import itertools
import math

import numpy as np
import scipy.fftpack

from mmfutils.containers import ObjectBase

from . import interfaces
from .interfaces import (
    implementer,
    IBasis,
    IBasisKx,
    IBasisLz,
    IBasisWithConvolution,
    BasisMixin,
)

from mmfutils.performance.fft import fft, ifft, fftn, ifftn, resample
from .utils import prod, dst, idst, get_xyz, get_kxyz
from mmfutils.math import bessel

sp = scipy

_TINY = np.finfo(float).tiny

__all__ = ["SphericalBasis", "PeriodicBasis", "CartesianBasis", "interfaces"]


@implementer(IBasisWithConvolution)
class SphericalBasis(ObjectBase, BasisMixin):
    """1-dimensional basis for radial problems.

    We represent exactly `N` positive abscissa, excluding the origin and use
    the discrete sine transform.  We represent the square-root of the
    wavefunctions here so that a factor of `r` is required to convert these
    into the radial functions.  Unlike the DVR techniques, this approach allows
    us to compute the Coulomb interaction for example.
    """

    def __init__(self, N, R):
        self.N = N
        self.R = R
        super().__init__()

    def init(self):
        dx = self.R / self.N
        r = np.arange(1, self.N + 1) * dx
        k = np.pi * (0.5 + np.arange(self.N)) / self.R
        self.xyz = [r]
        self._pxyz = [k]
        self.metric = 4 * np.pi * r ** 2 * dx
        self.k_max = k.max()

    def laplacian(self, y, factor=1.0, exp=False):
        """Return the laplacian of `y` times `factor` or the
        exponential of this.

        Arguments
        ---------
        factor : float
           Additional factor (mostly used with `exp=True`).  The
           implementation must be careful to allow the factor to
           broadcast across the components.
        exp : bool
           If `True`, then compute the exponential of the laplacian.
           This is used for split evolvers.
        """
        r = self.xyz[0]
        K = -factor * self._pxyz[0] ** 2
        if exp:
            K = np.exp(K)

        ys = [y.real, y.imag] if np.iscomplexobj(y) else [y]
        res = [idst(K * dst(r * _y)) / r for _y in ys]

        if np.iscomplexobj(y):
            res = res[0] + 1j * res[1]
        else:
            res = res[0]

        return res

    def coulomb_kernel(self, k):
        """Form for the truncated Coulomb kernel."""
        D = 2 * self.R
        return (
            4 * np.pi * np.ma.divide(1.0 - np.cos(k * D), k ** 2).filled(D ** 2 / 2.0)
        )

    def convolve_coulomb(self, y, form_factors=[]):
        """Modified Coulomb convolution to include form-factors (if provided).

        This version implemented a 3D spherically symmetric convolution.
        """
        y = np.asarray(y)
        r = self.xyz[0]
        N, R = self.N, self.R

        # Padded arrays with trailing _
        ry_ = np.concatenate([r * y, np.zeros(y.shape, dtype=y.dtype)], axis=-1)
        k_ = np.pi * (0.5 + np.arange(2 * N)) / (2 * R)
        K = prod([_K(k_) for _K in [self.coulomb_kernel] + form_factors])
        return idst(K * dst(ry_))[..., :N] / r

    def convolve(self, y, C=None, Ck=None):
        """Return the periodic convolution `int(C(x-r)*y(r),r)`.

        Note: this is the 3D convolution.
        """
        r = self.xyz[0]
        k = self._pxyz[0]
        N, R = self.N, self.R
        R_N = R / N
        if Ck is None:
            C0 = (self.metric * C).sum()
            Ck = np.ma.divide(2 * np.pi * R_N * dst(r * C), k).filled(C0)
        else:
            Ck = Ck(k)
        return idst(Ck * dst(r * y)) / r


@implementer(IBasisWithConvolution, IBasisKx, IBasisLz)
class PeriodicBasis(ObjectBase, BasisMixin):
    """dim-dimensional periodic bases.

    Parameters
    ----------
    Nxyz : (Nx, Ny, ...)
       Number of lattice points in basis.
    Lxyz : (Lx, Ly, ...)
       Size of each dimension (length of box and radius)
    symmetric_lattice: bool
       If True, then shift the lattice so that it is symmetric about
       the origin.  The default is to ensure that there is a lattice
       point at the origin which will make the lattice asymmetric for
       even Nxyz.
    axes : (int, int, ...)
       Axes in array y which correspond to the x, y, ... axes here.
       This is required for cases where y has additional dimensions.
       The default is the last dim axes (best for performance).
    boost_pxyz : float
       Momentum of moving frame.  Momenta are shifted by this, which
       corresponds to working in a boosted frame with velocity `vx = px/m`.
    smoothing_cutoff : float
       Fraction of maximum momentum used in the function smooth().
    """

    # Select operations are performed using self.xp instead of numpy.
    # This can be replaced cupy to provide gpu support with minimal
    # code changes.  Similarly with the fft functions and a generic
    # function to convert an array into a numpy array on the host.
    xp = np
    _fft = staticmethod(fft)
    _ifft = staticmethod(ifft)
    _fftn = staticmethod(fftn)
    _ifftn = staticmethod(ifftn)
    _asnumpy = staticmethod(np.asarray)  # Convert to numpy array

    def __init__(
        self,
        Nxyz,
        Lxyz,
        symmetric_lattice=False,
        axes=None,
        boost_pxyz=None,
        smoothing_cutoff=0.8,
    ):
        self.symmetric_lattice = symmetric_lattice
        self.Nxyz = np.asarray(Nxyz)
        self.Lxyz = np.asarray(Lxyz)
        self.smoothing_cutoff = smoothing_cutoff
        if boost_pxyz is None:
            boost_pxyz = np.zeros_like(self.Lxyz)
        self.boost_pxyz = np.asarray(boost_pxyz)
        if axes is None:
            axes = np.arange(-self.dim, 0)
        self.axes = np.asarray(axes)
        super().__init__()

    def init(self):
        self.xyz = tuple(
            map(
                self.xp.asarray,
                get_xyz(
                    Nxyz=self.Nxyz,
                    Lxyz=self.Lxyz,
                    symmetric_lattice=self.symmetric_lattice,
                ),
            )
        )
        self._pxyz = tuple(
            map(self.xp.asarray, get_kxyz(Nxyz=self.Nxyz, Lxyz=self.Lxyz))
        )
        self._pxyz_derivative = tuple(
            map(self.xp.asarray, get_kxyz(Nxyz=self.Nxyz, Lxyz=self.Lxyz))
        )

        # Zero out odd highest frequency component.
        for _N, _p in zip(self.Nxyz, self._pxyz_derivative):
            _p.ravel()[_N // 2] = 0.0

        # Add boosts
        self._pxyz = [
            _p - _b for (_p, _b) in zip(self._pxyz, self.xp.asarray(self.boost_pxyz))
        ]
        self.metric = np.prod(self.Lxyz / self.Nxyz)
        self.k_max = self._asnumpy([abs(_p).max() for _p in self._pxyz])

        p2_pc2 = sum(
            (_p / (self.smoothing_cutoff * _p).max()) ** 2 for _p in self._pxyz
        )
        self._smoothing_factor = self.xp.where(p2_pc2 < 1, 1, 0)
        # np.exp(-p2_pc2**4)
        # self._smoothing_factor = 1.0

        # Memoize momentum sums for speed
        _kx2 = self._pxyz[0] ** 2
        _kyz2 = sum(_p ** 2 for _p in self._pxyz[1:])
        _k2 = _kx2 + _kyz2
        self._k2_kx2_kyz2 = (_k2, _kx2, _kyz2)

    @property
    def kx(self):
        return self._pxyz[0]

    @property
    def Lx(self):
        return self.Lxyz[0]

    @property
    def Nx(self):
        return self.Nxyz[0]

    def laplacian(
        self, y, factor=1.0, exp=False, kx2=None, k2=None, kwz2=0, twist_phase_x=None
    ):
        """Return the laplacian of `y` times `factor` or the exponential of this.

        Arguments
        ---------
        factor : float
           Additional factor (mostly used with `exp=True`).  The
           implementation must be careful to allow the factor to
           broadcast across the components.
        exp : bool
           If `True`, then compute the exponential of the laplacian.
           This is used for split evolvers. Only allowed to be `True`
           if `kwz2 == 0`.
        kx2 : array, optional
           Replacement for the default `kx2=kx**2` used when computing the
           "laplacian".  This would allow you, for example, to implement a
           modified dispersion relationship like ``1-cos(kx)`` rather than
           ``kx**2``.
        kwz2 : None, float
           Angular velocity of the frame expressed as `kwz2 = m*omega_z/hbar`.
        k2 : array, optional
           Replacement for `k2 = kx**2 + ky**2 + kz**2`.
        twist_phase_x : array, optional
           To implement twisted boundary conditions, one needs to remove an
           overall phase from the wavefunction rendering it periodic for use
           the the FFT.  This the the phase that should be removed.  Note: to
           compensate, the momenta should be shifted as well::

              -factor * twist_phase_x*ifft((k+k_twist)**2*fft(y/twist_phase_x)
        """
        _k2, _kx2, _kyz2 = self._k2_kx2_kyz2
        if k2 is None:
            if kx2 is None:
                k2 = _k2
            else:
                kx2 = self.xp.asarray(kx2)
                k2 = kx2 + _kyz2
        else:
            k2 = self.xp.asarray(k2)
            assert kx2 is None

        K = -factor * k2
        if exp:
            if kwz2 != 0:
                raise NotImplementedError(
                    f"Cannot use exp=True if kwz2 != 0 (got {kwz2})."
                )
            K = self.xp.exp(K)

        if twist_phase_x is not None:
            twist_phase_x = self.xp.asarray(twist_phase_x)
            y = y / twist_phase_x

        yt = self.fftn(y)
        laplacian_y = self.ifftn(K * yt)

        if kwz2 != 0:
            laplacian_y += 2 * kwz2 * factor * self.apply_Lz_hbar(y, yt=yt)

        if twist_phase_x is not None:
            laplacian_y *= twist_phase_x
        return laplacian_y

    def apply_Lz_hbar(self, y, yt=None):
        """Apply `Lz/hbar` to `y`."""
        if yt is None:
            yt = self.fftn(y)
        x, y = self.xyz[:2]
        kx, ky = self._pxyz[:2]
        return x * self.ifftn(ky * yt) - y * self.ifftn(kx * yt)

    # We need these wrappers because the state may have additional
    # indices for components etc. in front.
    def fft(self, x, axis):
        """Perform the fft along self.axes[axis]"""
        axis = self.axes[axis] % len(x.shape)
        return self._fft(x, axis=axis)

    def ifft(self, x, axis):
        """Perform the ifft along self.axes[axis]"""
        axis = self.axes[axis] % len(x.shape)
        return self._ifft(x, axis=axis)

    def fftn(self, x):
        """Perform the fft along spatial axes"""
        axes = self.axes % len(x.shape)
        return self._fftn(x, axes=axes)

    def ifftn(self, x):
        """Perform the ifft along spatial axes"""
        axes = self.axes % len(x.shape)
        return self._ifftn(x, axes=axes)

    def smooth(self, x, frac=0.8):
        """Smooth the state by multiplying by form factor."""
        return self.ifftn(self._smoothing_factor * self.fftn(x))

    def get_gradient(self, y):
        # TODO: Check this for the highest momentum issue.
        return [
            self.ifft(1j * _p * self.fft(y, axis=_i), axis=_i)
            for _i, _p in enumerate(self._pxyz_derivative)
        ]

    def get_divergence(self, ys):
        # TODO: Check this for the highest momentum issue.
        return sum(
            self.ifft(1j * _p * self.fft(_y, axis=_i), axis=_i)
            for _i, (_p, _y) in enumerate(zip(self._pxy_derivativez, ys))
        )

    @staticmethod
    def _bcast(n, N):
        """Use this to broadcast a 1D array along the n'th of N dimensions"""
        inds = [None] * N
        inds[n] = slice(None)
        return inds

    def coulomb_kernel(self, k):
        """Form for the Coulomb kernel.

