a a &@sdZddlZddlmZddlmZmZddlm Z m Z gdZ e dZ dd d Zdd d ZdddZdddZdddZdS)zFast Hankel transforms using the FFTLog algorithm. The implementation closely follows the Fortran code of Hamilton (2000). added: 14/11/2020 Nicolas Tessore N)warn)rfftirfft)loggammapoch)fhtifht fhtoffsetc Cst|d}|dkrH|dd}t|}|t| |||}t|||||d}t||} |dkr| t| ||||9} | S)a 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) rrroffsetbiasnpshapeZarangeexpfhtcoeff_fhtq) adlnmurrnj_cjuAra/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/scipy/fft/_fftlog.pyr s]    r c Cst|d}|dkrJ|dd}t|}|t|||||}t|||||d}t||dd} |dkr| t| |||} | S)aCompute 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. r rrrrT)inverser) rrrrrrrrrrrrr r s/  r c CsV||}}|d|d}|d|d}tdtj|d|||dd} tj|ddtd} tj|ddtd} | | jdd<|| jdd<t| | d|| jdd<t| | d| dt|9} | j| j8_| jt|7_| j| j7_| j| 7_tj | | dd| jd<t | dsRd|t |||| d<| S)z?Compute the coefficient array for a fast Hankel transform. rrr)ZdtypeN)outr ) rZlinspacepiemptycompleximagrealrLN_2risfiniter) rrrrrlnkrqxpxmyrvrrr rs* (   rc Cs||}}|d|d}|d|d}tjd|}t|d|} t|d|} t||| j| jtj} || t| |S)aReturn 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) rry?)rr#rr(r&round) rrinitialrr*r+r,r-r.ZzpZzmargrrr r s# r FcCst|d}t|dr:|s:td|}d|d<n*|ddkrd|rdtd|}tj|d<t|dd}|s~||9}n ||}t||dd}|ddddf}|S)zUCompute the biased fast Hankel transform. This is the basic FFTLog routine. r rz.singular transform; consider changing the biasz6singular inverse transform; consider changing the bias)Zaxis.N) rrisinfrcopyinfrZconjr)rrr!rrrrr r's      r)r r )r r )r r )r r )F)__doc__ZnumpyrwarningsrZ_basicrrZspecialrr__all__logr(r r rr rrrrr s   t F ( .