cupy.exp(u, out=u)
# fix last coefficient to be real
u.imag[-1] = 0
# deal with special cases
if not cupy.isfinite(u[0]):
# write u_0 = 2^q Gamma(xp)/Gamma(xm) = 2^q poch(xm, xp-xm)
# poch() handles special cases for negative integers correctly
u[0] = 2**q * poch(xm, xp - xm)
# the coefficient may be inf or 0, meaning the transform or the
# inverse transform, respectively, is singular
return u
def _fhtq(a, u, inverse=False):
'''Compute the biased fast Hankel transform.
This is the basic FFTLog routine.
'''
# size of transform
n = a.shape[-1]
# check for singular transform or singular inverse transform
if cupy.isinf(u[0]) and not inverse:
warn('singular transform; consider changing the bias')
# fix coefficient to obtain (potentially correct) transform anyway
u = u.copy()
u[0] = 0
elif u[0] == 0 and inverse:
warn('singular inverse transform; consider changing the bias')
# fix coefficient to obtain (potentially correct) inverse anyway
u = u.copy()
u[0] = cupy.inf
# biased fast Hankel transform via real FFT
A = _fft.rfft(a, axis=-1)
if not inverse:
# forward transform
A *= u
else:
# backward transform
A /= u.conj()
A = _fft.irfft(A, n, axis=-1)
A = A[..., ::-1]
return A