        The normalization here is that the k=0 component is set to
        zero.  This means that the charge distribution has an overall
        constant background removed so that the net charge in the unit
        cell is zero.
        """
        return 4 * np.pi * np.ma.divide(1.0, k ** 2).filled(0.0)

    def convolve_coulomb(self, y, form_factors=[]):
        """Periodic convolution with the Coulomb kernel."""
        y = np.asarray(y)

        # This broadcasts to the appropriate size if there are
        # multiple components.
        # dim = len(np.asarray(self.Lxyz))
        # N = np.asarray(y.shape)
        # b_cast = [None] * (dim - len(N)) + [slice(None)]*dim

        k = np.sqrt(sum(_k ** 2 for _k in self._pxyz))
        Ck = prod([_K(k) for _K in [self.coulomb_kernel] + form_factors])
        return self.ifftn(Ck * self.fftn(y))

    def convolve(self, y, C=None, Ck=None):
        """Return the periodic convolution `int(C(x-r)*y(r),r)`.

        Arguments
        ---------
        y : array
           Usually the density, but can be any array
        C : array
           Convolution kernel. The convolution will be performed using the FFT.
        Ck : function (optional)
           If provided, then this function will be used instead directly in
           momentum space.  Assumed to be spherically symmetric (will be passed
           only the magnitude `k`)
        """
        if Ck is None:
            Ck = self.fftn(C)
        else:
            k = np.sqrt(sum(_k ** 2 for _k in self._pxyz))
            Ck = Ck(k)
        return self.ifftn(Ck * self.fftn(y))

    @property
    def dim(self):
        return len(self.Nxyz)


@implementer(IBasisWithConvolution)
class CartesianBasis(PeriodicBasis):
    """N-dimensional periodic bases but with Coulomb convolution that does not
    use periodic images.  Use this for nuclei in free space.

    Parameters
    ----------
    Nxyz : (Nx, Ny, ...)
       Number of lattice points in basis.
    Lxyz : (Lx, Ly, ...)
       Size of each dimension (length of box and radius)
    symmetric_lattice: bool
       If True, then shift the lattice so that it is symmetric about
       the origin.  The default is to ensure that there is a lattice
       point at the origin which will make the lattice asymmetric for
       even Nxyz.
    axes : (int, int, ...)
       Axes in array y which correspond to the x, y, ... axes here.
       This is required for cases where y has additional dimensions.
       The default is the last dim axes (best for performance).
    fast_coulomb : bool
       If `True`, use the fast Coulomb algorithm which is slightly less
       accurate but much faster.
    """

    def __init__(
        self, Nxyz, Lxyz, axes=None, symmetric_lattice=False, fast_coulomb=True
    ):
        self.fast_coulomb = fast_coulomb
        PeriodicBasis.__init__(
            self, Nxyz=Nxyz, Lxyz=Lxyz, axes=axes, symmetric_lattice=symmetric_lattice
        )

    def convolve_coulomb_fast(self, y, form_factors=[], correct=False):
        r"""Return the approximate convolution `int(C(x-r)*y(r),r)` where

        .. math::
           C(r) = 1/r

        is the Coulomb potential (without charges etc.)

        Arguments
        ---------
        y : array
           Usually the density, but can be any array
        correct : bool
           If `True`, then include the high frequency components via the
           periodic convolution.

        Notes
        -----
        This version uses the Truncated Kernel Expansion method which uses the
        Truncated Kernel

        .. math::
           4\pi(1-\cos\sqrt{3}Lk)/k^2

        on a padded array to remove the images, approximating the linear
        convolution without the highest frequency modes.  By choosing the
        smaller lattice to be at least 3 times smaller we can guarantee that
        the padded array will fit into memory.  This can be augmented by the
        periodic convolution to fill in the higher modes.

        There are two sources of error here:

        * We presently use the same periodic ``resample`` method to interpolate
          the linear convolution to the larger grid.  This assumes that the
          function is periodic -- which it is not -- and can introduce some
          aliasing artifacts.  Some preliminary experimentation shows, however,
          that these are generally small.  Perhaps cubic spline interpolation
          could be used to improve the interpolation, but this is not clear
          yet.
        * The contribution from the higher modes are computed from the periodic
          convolution which could in principle be contaminated by images.
          However, for smooth functions, there should be little amplitude here,
          and it should consist only of higher multipoles, so the contamination
          should be small.
        """
        y = np.asarray(y)
        L = np.asarray(self.Lxyz)
        dim = len(L)
        N = np.asarray(y.shape)
        N0 = N.copy()
        N0[-dim:] = N[-dim:] // 3

        y0 = resample(y, N0)
        V = resample(
            self.convolve_coulomb_exact(y0, form_factors=form_factors, method="pad"), N
        )
        if correct:
            k = np.sqrt(sum(_K ** 2 for _K in self._pxyz))
            C = 4 * np.pi * np.ma.divide(1.0, k ** 2).filled(0.0)
            for F in form_factors:
                C = C * F(k)
            dV = self.ifftn(C * self.fftn(y - resample(y0, N)))
            if np.issubdtype(V.dtype, np.complex128):
                V += dV
            else:
                assert np.allclose(0, V.imag)
                V += dV.real
        return V

    def convolve_coulomb_exact(self, y, form_factors=[], method="sum"):
        r"""Return the convolution `int(C(x-r)*y(r),r)` where

        .. math::
           C(r) = 1/r

        is the Coulomb potential (without charges etc.)

        Arguments
        ---------
        y : array
           Usually the density, but can be any array
        method : 'sum', 'pad'
           Either zero-pad the array (takes extra memory but can use multiple
           cores) or sum over the 27 small transforms (slow).

        This function is designed for computing the Coulomb potential of a
        charge distribution.  In this case, one would have the kernel:

        .. math::
           4\pi/k^2

        or in the case of non-periodic convolution to remove the images

        .. math::
           4\pi(1-\cos\sqrt{3}Lk)/k^2
        """
        y = np.asarray(y)
        L = np.asarray(self.Lxyz)
        dim = len(L)
        D = np.sqrt((L ** 2).sum())  # Diameter of cell

        def C(k):
            C = 4 * np.pi * np.ma.divide(1 - np.cos(D * k), k ** 2).filled(D ** 2 / 2.0)
            for F in form_factors:
                C = C * F(k)
            return C

        if method == "sum":
            # Sum with a loop.  Minimizes the memory usage, but will not
            # use multiple cores.
            K = self._pxyz
            X = self.xyz
            V = np.zeros(y.shape, dtype=y.dtype)
            for l in itertools.product(np.arange(3), repeat=dim):
                delta = [2 * np.pi * _l / 3.0 / _L for _l, _L in zip(l, L)]
                exp_delta = np.exp(1j * sum(_d * _x for _x, _d in zip(X, delta)))
                y_delta = exp_delta.conj() * y
                k = np.sqrt(sum((_k + _d) ** 2 for _k, _d in zip(K, delta)))
                dV = exp_delta * self.ifftn(C(k) * self.fftn(y_delta))
                if np.issubdtype(V.dtype, np.complex128):
                    V += dV
                else:
                    assert np.allclose(0, V.imag)
                    V += dV.real
            return V / dim ** 3
        elif method == "pad":
            N = np.asarray(y.shape[-dim:])
            N_padded = 3 * N
            L_padded = 3 * L
            shape = np.asarray(y.shape)
            shape_padded = shape.copy()
            shape_padded[-dim:] = N_padded
            y_padded = np.zeros(shape_padded, dtype=y.dtype)
            inds = tuple(slice(0, _N) for _N in shape)
            y_padded[inds] = y
            k = np.sqrt(sum(_K ** 2 for _K in get_kxyz(N_padded, L_padded)))

            # This broadcasts to the appropriate size
            b_cast = (None,) * (dim - len(N)) + (slice(None),) * dim
            return self.ifftn(C(k)[b_cast] * self.fftn(y_padded))[inds]
        else:
            raise NotImplementedError(
                "method=%s not implemented: use 'sum' or 'pad'" % (method,)
            )

    def convolve_coulomb(self, y, form_factors=[], **kw):
        if self.fast_coulomb:
            return self.convolve_coulomb_fast(y, form_factors=form_factors, **kw)
        else:
            return self.convolve_coulomb_exact(y, form_factors=form_factors, **kw)


@implementer(IBasis, IBasisKx)
class CylindricalBasis(ObjectBase, BasisMixin):
    r"""2D basis for Cylindrical coordinates via a DVR basis.

    This represents 3-dimensional problems with axial symmetry, but only has
    two dimensions (x, r).

    Parameters
    ----------
    Nxr : (Nx, Nr)
       Number of lattice points in basis.
    Lxr : (L, R)
       Size of each dimension (length of box and radius)
    twist : float
       Twist (angle) in periodic dimension.  This adds a constant offset to the
       momenta allowing one to study Bloch waves.
    boost_px : float
       Momentum of moving frame (along the x axis).  Momenta are shifted by
       this, which corresponds to working in a boosted frame with velocity
       `vx = boost_px/m`.
    axes : (int, int)
       Axes in array y which correspond to the x and r axes here.
       This is required for cases where y has additional dimensions.
       The default is the last two axes (best for performance).
    """
    _d = 2  # Dimension of spherical part (see nu())

    def __init__(self, Nxr, Lxr, twist=0, boost_px=0, axes=(-2, -1), symmetric_x=True):
        self.twist = twist
        self.boost_px = np.asarray(boost_px)
        self.Nxr = np.asarray(Nxr)
        self.Lxr = np.asarray(Lxr)
        self.symmetric_x = symmetric_x
        self.axes = np.asarray(axes)
        super().__init__()

    def init(self):
        Lx, R = self.Lxr
        x = get_xyz(Nxyz=self.Nxr, Lxyz=self.Lxr, symmetric_lattice=self.symmetric_x)[0]
        kx0 = get_kxyz(Nxyz=self.Nxr, Lxyz=self.Lxr)[0]
        self.kx = kx0 + float(self.twist) / Lx - self.boost_px
        self._kx0 = kx0
        self._kx2 = self.kx ** 2

        self.y_twist = np.exp(1j * self.twist * x / Lx)

        Nx, Nr = self.Nxr

        # For large n, the roots of the bessel function are approximately
        # z[n] = (n + 0.75)*pi, so R = r_max = z_max/k_max = (N-0.25)*pi/kmax
        # This self._kmax defines the DVR basis, not self.k_max
        self._kmax = (Nr - 0.25) * np.pi / R

        # This is just the maximum momentum for diagnostics,
        # determining cutoffs etc.
        self.k_max = np.array([abs(self.kx).max(), self._kmax])

        nr = np.arange(Nr)[None, :]
        r = self._r(Nr)[None, :]  # Do this after setting _kmax
        self.xyz = [x, r]

        _lambda = np.asarray(
            [1.0 / (self._F(_nr, _r)) ** 2 for _nr, _r in zip(nr.ravel(), r.ravel())]
        )[None, :]
        self.metric = 2 * np.pi * r * _lambda * (Lx / Nx)
        self.metric.setflags(write=False)
        # Get the DVR kinetic piece for radial component
        K, r1, r2, w = self._get_K()

        # We did not apply the sqrt(r) factors so at this point, K is still
        # Hermitian and we can diagonalize for later exponentiation.
        d, V = sp.linalg.eigh(K)  # K = np.dot(V*d, V.T)

        # Here we convert from the wavefunction Psi(r) to the radial
        # function u(r) = sqrt(r)*Psi(r) and back with factors of sqrt(r).
        K *= r1
        K *= r2

        self.weights = w
        self._Kr = K
        self._Kr_diag = (r1, r2, V, d)  # For use when exponentiating

        # And factor for x.
        self._Kx = self._kx2

        # Cache for K_data from apply_exp_K.
        self._K_data = []

    @property
    def Lx(self):
        return self.Lxr[0]

    @property
    def Nx(self):
        return self.Nxr[0]

    ######################################################################
    # IBasisMinimal: Required methods
    def laplacian(self, y, factor=1.0, exp=False, kx2=None, twist_phase_x=None):
        r"""Return the laplacian of y.

