/
_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).
added: 14/11/2020 Nicolas Tessore <n.tessore@ucl.ac.uk>
'''
import numpy as np
from warnings import warn
from ._basic import rfft, irfft
from ..special import loggamma, poch
__all__ = [
'fht', 'ifht',
'fhtoffset',
]
# constants
LN_2 = np.log(2)
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.
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)
'''
# size of transform
n = np.shape(a)[-1]
# bias input array
if bias != 0:
# a_q(r) = a(r) (r/r_c)^{-q}
j_c = (n-1)/2
j = np.arange(n)
a = a * np.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 *= np.exp(-bias*((j - j_c)*dln + offset))
return A
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.
See `fht` for further details.
'''
# size of transform
n = np.shape(A)[-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 = np.arange(n)
A = A * np.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 /= np.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 = np.linspace(0, np.pi*(n//2)/(n*dln), n//2+1)
u = np.empty(n//2+1, dtype=complex)
v = np.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
np.exp(u, out=u)
# fix last coefficient to be real
u.imag[-1] = 0
# deal with special cases
if not np.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 fhtoffset(dln, mu, initial=0.0, bias=0.0):
'''Return optimal offset for a fast Hankel transform.
Returns an offset close to `initial` that fulfils the low-ringing
condition of [1]_ for the fast Hankel transform `fht` with logarithmic
spacing `dln`, order `mu` and bias `bias`.
Parameters
----------
dln : float
Uniform logarithmic spacing of the transform.
mu : float
Order of the Hankel transform, any positive or negative real number.
initial : float, optional
Initial value for the offset. Returns the closest value that fulfils
the low-ringing condition.
bias : float, optional
Exponent of power law bias, any positive or negative real number.
Returns
-------
offset : float
Optimal offset of the uniform logarithmic spacing of the transform that
fulfils a low-ringing condition.
See also
--------
fht : Definition of the fast Hankel transform.
References
----------
.. [1] Hamilton A. J. S., 2000, MNRAS, 312, 257 (astro-ph/9905191)
'''
lnkr, q = initial, bias
xp = (mu+1+q)/2
xm = (mu+1-q)/2
y = np.pi/(2*dln)
zp = loggamma(xp + 1j*y)
zm = loggamma(xm + 1j*y)
arg = (LN_2 - lnkr)/dln + (zp.imag + zm.imag)/np.pi
return lnkr + (arg - np.round(arg))*dln
def _fhtq(a, u, inverse=False):
'''Compute the biased fast Hankel transform.
This is the basic FFTLog routine.
'''
# size of transform
n = np.shape(a)[-1]
# check for singular transform or singular inverse transform
if np.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] = np.inf
# biased fast Hankel transform via real FFT
A = rfft(a, axis=-1)
if not inverse:
# forward transform
A *= u
else:
# backward transform
A /= u.conj()
A = irfft(A, n, axis=-1)
A = A[..., ::-1]
return A