        Arguments
        ---------
        factor : float
           Additional factor (mostly used with `exp=True`).  The
           implementation must be careful to allow the factor to
           broadcast across the components.
        exp : bool
           If `True`, then compute the exponential of the laplacian.
           This is used for split evolvers. Only allowed to be `True`
           if `kwz2 == 0`.
        kx2 : array, optional
           Replacement for the default `kx2=kx**2` used when computing the
           "laplacian".  This would allow you, for example, to implement a
           modified dispersion relationship like ``1-cos(kx)`` rather than
           ``kx**2``.
        kwz2 : float
           Angular velocity of the frame expressed as `kwz2 = m*omega_z/hbar`.
        twist_phase_x : array, optional
           To implement twisted boundary conditions, one needs to remove an
           overall phase from the wavefunction rendering it periodic for use
           the the FFT.  This the the phase that should be removed.  Note: to
           compensate, the momenta should be shifted as well::

              -factor * twist_phase_x*ifft((k+k_twist)**2*fft(y/twist_phase_x)
        """
        if not exp:
            return self.apply_K(y=y, kx2=kx2, twist_phase_x=twist_phase_x) * (-factor)
        else:
            return self.apply_exp_K(
                y=y, factor=-factor, kx2=kx2, twist_phase_x=twist_phase_x
            )

    ######################################################################

    def get_gradient(self, y):
        """Returns the gradient along the x axis."""
        kx = self.kx
        return [self.ifft(1j * kx * self.fft(y)), NotImplemented]

    def apply_Lz(self, y, hermitian=False):
        raise NotImplementedError

    def apply_Px(self, y, hermitian=False):
        r"""Apply the Pz operator to y without any px.

        Requires :attr:`_pxyz` to be defined.
        """
        return self.y_twist * self.ifft(self._kx0 * self.fft(y / self.y_twist))

    def apply_exp_K(self, y, factor, kx2=None, twist_phase_x=None):
        r"""Return `exp(K*factor)*y` or return precomputed data if
        `K_data` is `None`.
        """
        if kx2 is None:
            kx2 = self._Kx
        _K_data_max_len = 3
        ind = None
        for _i, (_f, _d) in enumerate(self._K_data):
            if np.allclose(factor, _f):
                ind = _i
        if ind is None:
            _r1, _r2, V, d = self._Kr_diag
            exp_K_r = _r1 * np.dot(V * np.exp(factor * d), V.T) * _r2
            exp_K_x = np.exp(factor * kx2)
            K_data = (exp_K_r, exp_K_x)
            self._K_data.append((factor, K_data))
            ind = -1
            while len(self._K_data) > _K_data_max_len:
                # Reduce storage
                self._K_data.pop(0)

        K_data = self._K_data[ind][1]
        exp_K_r, exp_K_x = K_data
        if twist_phase_x is None or self.twist == 0:
            tmp = self.ifft(exp_K_x * self.fft(y))
        else:
            if twist_phase_x is None:
                twist_phase_x = self.y_twist
            tmp = twist_phase_x * self.ifft(exp_K_x * self.fft(y / twist_phase_x))
        return np.einsum("...ij,...yj->...yi", exp_K_r, tmp)

    def apply_K(self, y, kx2=None, twist_phase_x=None):
        r"""Return `K*y` where `K = k**2/2`"""
        # Here is how the indices work:
        if kx2 is None:
            kx2 = self._Kx

        if twist_phase_x is None or self.twist == 0:
            yt = self.fft(y)
            yt *= kx2
            yt = self.ifft(yt)
        else:
            if twist_phase_x is None:
                twist_phase_x = self.y_twist
            yt = self.fft(y / twist_phase_x)
            yt *= kx2
            yt = self.ifft(yt)
            yt *= twist_phase_x

        # C <- alpha*B*A + beta*C    A = A^T  zSYMM or zHYMM but not supported
        # maybe cvxopt.blas?  Actually, A is not symmetric... so be careful!
        yt += np.dot(y, self._Kr.T)
        return yt

    ######################################################################
    # FFT and DVR Helper functions.
    #
    # These are specific to the basis, defining the kinetic energy
    # matrix for example.

    # We need these wrappers because the state may have additional
    # indices for components etc. in front.
    def fft(self, x):
        """Perform the fft along the x axes"""
        # Makes sure that
        axis = (self.axes % len(x.shape))[0]
        return fft(x, axis=axis)

    def ifft(self, x):
        """Perform the fft along the x axes"""
        axis = (self.axes % len(x.shape))[0]
        return ifft(x, axis=axis)

    def _get_K(self, l=0):
        r"""Return `(K, r1, r2, w)`: the DVR kinetic term for the radial function
        and the appropriate factors for converting to the radial coordinates.

        This term effects the $-d^2/dr^2 - (\nu^2 - 1/4)/r^2$ term.

        Returns
        -------
        K : array
           Operates on radial wavefunctions
        r1, r2 : array
           K*r1*r2 operators on the full wavefunction (but is no longer
           Hermitian)
        w : array
           Quadrature integration weights.
        """
        nu = self.nu(l=l)
        if l == 0:
            r = self.xyz[1].ravel()
        else:
            r = self._r(self.Nxr[1], l=l)
        z = self._kmax * r
        n = np.arange(len(z))
        i1 = (slice(None), None)
        i2 = (None, slice(None))

        # Quadrature weights
        w = 2.0 / (self._kmax * z * bessel.J(nu=nu, d=1)(z) ** 2)

        # DVR kinetic term for radial function:
        K = np.ma.divide(
            (-1.0) ** (n[i1] - n[i2]) * 8.0 * z[i1] * z[i2],
            (z[i1] ** 2 - z[i2] ** 2) ** 2,
        ).filled(0)
        K[n, n] = 1.0 / 3.0 * (1.0 + 2.0 * (nu ** 2 - 1.0) / z ** 2)
        K *= self._kmax ** 2

        # Here we convert from the wavefunction Psi(r) to the radial
        # function u(r) = sqrt(r)*Psi(r) and back with factors of
        # sqrt(wr).  This includes the integration weights (since K is
        # defined acting on the basis functions).
        # Note: this makes the matrix non-hermitian, so don't do this if you
        # want to diagonalize.
        _tmp = np.sqrt(w * r)
        r2 = _tmp[i2]
        r1 = 1.0 / _tmp[i1]

        return K, r1, r2, w

    def nu(self, l):
        """Return `nu = l + d/2 - 1` for the centrifugal term.

        Arguments
        ---------
        l : int
           Angular quantum number.
        """
        nu = l + self._d / 2 - 1
        return nu

    def _r(self, N, l=0):
        r"""Return the abscissa."""
        # l=0 cylindrical: nu = l + d/2 - 1
        return bessel.j_root(nu=self.nu(l=l), N=N) / self._kmax

    def _F(self, n, r, d=0):
        r"""Return the dth derivative of the n'th basis function."""
        nu = 0.0  # l=0 cylindrical: nu = l + d/2 - 1
        rn = self.xyz[1].ravel()[n]
        zn = self._kmax * rn
        z = self._kmax * r
        H = bessel.J_sqrt_pole(nu=nu, zn=zn, d=0)
        coeff = math.sqrt(2.0 * self._kmax) * (-1.0) ** (n + 1) / (1.0 + r / rn)
        if 0 == d:
            return coeff * H(z)
        elif 1 == d:
            dH = bessel.J_sqrt_pole(nu=nu, zn=zn, d=1)
            return coeff * (dH(z) - H(z) / (z + zn)) * self._kmax
        else:
            raise NotImplementedError

    def get_F(self, r):
        """Return a function that can extrapolate a radial
        wavefunction to a new set of abscissa (x, r)."""
        x, r0 = self.xyz
        n = np.arange(r0.size)[:, None]

        # Here is the transform matrix
        _F = self._F(n, r) / self._F(n, r0.T)

        def F(u):
            return np.dot(u, _F)

        return F

    def F(self, u, xr):
        r"""Return u evaluated on the new abscissa (Assumes x does not
        change for now)"""
        x0, r0 = self.xyz
        x, r = xr
        assert np.allclose(x, x0)

        return self.get_F(r)(u)

    def get_Psi(self, r, return_matrix=False):
        r"""Return a function that can extrapolate a wavefunction to a
        new set of abscissa (x, r).

        This includes the factor of $\sqrt{r}$ that converts the
        wavefunction to the radial function, then uses the basis to
        extrapolate the radial function.

        Arguments
        ---------
        r : array
           The new abscissa in the radial direction (the $x$ values
           stay the same.)
        return_matrix : bool
           If True, then return the extrapolation matrix F so that
           ``Psi = np.dot(psi, F)``
        """
        x, r0 = self.xyz
        n = np.arange(r0.size)[:, None]

        # Here is the transform matrix
        _F = (np.sqrt(r) * self._F(n, r)) / (np.sqrt(r0.T) * self._F(n, r0.T))

        if return_matrix:
            return _F

        def Psi(psi):
            return np.dot(psi, _F)

        return Psi

    def Psi(self, psi, xr):
        r"""Return psi evaluated on the new abscissa (Assumes x does not
        change for now)"""
        x0, r0 = self.xyz
        x, r = xr
        assert np.allclose(x, x0)

        return self.get_Psi(r)(psi)

    def integrate1(self, n):
        """Return the integral of n over y and z."""
        n = np.asarray(n)
        x, r = self.xyz
        x_axis, r_axis = self.axes
        bcast = [None] * len(n.shape)
        bcast[x_axis] = slice(None)
        bcast[r_axis] = slice(None)
        return ((2 * np.pi * r * self.weights)[tuple(bcast)] * n).sum(axis=r_axis)

    def integrate2(self, n, y=None, Nz=100):
        """Return the integral of n over z (line-of-sight integral) at y.

        This is an Abel transform, and is used to compute the 1D
        line-of-sight integral as would be seen by a photographic
        image through an axial cloud.

        Arguments
        ---------
        n : array
           (Nx, Nr) array of the function to be integrated tabulated
           on the abscissa.  Note: the extrapolation assumes that `n =
           abs(psi)**2` where `psi` is well represented in the basis.
        y : array, None
           Ny points at which the resulting integral should be
           returned.  If not provided, then the function will be
           tabulated at the radial abscissa.
        Nz : int
           Number of points to use in z integral.
        """
        n = np.asarray(n)
        x, r = self.xyz
        if y is None:
            y = r

        y = y.ravel()
        Ny = len(y)

        x_axis, r_axis = self.axes
        y_axis = r_axis
        bcast_y = [None] * len(n.shape)
        bcast_z = [None] * len(n.shape)
        bcast_y[y_axis] = slice(None)
        bcast_y.append(None)
        bcast_z.append(slice(None))

        bcast_y, bcast_z = tuple(bcast_y), tuple(bcast_z)

        z = np.linspace(0, r.max(), Nz)
        shape_xyz = n.shape[:-1] + (Ny, Nz)
        rs = np.sqrt(y.ravel()[bcast_y] ** 2 + z[bcast_z] ** 2)
        n_xyz = (abs(self.Psi(np.sqrt(n), (x, rs.ravel()))) ** 2).reshape(shape_xyz)
        n_2D = 2 * np.trapz(n_xyz, z, axis=-1)
        return n_2D


'''
Incomplete!
@implementer(IBasis, IBasisKx)
class SphericalDVRBasis(ObjectBase, BasisMixin):
    r"""3D Spherical DVR basis.

    This represents 3-dimensional problems with spherical symmetry.

    Parameters
    ----------
    N : int
       Number of lattice points in basis.
    R : float
       Radius of the basis.
    """
    _d = 3  # Dimension of spherical part (see nu())

    def __init__(self, N, R):
        self.N = N
        self.R = R
        super().__init__()

    def init(self):
        R = self.R
        N = self.N

        # For large n, the roots of the bessel function are approximately
        # z[n] = (n + 0.75)*pi, so R = r_max = z_max/k_max = (N-0.25)*pi/kmax
        # This self._kmax defines the DVR basis, not self.k_max
        self._kmax = (Nr - 0.25) * np.pi / R

        # This is just the maximum momentum for diagnostics,
        # determining cutoffs etc.
        self.k_max = np.array([abs(self.kx).max(), self._kmax])

        nr = np.arange(Nr)[None, :]
        r = self._r(Nr)[None, :]  # Do this after setting _kmax
        self.xyz = [x, r]

        _lambda = np.asarray(
            [1.0 / (self._F(_nr, _r)) ** 2 for _nr, _r in zip(nr.ravel(), r.ravel())]
        )[None, :]
        self.metric = 2 * np.pi * r * _lambda * (Lx / Nx)
        self.metric.setflags(write=False)
        # Get the DVR kinetic piece for radial component
        K, r1, r2, w = self._get_K()

        # We did not apply the sqrt(r) factors so at this point, K is still
        # Hermitian and we can diagonalize for later exponentiation.
        d, V = sp.linalg.eigh(K)  # K = np.dot(V*d, V.T)

        # Here we convert from the wavefunction Psi(r) to the radial
        # function u(r) = sqrt(r)*Psi(r) and back with factors of sqrt(r).
        K *= r1
        K *= r2

        self.weights = w
        self._Kr = K
        self._Kr_diag = (r1, r2, V, d)  # For use when exponentiating

        # And factor for x.
        self._Kx = self._kx2

        # Cache for K_data from apply_exp_K.
        self._K_data = []

    @property
    def Lx(self):
        return self.Lxr[0]

    @property
    def Nx(self):
        return self.Nxr[0]

    ######################################################################
    # IBasisMinimal: Required methods
    def laplacian(self, y, factor=1.0, exp=False, kx2=None, twist_phase_x=None):
        r"""Return the laplacian of y.

        Arguments
        ---------
        factor : float
           Additional factor (mostly used with `exp=True`).  The
           implementation must be careful to allow the factor to
           broadcast across the components.
        exp : bool
           If `True`, then compute the exponential of the laplacian.
           This is used for split evolvers. Only allowed to be `True`
           if `kwz2 == 0`.
        kx2 : array, optional
           Replacement for the default `kx2=kx**2` used when computing the
           "laplacian".  This would allow you, for example, to implement a
           modified dispersion relationship like ``1-cos(kx)`` rather than
           ``kx**2``.
        kwz2 : float
           Angular velocity of the frame expressed as `kwz2 = m*omega_z/hbar`.
        twist_phase_x : array, optional
           To implement twisted boundary conditions, one needs to remove an
           overall phase from the wavefunction rendering it periodic for use
           the the FFT.  This the the phase that should be removed.  Note: to
           compensate, the momenta should be shifted as well::

              -factor * twist_phase_x*ifft((k+k_twist)**2*fft(y/twist_phase_x)
        """
        if not exp:
            return self.apply_K(y=y, kx2=kx2, twist_phase_x=twist_phase_x) * (-factor)
        else:
            return self.apply_exp_K(
                y=y, factor=-factor, kx2=kx2, twist_phase_x=twist_phase_x
            )

    ######################################################################

    def get_gradient(self, y):
        """Returns the gradient along the x axis."""
        kx = self.kx
        return [self.ifft(1j * kx * self.fft(y)), NotImplemented]

    def apply_Lz(self, y, hermitian=False):
        raise NotImplementedError

    def apply_Px(self, y, hermitian=False):
        r"""Apply the Pz operator to y without any px.

        Requires :attr:`_pxyz` to be defined.
        """
        return self.y_twist * self.ifft(self._kx0 * self.fft(y / self.y_twist))

    def apply_exp_K(self, y, factor, kx2=None, twist_phase_x=None):
        r"""Return `exp(K*factor)*y` or return precomputed data if
        `K_data` is `None`.
        """
        if kx2 is None:
            kx2 = self._Kx
        _K_data_max_len = 3
        ind = None
        for _i, (_f, _d) in enumerate(self._K_data):
            if np.allclose(factor, _f):
                ind = _i
        if ind is None:
            _r1, _r2, V, d = self._Kr_diag
            exp_K_r = _r1 * np.dot(V * np.exp(factor * d), V.T) * _r2
            exp_K_x = np.exp(factor * kx2)
            K_data = (exp_K_r, exp_K_x)
            self._K_data.append((factor, K_data))
            ind = -1
            while len(self._K_data) > _K_data_max_len:
                # Reduce storage
                self._K_data.pop(0)

        K_data = self._K_data[ind][1]
        exp_K_r, exp_K_x = K_data
        if twist_phase_x is None or self.twist == 0:
            tmp = self.ifft(exp_K_x * self.fft(y))
        else:
            if twist_phase_x is None:
                twist_phase_x = self.y_twist
            tmp = twist_phase_x * self.ifft(exp_K_x * self.fft(y / twist_phase_x))
        return np.einsum("...ij,...yj->...yi", exp_K_r, tmp)

    def apply_K(self, y, kx2=None, twist_phase_x=None):
        r"""Return `K*y` where `K = k**2/2`"""
        # Here is how the indices work:
        if kx2 is None:
            kx2 = self._Kx

        if twist_phase_x is None or self.twist == 0:
            yt = self.fft(y)
            yt *= kx2
            yt = self.ifft(yt)
        else:
            if twist_phase_x is None:
                twist_phase_x = self.y_twist
            yt = self.fft(y / twist_phase_x)
            yt *= kx2
            yt = self.ifft(yt)
            yt *= twist_phase_x

        # C <- alpha*B*A + beta*C    A = A^T  zSYMM or zHYMM but not supported
        # maybe cvxopt.blas?  Actually, A is not symmetric... so be careful!
        yt += np.dot(y, self._Kr.T)
        return yt

    ######################################################################
    # FFT and DVR Helper functions.
    #
    # These are specific to the basis, defining the kinetic energy
    # matrix for example.

    # We need these wrappers because the state may have additional
    # indices for components etc. in front.
    def fft(self, x):
        """Perform the fft along the x axes"""
        # Makes sure that
        axis = (self.axes % len(x.shape))[0]
        return fft(x, axis=axis)

    def ifft(self, x):
        """Perform the fft along the x axes"""
        axis = (self.axes % len(x.shape))[0]
        return ifft(x, axis=axis)

    def _get_K(self, l=0):
        r"""Return `(K, r1, r2, w)`: the DVR kinetic term for the radial function
        and the appropriate factors for converting to the radial coordinates.

        This term effects the $-d^2/dr^2 - (\nu^2 - 1/4)/r^2$ term.

        Returns
        -------
        K : array
           Operates on radial wavefunctions
        r1, r2 : array
           K*r1*r2 operators on the full wavefunction (but is no longer
           Hermitian)
        w : array
           Quadrature integration weights.
        """
        nu = self.nu(l=l)
        if l == 0:
            r = self.xyz[1].ravel()
        else:
            r = self._r(self.Nxr[1], l=l)
        z = self._kmax * r
        n = np.arange(len(z))
        i1 = (slice(None), None)
        i2 = (None, slice(None))

        # Quadrature weights
        w = 2.0 / (self._kmax * z * bessel.J(nu=nu, d=1)(z) ** 2)

        # DVR kinetic term for radial function:
        K = np.ma.divide(
            (-1.0) ** (n[i1] - n[i2]) * 8.0 * z[i1] * z[i2],
            (z[i1] ** 2 - z[i2] ** 2) ** 2,
        ).filled(0)
        K[n, n] = 1.0 / 3.0 * (1.0 + 2.0 * (nu ** 2 - 1.0) / z ** 2)
        K *= self._kmax ** 2

        # Here we convert from the wavefunction Psi(r) to the radial
        # function u(r) = sqrt(r)*Psi(r) and back with factors of
        # sqrt(wr).  This includes the integration weights (since K is
        # defined acting on the basis functions).
        # Note: this makes the matrix non-hermitian, so don't do this if you
        # want to diagonalize.
        _tmp = np.sqrt(w * r)
        r2 = _tmp[i2]
        r1 = 1.0 / _tmp[i1]

        return K, r1, r2, w

    def nu(self, l):
        """Return `nu = l + d/2 - 1` for the centrifugal term.

        Arguments
        ---------
        l : int
           Angular quantum number.
        """
        nu = l + self._d / 2 - 1
        return nu

    def _r(self, N, l=0):
        r"""Return the abscissa."""
        # l=0 cylindrical: nu = l + d/2 - 1
        return bessel.j_root(nu=self.nu(l=l), N=N) / self._kmax

    def _F(self, n, r, d=0):
        r"""Return the dth derivative of the n'th basis function."""
        nu = 0.0  # l=0 cylindrical: nu = l + d/2 - 1
        rn = self.xyz[1].ravel()[n]
        zn = self._kmax * rn
        z = self._kmax * r
        H = bessel.J_sqrt_pole(nu=nu, zn=zn, d=0)
        coeff = math.sqrt(2.0 * self._kmax) * (-1.0) ** (n + 1) / (1.0 + r / rn)
        if 0 == d:
            return coeff * H(z)
        elif 1 == d:
            dH = bessel.J_sqrt_pole(nu=nu, zn=zn, d=1)
            return coeff * (dH(z) - H(z) / (z + zn)) * self._kmax
        else:
            raise NotImplementedError

    def get_F(self, r):
        """Return a function that can extrapolate a radial
        wavefunction to a new set of abscissa (x, r)."""
        x, r0 = self.xyz
        n = np.arange(r0.size)[:, None]

        # Here is the transform matrix
        _F = self._F(n, r) / self._F(n, r0.T)

        def F(u):
            return np.dot(u, _F)

        return F

    def F(self, u, xr):
        r"""Return u evaluated on the new abscissa (Assumes x does not
        change for now)"""
        x0, r0 = self.xyz
        x, r = xr
        assert np.allclose(x, x0)

        return self.get_F(r)(u)

    def get_Psi(self, r, return_matrix=False):
        r"""Return a function that can extrapolate a wavefunction to a
        new set of abscissa (x, r).

        This includes the factor of $\sqrt{r}$ that converts the
        wavefunction to the radial function, then uses the basis to
        extrapolate the radial function.

        Arguments
        ---------
        r : array
           The new abscissa in the radial direction (the $x$ values
           stay the same.)
        return_matrix : bool
           If True, then return the extrapolation matrix F so that
           ``Psi = np.dot(psi, F)``
        """
        x, r0 = self.xyz
        n = np.arange(r0.size)[:, None]

        # Here is the transform matrix
        _F = (np.sqrt(r) * self._F(n, r)) / (np.sqrt(r0.T) * self._F(n, r0.T))

        if return_matrix:
            return _F

        def Psi(psi):
            return np.dot(psi, _F)

        return Psi

    def Psi(self, psi, xr):
        r"""Return psi evaluated on the new abscissa (Assumes x does not
        change for now)"""
        x0, r0 = self.xyz
        x, r = xr
        assert np.allclose(x, x0)

        return self.get_Psi(r)(psi)

    def integrate1(self, n):
        """Return the integral of n over y and z."""
        n = np.asarray(n)
        x, r = self.xyz
        x_axis, r_axis = self.axes
        bcast = [None] * len(n.shape)
        bcast[x_axis] = slice(None)
        bcast[r_axis] = slice(None)
        return ((2 * np.pi * r * self.weights)[tuple(bcast)] * n).sum(axis=r_axis)

    def integrate2(self, n, y=None, Nz=100):
        """Return the integral of n over z (line-of-sight integral) at y.

        This is an Abel transform, and is used to compute the 1D
        line-of-sight integral as would be seen by a photographic
        image through an axial cloud.

        Arguments
        ---------
        n : array
           (Nx, Nr) array of the function to be integrated tabulated
           on the abscissa.  Note: the extrapolation assumes that `n =
           abs(psi)**2` where `psi` is well represented in the basis.
        y : array, None
           Ny points at which the resulting integral should be
           returned.  If not provided, then the function will be
           tabulated at the radial abscissa.
        Nz : int
           Number of points to use in z integral.
        """
        n = np.asarray(n)
        x, r = self.xyz
        if y is None:
            y = r

        y = y.ravel()
        Ny = len(y)

        x_axis, r_axis = self.axes
        y_axis = r_axis
        bcast_y = [None] * len(n.shape)
        bcast_z = [None] * len(n.shape)
        bcast_y[y_axis] = slice(None)
        bcast_y.append(None)
        bcast_z.append(slice(None))

        bcast_y, bcast_z = tuple(bcast_y), tuple(bcast_z)

        z = np.linspace(0, r.max(), Nz)
        shape_xyz = n.shape[:-1] + (Ny, Nz)
        rs = np.sqrt(y.ravel()[bcast_y] ** 2 + z[bcast_z] ** 2)
        n_xyz = (abs(self.Psi(np.sqrt(n), (x, rs.ravel()))) ** 2).reshape(shape_xyz)
        n_2D = 2 * np.trapz(n_xyz, z, axis=-1)
        return n_2D
'''

import itertools
import math

import numpy as np
import scipy.fftpack

from mmfutils.containers import ObjectBase

from . import interfaces
from .interfaces import (
    implementer,
    IBasis,
    IBasisKx,
    IBasisLz,
    IBasisWithConvolution,
    BasisMixin,
)

from mmfutils.performance.fft import fft, ifft, fftn, ifftn, resample
from .utils import prod, dst, idst, get_xyz, get_kxyz
from mmfutils.math import bessel

sp = scipy

_TINY = np.finfo(float).tiny

__all__ = ["SphericalBasis", "PeriodicBasis", "CartesianBasis", "interfaces"]


@implementer(IBasisWithConvolution)
class SphericalBasis(ObjectBase, BasisMixin):
    """1-dimensional basis for radial problems.

    We represent exactly `N` positive abscissa, excluding the origin and use
    the discrete sine transform.  We represent the square-root of the
    wavefunctions here so that a factor of `r` is required to convert these
    into the radial functions.  Unlike the DVR techniques, this approach allows
    us to compute the Coulomb interaction for example.
    """

    def __init__(self, N, R):
        self.N = N
        self.R = R
        super().__init__()

    def init(self):
        dx = self.R / self.N
        r = np.arange(1, self.N + 1) * dx
        k = np.pi * (0.5 + np.arange(self.N)) / self.R
        self.xyz = [r]
        self._pxyz = [k]
        self.metric = 4 * np.pi * r ** 2 * dx
        self.k_max = k.max()

    def laplacian(self, y, factor=1.0, exp=False):
        """Return the laplacian of `y` times `factor` or the
        exponential of this.

        Arguments
        ---------
        factor : float
           Additional factor (mostly used with `exp=True`).  The
           implementation must be careful to allow the factor to
           broadcast across the components.
        exp : bool
           If `True`, then compute the exponential of the laplacian.
           This is used for split evolvers.
        """
        r = self.xyz[0]
        K = -factor * self._pxyz[0] ** 2
        if exp:
            K = np.exp(K)

        ys = [y.real, y.imag] if np.iscomplexobj(y) else [y]
        res = [idst(K * dst(r * _y)) / r for _y in ys]

        if np.iscomplexobj(y):
            res = res[0] + 1j * res[1]
        else:
            res = res[0]

        return res

    def coulomb_kernel(self, k):
        """Form for the truncated Coulomb kernel."""
        D = 2 * self.R
        return (
            4 * np.pi * np.ma.divide(1.0 - np.cos(k * D), k ** 2).filled(D ** 2 / 2.0)
        )

    def convolve_coulomb(self, y, form_factors=[]):
        """Modified Coulomb convolution to include form-factors (if provided).

        This version implemented a 3D spherically symmetric convolution.
        """
        y = np.asarray(y)
        r = self.xyz[0]
        N, R = self.N, self.R

        # Padded arrays with trailing _
        ry_ = np.concatenate([r * y, np.zeros(y.shape, dtype=y.dtype)], axis=-1)
        k_ = np.pi * (0.5 + np.arange(2 * N)) / (2 * R)
        K = prod([_K(k_) for _K in [self.coulomb_kernel] + form_factors])
        return idst(K * dst(ry_))[..., :N] / r

    def convolve(self, y, C=None, Ck=None):
        """Return the periodic convolution `int(C(x-r)*y(r),r)`.

        Note: this is the 3D convolution.
        """
        r = self.xyz[0]
        k = self._pxyz[0]
        N, R = self.N, self.R
        R_N = R / N
        if Ck is None:
            C0 = (self.metric * C).sum()
            Ck = np.ma.divide(2 * np.pi * R_N * dst(r * C), k).filled(C0)
        else:
            Ck = Ck(k)
        return idst(Ck * dst(r * y)) / r


@implementer(IBasisWithConvolution, IBasisKx, IBasisLz)
class PeriodicBasis(ObjectBase, BasisMixin):
    """dim-dimensional periodic bases.

    Parameters
    ----------
    Nxyz : (Nx, Ny, ...)
       Number of lattice points in basis.
    Lxyz : (Lx, Ly, ...)
       Size of each dimension (length of box and radius)
    symmetric_lattice: bool
       If True, then shift the lattice so that it is symmetric about
       the origin.  The default is to ensure that there is a lattice
       point at the origin which will make the lattice asymmetric for
       even Nxyz.
    axes : (int, int, ...)
       Axes in array y which correspond to the x, y, ... axes here.
       This is required for cases where y has additional dimensions.
       The default is the last dim axes (best for performance).
    boost_pxyz : float
       Momentum of moving frame.  Momenta are shifted by this, which
       corresponds to working in a boosted frame with velocity `vx = px/m`.
    smoothing_cutoff : float
       Fraction of maximum momentum used in the function smooth().
    """

    # Select operations are performed using self.xp instead of numpy.
    # This can be replaced cupy to provide gpu support with minimal
    # code changes.  Similarly with the fft functions and a generic
    # function to convert an array into a numpy array on the host.
    xp = np
    _fft = staticmethod(fft)
    _ifft = staticmethod(ifft)
    _fftn = staticmethod(fftn)
    _ifftn = staticmethod(ifftn)
    _asnumpy = staticmethod(np.asarray)  # Convert to numpy array

    def __init__(
        self,
        Nxyz,
        Lxyz,
        symmetric_lattice=False,
        axes=None,
        boost_pxyz=None,
        smoothing_cutoff=0.8,
    ):
        self.symmetric_lattice = symmetric_lattice
        self.Nxyz = np.asarray(Nxyz)
        self.Lxyz = np.asarray(Lxyz)
        self.smoothing_cutoff = smoothing_cutoff
        if boost_pxyz is None:
            boost_pxyz = np.zeros_like(self.Lxyz)
        self.boost_pxyz = np.asarray(boost_pxyz)
        if axes is None:
            axes = np.arange(-self.dim, 0)
        self.axes = np.asarray(axes)
        super().__init__()

    def init(self):
        self.xyz = tuple(
            map(
                self.xp.asarray,
                get_xyz(
                    Nxyz=self.Nxyz,
                    Lxyz=self.Lxyz,
                    symmetric_lattice=self.symmetric_lattice,
                ),
            )
        )
        self._pxyz = tuple(
            map(self.xp.asarray, get_kxyz(Nxyz=self.Nxyz, Lxyz=self.Lxyz))
        )
        self._pxyz_derivative = tuple(
            map(self.xp.asarray, get_kxyz(Nxyz=self.Nxyz, Lxyz=self.Lxyz))
        )

        # Zero out odd highest frequency component.
        for _N, _p in zip(self.Nxyz, self._pxyz_derivative):
            _p.ravel()[_N // 2] = 0.0

        # Add boosts
        self._pxyz = [
            _p - _b for (_p, _b) in zip(self._pxyz, self.xp.asarray(self.boost_pxyz))
        ]
        self.metric = np.prod(self.Lxyz / self.Nxyz)
        self.k_max = self._asnumpy([abs(_p).max() for _p in self._pxyz])

        p2_pc2 = sum(
            (_p / (self.smoothing_cutoff * _p).max()) ** 2 for _p in self._pxyz
        )
        self._smoothing_factor = self.xp.where(p2_pc2 < 1, 1, 0)
        # np.exp(-p2_pc2**4)
        # self._smoothing_factor = 1.0

        # Memoize momentum sums for speed
        _kx2 = self._pxyz[0] ** 2
        _kyz2 = sum(_p ** 2 for _p in self._pxyz[1:])
        _k2 = _kx2 + _kyz2
        self._k2_kx2_kyz2 = (_k2, _kx2, _kyz2)

    @property
    def kx(self):
        return self._pxyz[0]

    @property
    def Lx(self):
        return self.Lxyz[0]

    @property
    def Nx(self):
        return self.Nxyz[0]

    def laplacian(
        self, y, factor=1.0, exp=False, kx2=None, k2=None, kwz2=0, twist_phase_x=None
    ):
        """Return the laplacian of `y` times `factor` or the exponential of this.

        Arguments
        ---------
        factor : float
           Additional factor (mostly used with `exp=True`).  The
           implementation must be careful to allow the factor to
           broadcast across the components.
        exp : bool
           If `True`, then compute the exponential of the laplacian.
           This is used for split evolvers. Only allowed to be `True`
           if `kwz2 == 0`.
        kx2 : array, optional
           Replacement for the default `kx2=kx**2` used when computing the
           "laplacian".  This would allow you, for example, to implement a
           modified dispersion relationship like ``1-cos(kx)`` rather than
           ``kx**2``.
        kwz2 : None, float
           Angular velocity of the frame expressed as `kwz2 = m*omega_z/hbar`.
        k2 : array, optional
           Replacement for `k2 = kx**2 + ky**2 + kz**2`.
        twist_phase_x : array, optional
           To implement twisted boundary conditions, one needs to remove an
           overall phase from the wavefunction rendering it periodic for use
           the the FFT.  This the the phase that should be removed.  Note: to
           compensate, the momenta should be shifted as well::

              -factor * twist_phase_x*ifft((k+k_twist)**2*fft(y/twist_phase_x)
        """
        _k2, _kx2, _kyz2 = self._k2_kx2_kyz2
        if k2 is None:
            if kx2 is None:
                k2 = _k2
            else:
                kx2 = self.xp.asarray(kx2)
                k2 = kx2 + _kyz2
        else:
            k2 = self.xp.asarray(k2)
            assert kx2 is None

        K = -factor * k2
        if exp:
            if kwz2 != 0:
                raise NotImplementedError(
                    f"Cannot use exp=True if kwz2 != 0 (got {kwz2})."
                )
            K = self.xp.exp(K)

        if twist_phase_x is not None:
            twist_phase_x = self.xp.asarray(twist_phase_x)
            y = y / twist_phase_x

        yt = self.fftn(y)
        laplacian_y = self.ifftn(K * yt)

        if kwz2 != 0:
            laplacian_y += 2 * kwz2 * factor * self.apply_Lz_hbar(y, yt=yt)

        if twist_phase_x is not None:
            laplacian_y *= twist_phase_x
        return laplacian_y

    def apply_Lz_hbar(self, y, yt=None):
        """Apply `Lz/hbar` to `y`."""
        if yt is None:
            yt = self.fftn(y)
        x, y = self.xyz[:2]
        kx, ky = self._pxyz[:2]
        return x * self.ifftn(ky * yt) - y * self.ifftn(kx * yt)

    # We need these wrappers because the state may have additional
    # indices for components etc. in front.
    def fft(self, x, axis):
        """Perform the fft along self.axes[axis]"""
        axis = self.axes[axis] % len(x.shape)
        return self._fft(x, axis=axis)

    def ifft(self, x, axis):
        """Perform the ifft along self.axes[axis]"""
        axis = self.axes[axis] % len(x.shape)
        return self._ifft(x, axis=axis)

    def fftn(self, x):
        """Perform the fft along spatial axes"""
        axes = self.axes % len(x.shape)
        return self._fftn(x, axes=axes)

    def ifftn(self, x):
        """Perform the ifft along spatial axes"""
        axes = self.axes % len(x.shape)
        return self._ifftn(x, axes=axes)

    def smooth(self, x, frac=0.8):
        """Smooth the state by multiplying by form factor."""
        return self.ifftn(self._smoothing_factor * self.fftn(x))

    def get_gradient(self, y):
        # TODO: Check this for the highest momentum issue.
        return [
            self.ifft(1j * _p * self.fft(y, axis=_i), axis=_i)
            for _i, _p in enumerate(self._pxyz_derivative)
        ]

    def get_divergence(self, ys):
        # TODO: Check this for the highest momentum issue.
        return sum(
            self.ifft(1j * _p * self.fft(_y, axis=_i), axis=_i)
            for _i, (_p, _y) in enumerate(zip(self._pxy_derivativez, ys))
        )

    @staticmethod
    def _bcast(n, N):
        """Use this to broadcast a 1D array along the n'th of N dimensions"""
        inds = [None] * N
        inds[n] = slice(None)
        return inds

    def coulomb_kernel(self, k):
        """Form for the Coulomb kernel.

        The normalization here is that the k=0 component is set to
        zero.  This means that the charge distribution has an overall
        constant background removed so that the net charge in the unit
        cell is zero.
        """
        return 4 * np.pi * np.ma.divide(1.0, k ** 2).filled(0.0)

    def convolve_coulomb(self, y, form_factors=[]):
        """Periodic convolution with the Coulomb kernel."""
        y = np.asarray(y)

        # This broadcasts to the appropriate size if there are
        # multiple components.
        # dim = len(np.asarray(self.Lxyz))
        # N = np.asarray(y.shape)
        # b_cast = [None] * (dim - len(N)) + [slice(None)]*dim

        k = np.sqrt(sum(_k ** 2 for _k in self._pxyz))
        Ck = prod([_K(k) for _K in [self.coulomb_kernel] + form_factors])
        return self.ifftn(Ck * self.fftn(y))

    def convolve(self, y, C=None, Ck=None):
        """Return the periodic convolution `int(C(x-r)*y(r),r)`.

        Arguments
        ---------
        y : array
           Usually the density, but can be any array
        C : array
           Convolution kernel. The convolution will be performed using the FFT.
        Ck : function (optional)
           If provided, then this function will be used instead directly in
           momentum space.  Assumed to be spherically symmetric (will be passed
           only the magnitude `k`)
        """
        if Ck is None:
            Ck = self.fftn(C)
        else:
            k = np.sqrt(sum(_k ** 2 for _k in self._pxyz))
            Ck = Ck(k)
        return self.ifftn(Ck * self.fftn(y))

    @property
    def dim(self):
        return len(self.Nxyz)


@implementer(IBasisWithConvolution)
class CartesianBasis(PeriodicBasis):
    """N-dimensional periodic bases but with Coulomb convolution that does not
    use periodic images.  Use this for nuclei in free space.

    Parameters
    ----------
    Nxyz : (Nx, Ny, ...)
       Number of lattice points in basis.
    Lxyz : (Lx, Ly, ...)
       Size of each dimension (length of box and radius)
    symmetric_lattice: bool
       If True, then shift the lattice so that it is symmetric about
       the origin.  The default is to ensure that there is a lattice
       point at the origin which will make the lattice asymmetric for
       even Nxyz.
    axes : (int, int, ...)
       Axes in array y which correspond to the x, y, ... axes here.
       This is required for cases where y has additional dimensions.
       The default is the last dim axes (best for performance).
    fast_coulomb : bool
       If `True`, use the fast Coulomb algorithm which is slightly less
       accurate but much faster.
    """

    def __init__(
        self, Nxyz, Lxyz, axes=None, symmetric_lattice=False, fast_coulomb=True
    ):
        self.fast_coulomb = fast_coulomb
        PeriodicBasis.__init__(
            self, Nxyz=Nxyz, Lxyz=Lxyz, axes=axes, symmetric_lattice=symmetric_lattice
        )

    def convolve_coulomb_fast(self, y, form_factors=[], correct=False):
        r"""Return the approximate convolution `int(C(x-r)*y(r),r)` where

        .. math::
           C(r) = 1/r

        is the Coulomb potential (without charges etc.)

        Arguments
        ---------
        y : array
           Usually the density, but can be any array
        correct : bool
           If `True`, then include the high frequency components via the
           periodic convolution.

        Notes
        -----
        This version uses the Truncated Kernel Expansion method which uses the
        Truncated Kernel

        .. math::
           4\pi(1-\cos\sqrt{3}Lk)/k^2

        on a padded array to remove the images, approximating the linear
        convolution without the highest frequency modes.  By choosing the
        smaller lattice to be at least 3 times smaller we can guarantee that
        the padded array will fit into memory.  This can be augmented by the
        periodic convolution to fill in the higher modes.

        There are two sources of error here:

        * We presently use the same periodic ``resample`` method to interpolate
          the linear convolution to the larger grid.  This assumes that the
          function is periodic -- which it is not -- and can introduce some
          aliasing artifacts.  Some preliminary experimentation shows, however,
          that these are generally small.  Perhaps cubic spline interpolation
          could be used to improve the interpolation, but this is not clear
          yet.
        * The contribution from the higher modes are computed from the periodic
          convolution which could in principle be contaminated by images.
          However, for smooth functions, there should be little amplitude here,
          and it should consist only of higher multipoles, so the contamination
          should be small.
        """
        y = np.asarray(y)
        L = np.asarray(self.Lxyz)
        dim = len(L)
        N = np.asarray(y.shape)
        N0 = N.copy()
        N0[-dim:] = N[-dim:] // 3

        y0 = resample(y, N0)
        V = resample(
            self.convolve_coulomb_exact(y0, form_factors=form_factors, method="pad"), N
        )
        if correct:
            k = np.sqrt(sum(_K ** 2 for _K in self._pxyz))
            C = 4 * np.pi * np.ma.divide(1.0, k ** 2).filled(0.0)
            for F in form_factors:
                C = C * F(k)
            dV = self.ifftn(C * self.fftn(y - resample(y0, N)))
            if np.issubdtype(V.dtype, np.complex128):
                V += dV
            else:
                assert np.allclose(0, V.imag)
                V += dV.real
        return V

    def convolve_coulomb_exact(self, y, form_factors=[], method="sum"):
        r"""Return the convolution `int(C(x-r)*y(r),r)` where

        .. math::
           C(r) = 1/r

        is the Coulomb potential (without charges etc.)

        Arguments
        ---------
        y : array
           Usually the density, but can be any array
        method : 'sum', 'pad'
           Either zero-pad the array (takes extra memory but can use multiple
           cores) or sum over the 27 small transforms (slow).

        This function is designed for computing the Coulomb potential of a
        charge distribution.  In this case, one would have the kernel:

        .. math::
           4\pi/k^2

        or in the case of non-periodic convolution to remove the images

        .. math::
           4\pi(1-\cos\sqrt{3}Lk)/k^2
        """
        y = np.asarray(y)
        L = np.asarray(self.Lxyz)
        dim = len(L)
        D = np.sqrt((L ** 2).sum())  # Diameter of cell

        def C(k):
            C = 4 * np.pi * np.ma.divide(1 - np.cos(D * k), k ** 2).filled(D ** 2 / 2.0)
            for F in form_factors:
                C = C * F(k)
            return C

        if method == "sum":
            # Sum with a loop.  Minimizes the memory usage, but will not
            # use multiple cores.
            K = self._pxyz
            X = self.xyz
            V = np.zeros(y.shape, dtype=y.dtype)
            for l in itertools.product(np.arange(3), repeat=dim):
                delta = [2 * np.pi * _l / 3.0 / _L for _l, _L in zip(l, L)]
                exp_delta = np.exp(1j * sum(_d * _x for _x, _d in zip(X, delta)))
                y_delta = exp_delta.conj() * y
                k = np.sqrt(sum((_k + _d) ** 2 for _k, _d in zip(K, delta)))
                dV = exp_delta * self.ifftn(C(k) * self.fftn(y_delta))
                if np.issubdtype(V.dtype, np.complex128):
                    V += dV
                else:
                    assert np.allclose(0, V.imag)
                    V += dV.real
            return V / dim ** 3
        elif method == "pad":
            N = np.asarray(y.shape[-dim:])
            N_padded = 3 * N
            L_padded = 3 * L
            shape = np.asarray(y.shape)
            shape_padded = shape.copy()
            shape_padded[-dim:] = N_padded
            y_padded = np.zeros(shape_padded, dtype=y.dtype)
            inds = tuple(slice(0, _N) for _N in shape)
            y_padded[inds] = y
            k = np.sqrt(sum(_K ** 2 for _K in get_kxyz(N_padded, L_padded)))

            # This broadcasts to the appropriate size
            b_cast = (None,) * (dim - len(N)) + (slice(None),) * dim
            return self.ifftn(C(k)[b_cast] * self.fftn(y_padded))[inds]
        else:
            raise NotImplementedError(
                "method=%s not implemented: use 'sum' or 'pad'" % (method,)
            )

    def convolve_coulomb(self, y, form_factors=[], **kw):
        if self.fast_coulomb:
            return self.convolve_coulomb_fast(y, form_factors=form_factors, **kw)
        else:
            return self.convolve_coulomb_exact(y, form_factors=form_factors, **kw)


@implementer(IBasis, IBasisKx)
class CylindricalBasis(ObjectBase, BasisMixin):
    r"""2D basis for Cylindrical coordinates via a DVR basis.

    This represents 3-dimensional problems with axial symmetry, but only has
    two dimensions (x, r).

    Parameters
    ----------
    Nxr : (Nx, Nr)
       Number of lattice points in basis.
    Lxr : (L, R)
       Size of each dimension (length of box and radius)
    twist : float
       Twist (angle) in periodic dimension.  This adds a constant offset to the
       momenta allowing one to study Bloch waves.
    boost_px : float
       Momentum of moving frame (along the x axis).  Momenta are shifted by
       this, which corresponds to working in a boosted frame with velocity
       `vx = boost_px/m`.
    axes : (int, int)
       Axes in array y which correspond to the x and r axes here.
       This is required for cases where y has additional dimensions.
       The default is the last two axes (best for performance).
    """
    _d = 2  # Dimension of spherical part (see nu())

    def __init__(self, Nxr, Lxr, twist=0, boost_px=0, axes=(-2, -1), symmetric_x=True):
        self.twist = twist
        self.boost_px = np.asarray(boost_px)
        self.Nxr = np.asarray(Nxr)
        self.Lxr = np.asarray(Lxr)
        self.symmetric_x = symmetric_x
        self.axes = np.asarray(axes)
        super().__init__()

    def init(self):
        Lx, R = self.Lxr
        x = get_xyz(Nxyz=self.Nxr, Lxyz=self.Lxr, symmetric_lattice=self.symmetric_x)[0]
        kx0 = get_kxyz(Nxyz=self.Nxr, Lxyz=self.Lxr)[0]
        self.kx = kx0 + float(self.twist) / Lx - self.boost_px
        self._kx0 = kx0
        self._kx2 = self.kx ** 2

        self.y_twist = np.exp(1j * self.twist * x / Lx)

        Nx, Nr = self.Nxr

        # For large n, the roots of the bessel function are approximately
        # z[n] = (n + 0.75)*pi, so R = r_max = z_max/k_max = (N-0.25)*pi/kmax
        # This self._kmax defines the DVR basis, not self.k_max
        self._kmax = (Nr - 0.25) * np.pi / R

        # This is just the maximum momentum for diagnostics,
        # determining cutoffs etc.
        self.k_max = np.array([abs(self.kx).max(), self._kmax])

        nr = np.arange(Nr)[None, :]
        r = self._r(Nr)[None, :]  # Do this after setting _kmax
        self.xyz = [x, r]

        _lambda = np.asarray(
            [1.0 / (self._F(_nr, _r)) ** 2 for _nr, _r in zip(nr.ravel(), r.ravel())]
        )[None, :]
        self.metric = 2 * np.pi * r * _lambda * (Lx / Nx)
        self.metric.setflags(write=False)
        # Get the DVR kinetic piece for radial component
        K, r1, r2, w = self._get_K()

        # We did not apply the sqrt(r) factors so at this point, K is still
        # Hermitian and we can diagonalize for later exponentiation.
        d, V = sp.linalg.eigh(K)  # K = np.dot(V*d, V.T)

        # Here we convert from the wavefunction Psi(r) to the radial
        # function u(r) = sqrt(r)*Psi(r) and back with factors of sqrt(r).
        K *= r1
        K *= r2

        self.weights = w
        self._Kr = K
        self._Kr_diag = (r1, r2, V, d)  # For use when exponentiating

        # And factor for x.
        self._Kx = self._kx2

        # Cache for K_data from apply_exp_K.
        self._K_data = []

    @property
    def Lx(self):
        return self.Lxr[0]

    @property
    def Nx(self):
        return self.Nxr[0]

    ######################################################################
    # IBasisMinimal: Required methods
    def laplacian(self, y, factor=1.0, exp=False, kx2=None, twist_phase_x=None):
        r"""Return the laplacian of y.

        Arguments
        ---------
        factor : float
           Additional factor (mostly used with `exp=True`).  The
           implementation must be careful to allow the factor to
           broadcast across the components.
        exp : bool
           If `True`, then compute the exponential of the laplacian.
           This is used for split evolvers. Only allowed to be `True`
           if `kwz2 == 0`.
        kx2 : array, optional
           Replacement for the default `kx2=kx**2` used when computing the
           "laplacian".  This would allow you, for example, to implement a
           modified dispersion relationship like ``1-cos(kx)`` rather than
           ``kx**2``.
        kwz2 : float
           Angular velocity of the frame expressed as `kwz2 = m*omega_z/hbar`.
        twist_phase_x : array, optional
           To implement twisted boundary conditions, one needs to remove an
           overall phase from the wavefunction rendering it periodic for use
           the the FFT.  This the the phase that should be removed.  Note: to
           compensate, the momenta should be shifted as well::

              -factor * twist_phase_x*ifft((k+k_twist)**2*fft(y/twist_phase_x)
        """
        if not exp:
            return self.apply_K(y=y, kx2=kx2, twist_phase_x=twist_phase_x) * (-factor)
        else:
            return self.apply_exp_K(
                y=y, factor=-factor, kx2=kx2, twist_phase_x=twist_phase_x
            )

    ######################################################################

    def get_gradient(self, y):
        """Returns the gradient along the x axis."""
        kx = self.kx
        return [self.ifft(1j * kx * self.fft(y)), NotImplemented]

    def apply_Lz(self, y, hermitian=False):
        raise NotImplementedError

    def apply_Px(self, y, hermitian=False):
        r"""Apply the Pz operator to y without any px.

        Requires :attr:`_pxyz` to be defined.
        """
        return self.y_twist * self.ifft(self._kx0 * self.fft(y / self.y_twist))

    def apply_exp_K(self, y, factor, kx2=None, twist_phase_x=None):
        r"""Return `exp(K*factor)*y` or return precomputed data if
        `K_data` is `None`.
        """
        if kx2 is None:
            kx2 = self._Kx
        _K_data_max_len = 3
        ind = None
        for _i, (_f, _d) in enumerate(self._K_data):
            if np.allclose(factor, _f):
                ind = _i
        if ind is None:
            _r1, _r2, V, d = self._Kr_diag
            exp_K_r = _r1 * np.dot(V * np.exp(factor * d), V.T) * _r2
            exp_K_x = np.exp(factor * kx2)
            K_data = (exp_K_r, exp_K_x)
            self._K_data.append((factor, K_data))
            ind = -1
            while len(self._K_data) > _K_data_max_len:
                # Reduce storage
                self._K_data.pop(0)

        K_data = self._K_data[ind][1]
        exp_K_r, exp_K_x = K_data
        if twist_phase_x is None or self.twist == 0:
            tmp = self.ifft(exp_K_x * self.fft(y))
        else:
            if twist_phase_x is None:
                twist_phase_x = self.y_twist
            tmp = twist_phase_x * self.ifft(exp_K_x * self.fft(y / twist_phase_x))
        return np.einsum("...ij,...yj->...yi", exp_K_r, tmp)

    def apply_K(self, y, kx2=None, twist_phase_x=None):
        r"""Return `K*y` where `K = k**2/2`"""
        # Here is how the indices work:
        if kx2 is None:
            kx2 = self._Kx

        if twist_phase_x is None or self.twist == 0:
            yt = self.fft(y)
            yt *= kx2
            yt = self.ifft(yt)
        else:
            if twist_phase_x is None:
                twist_phase_x = self.y_twist
            yt = self.fft(y / twist_phase_x)
            yt *= kx2
            yt = self.ifft(yt)
            yt *= twist_phase_x

        # C <- alpha*B*A + beta*C    A = A^T  zSYMM or zHYMM but not supported
        # maybe cvxopt.blas?  Actually, A is not symmetric... so be careful!
        yt += np.dot(y, self._Kr.T)
        return yt

    ######################################################################
    # FFT and DVR Helper functions.
    #
    # These are specific to the basis, defining the kinetic energy
    # matrix for example.

    # We need these wrappers because the state may have additional
    # indices for components etc. in front.
    def fft(self, x):
        """Perform the fft along the x axes"""
        # Makes sure that
        axis = (self.axes % len(x.shape))[0]
        return fft(x, axis=axis)

    def ifft(self, x):
        """Perform the fft along the x axes"""
        axis = (self.axes % len(x.shape))[0]
        return ifft(x, axis=axis)

    def _get_K(self, l=0):
        r"""Return `(K, r1, r2, w)`: the DVR kinetic term for the radial function
        and the appropriate factors for converting to the radial coordinates.

        This term effects the $-d^2/dr^2 - (\nu^2 - 1/4)/r^2$ term.

        Returns
        -------
        K : array
           Operates on radial wavefunctions
        r1, r2 : array
           K*r1*r2 operators on the full wavefunction (but is no longer
           Hermitian)
        w : array
           Quadrature integration weights.
        """
        nu = self.nu(l=l)
        if l == 0:
            r = self.xyz[1].ravel()
        else:
            r = self._r(self.Nxr[1], l=l)
        z = self._kmax * r
        n = np.arange(len(z))
        i1 = (slice(None), None)
        i2 = (None, slice(None))

        # Quadrature weights
        w = 2.0 / (self._kmax * z * bessel.J(nu=nu, d=1)(z) ** 2)

        # DVR kinetic term for radial function:
        K = np.ma.divide(
            (-1.0) ** (n[i1] - n[i2]) * 8.0 * z[i1] * z[i2],
            (z[i1] ** 2 - z[i2] ** 2) ** 2,
        ).filled(0)
        K[n, n] = 1.0 / 3.0 * (1.0 + 2.0 * (nu ** 2 - 1.0) / z ** 2)
        K *= self._kmax ** 2

        # Here we convert from the wavefunction Psi(r) to the radial
        # function u(r) = sqrt(r)*Psi(r) and back with factors of
        # sqrt(wr).  This includes the integration weights (since K is
        # defined acting on the basis functions).
        # Note: this makes the matrix non-hermitian, so don't do this if you
        # want to diagonalize.
        _tmp = np.sqrt(w * r)
        r2 = _tmp[i2]
        r1 = 1.0 / _tmp[i1]

        return K, r1, r2, w

    def nu(self, l):
        """Return `nu = l + d/2 - 1` for the centrifugal term.

        Arguments
        ---------
        l : int
           Angular quantum number.
        """
        nu = l + self._d / 2 - 1
        return nu

    def _r(self, N, l=0):
        r"""Return the abscissa."""
        # l=0 cylindrical: nu = l + d/2 - 1
        return bessel.j_root(nu=self.nu(l=l), N=N) / self._kmax

    def _F(self, n, r, d=0):
        r"""Return the dth derivative of the n'th basis function."""
        nu = 0.0  # l=0 cylindrical: nu = l + d/2 - 1
        rn = self.xyz[1].ravel()[n]
        zn = self._kmax * rn
        z = self._kmax * r
        H = bessel.J_sqrt_pole(nu=nu, zn=zn, d=0)
        coeff = math.sqrt(2.0 * self._kmax) * (-1.0) ** (n + 1) / (1.0 + r / rn)
        if 0 == d:
            return coeff * H(z)
        elif 1 == d:
            dH = bessel.J_sqrt_pole(nu=nu, zn=zn, d=1)
            return coeff * (dH(z) - H(z) / (z + zn)) * self._kmax
        else:
            raise NotImplementedError

    def get_F(self, r):
        """Return a function that can extrapolate a radial
        wavefunction to a new set of abscissa (x, r)."""
        x, r0 = self.xyz
        n = np.arange(r0.size)[:, None]

        # Here is the transform matrix
        _F = self._F(n, r) / self._F(n, r0.T)

        def F(u):
            return np.dot(u, _F)

        return F

    def F(self, u, xr):
        r"""Return u evaluated on the new abscissa (Assumes x does not
        change for now)"""
        x0, r0 = self.xyz
        x, r = xr
        assert np.allclose(x, x0)

        return self.get_F(r)(u)

    def get_Psi(self, r, return_matrix=False):
        r"""Return a function that can extrapolate a wavefunction to a
        new set of abscissa (x, r).

        This includes the factor of $\sqrt{r}$ that converts the
        wavefunction to the radial function, then uses the basis to
        extrapolate the radial function.

        Arguments
        ---------
        r : array
           The new abscissa in the radial direction (the $x$ values
           stay the same.)
        return_matrix : bool
           If True, then return the extrapolation matrix F so that
           ``Psi = np.dot(psi, F)``
        """
        x, r0 = self.xyz
        n = np.arange(r0.size)[:, None]

        # Here is the transform matrix
        _F = (np.sqrt(r) * self._F(n, r)) / (np.sqrt(r0.T) * self._F(n, r0.T))

        if return_matrix:
            return _F

        def Psi(psi):
            return np.dot(psi, _F)

        return Psi

    def Psi(self, psi, xr):
        r"""Return psi evaluated on the new abscissa (Assumes x does not
        change for now)"""
        x0, r0 = self.xyz
        x, r = xr
        assert np.allclose(x, x0)

        return self.get_Psi(r)(psi)

    def integrate1(self, n):
        """Return the integral of n over y and z."""
        n = np.asarray(n)
        x, r = self.xyz
        x_axis, r_axis = self.axes
        bcast = [None] * len(n.shape)
        bcast[x_axis] = slice(None)
        bcast[r_axis] = slice(None)
        return ((2 * np.pi * r * self.weights)[tuple(bcast)] * n).sum(axis=r_axis)

    def integrate2(self, n, y=None, Nz=100):
        """Return the integral of n over z (line-of-sight integral) at y.

        This is an Abel transform, and is used to compute the 1D
        line-of-sight integral as would be seen by a photographic
        image through an axial cloud.

        Arguments
        ---------
        n : array
           (Nx, Nr) array of the function to be integrated tabulated
           on the abscissa.  Note: the extrapolation assumes that `n =
           abs(psi)**2` where `psi` is well represented in the basis.
        y : array, None
           Ny points at which the resulting integral should be
           returned.  If not provided, then the function will be
           tabulated at the radial abscissa.
        Nz : int
           Number of points to use in z integral.
        """
        n = np.asarray(n)
        x, r = self.xyz
        if y is None:
            y = r

        y = y.ravel()
        Ny = len(y)

        x_axis, r_axis = self.axes
        y_axis = r_axis
        bcast_y = [None] * len(n.shape)
        bcast_z = [None] * len(n.shape)
        bcast_y[y_axis] = slice(None)
        bcast_y.append(None)
        bcast_z.append(slice(None))

        bcast_y, bcast_z = tuple(bcast_y), tuple(bcast_z)

        z = np.linspace(0, r.max(), Nz)
        shape_xyz = n.shape[:-1] + (Ny, Nz)
        rs = np.sqrt(y.ravel()[bcast_y] ** 2 + z[bcast_z] ** 2)
        n_xyz = (abs(self.Psi(np.sqrt(n), (x, rs.ravel()))) ** 2).reshape(shape_xyz)
        n_2D = 2 * np.trapz(n_xyz, z, axis=-1)
        return n_2D


'''
Incomplete!
@implementer(IBasis, IBasisKx)
class SphericalDVRBasis(ObjectBase, BasisMixin):
    r"""3D Spherical DVR basis.

    This represents 3-dimensional problems with spherical symmetry.

    Parameters
    ----------
    N : int
       Number of lattice points in basis.
    R : float
       Radius of the basis.
    """
    _d = 3  # Dimension of spherical part (see nu())

    def __init__(self, N, R):
        self.N = N
        self.R = R
        super().__init__()

    def init(self):
        R = self.R
        N = self.N

        # For large n, the roots of the bessel function are approximately
        # z[n] = (n + 0.75)*pi, so R = r_max = z_max/k_max = (N-0.25)*pi/kmax
        # This self._kmax defines the DVR basis, not self.k_max
        self._kmax = (Nr - 0.25) * np.pi / R

        # This is just the maximum momentum for diagnostics,
        # determining cutoffs etc.
        self.k_max = np.array([abs(self.kx).max(), self._kmax])

        nr = np.arange(Nr)[None, :]
        r = self._r(Nr)[None, :]  # Do this after setting _kmax
        self.xyz = [x, r]

        _lambda = np.asarray(
            [1.0 / (self._F(_nr, _r)) ** 2 for _nr, _r in zip(nr.ravel(), r.ravel())]
        )[None, :]
        self.metric = 2 * np.pi * r * _lambda * (Lx / Nx)
        self.metric.setflags(write=False)
        # Get the DVR kinetic piece for radial component
        K, r1, r2, w = self._get_K()

        # We did not apply the sqrt(r) factors so at this point, K is still
        # Hermitian and we can diagonalize for later exponentiation.
        d, V = sp.linalg.eigh(K)  # K = np.dot(V*d, V.T)

        # Here we convert from the wavefunction Psi(r) to the radial
        # function u(r) = sqrt(r)*Psi(r) and back with factors of sqrt(r).
        K *= r1
        K *= r2

        self.weights = w
        self._Kr = K
        self._Kr_diag = (r1, r2, V, d)  # For use when exponentiating

        # And factor for x.
        self._Kx = self._kx2

        # Cache for K_data from apply_exp_K.
        self._K_data = []

    @property
    def Lx(self):
        return self.Lxr[0]

    @property
    def Nx(self):
        return self.Nxr[0]

    ######################################################################
    # IBasisMinimal: Required methods
    def laplacian(self, y, factor=1.0, exp=False, kx2=None, twist_phase_x=None):
        r"""Return the laplacian of y.

        Arguments
        ---------
        factor : float
           Additional factor (mostly used with `exp=True`).  The
           implementation must be careful to allow the factor to
           broadcast across the components.
        exp : bool
           If `True`, then compute the exponential of the laplacian.
           This is used for split evolvers. Only allowed to be `True`
           if `kwz2 == 0`.
        kx2 : array, optional
           Replacement for the default `kx2=kx**2` used when computing the
           "laplacian".  This would allow you, for example, to implement a
           modified dispersion relationship like ``1-cos(kx)`` rather than
           ``kx**2``.
        kwz2 : float
           Angular velocity of the frame expressed as `kwz2 = m*omega_z/hbar`.
        twist_phase_x : array, optional
           To implement twisted boundary conditions, one needs to remove an
           overall phase from the wavefunction rendering it periodic for use
           the the FFT.  This the the phase that should be removed.  Note: to
           compensate, the momenta should be shifted as well::

              -factor * twist_phase_x*ifft((k+k_twist)**2*fft(y/twist_phase_x)
        """
        if not exp:
            return self.apply_K(y=y, kx2=kx2, twist_phase_x=twist_phase_x) * (-factor)
        else:
            return self.apply_exp_K(
                y=y, factor=-factor, kx2=kx2, twist_phase_x=twist_phase_x
            )

    ######################################################################

    def get_gradient(self, y):
        """Returns the gradient along the x axis."""
        kx = self.kx
        return [self.ifft(1j * kx * self.fft(y)), NotImplemented]

    def apply_Lz(self, y, hermitian=False):
        raise NotImplementedError

    def apply_Px(self, y, hermitian=False):
        r"""Apply the Pz operator to y without any px.

        Requires :attr:`_pxyz` to be defined.
        """
        return self.y_twist * self.ifft(self._kx0 * self.fft(y / self.y_twist))

    def apply_exp_K(self, y, factor, kx2=None, twist_phase_x=None):
        r"""Return `exp(K*factor)*y` or return precomputed data if
        `K_data` is `None`.
        """
        if kx2 is None:
            kx2 = self._Kx
        _K_data_max_len = 3
        ind = None
        for _i, (_f, _d) in enumerate(self._K_data):
            if np.allclose(factor, _f):
                ind = _i
        if ind is None:
            _r1, _r2, V, d = self._Kr_diag
            exp_K_r = _r1 * np.dot(V * np.exp(factor * d), V.T) * _r2
            exp_K_x = np.exp(factor * kx2)
            K_data = (exp_K_r, exp_K_x)
            self._K_data.append((factor, K_data))
            ind = -1
            while len(self._K_data) > _K_data_max_len:
                # Reduce storage
                self._K_data.pop(0)

        K_data = self._K_data[ind][1]
        exp_K_r, exp_K_x = K_data
        if twist_phase_x is None or self.twist == 0:
            tmp = self.ifft(exp_K_x * self.fft(y))
        else:
            if twist_phase_x is None:
                twist_phase_x = self.y_twist
            tmp = twist_phase_x * self.ifft(exp_K_x * self.fft(y / twist_phase_x))
        return np.einsum("...ij,...yj->...yi", exp_K_r, tmp)

    def apply_K(self, y, kx2=None, twist_phase_x=None):
        r"""Return `K*y` where `K = k**2/2`"""
        # Here is how the indices work:
        if kx2 is None:
            kx2 = self._Kx

        if twist_phase_x is None or self.twist == 0:
            yt = self.fft(y)
            yt *= kx2
            yt = self.ifft(yt)
        else:
            if twist_phase_x is None:
                twist_phase_x = self.y_twist
            yt = self.fft(y / twist_phase_x)
            yt *= kx2
            yt = self.ifft(yt)
            yt *= twist_phase_x

        # C <- alpha*B*A + beta*C    A = A^T  zSYMM or zHYMM but not supported
        # maybe cvxopt.blas?  Actually, A is not symmetric... so be careful!
        yt += np.dot(y, self._Kr.T)
        return yt

    ######################################################################
    # FFT and DVR Helper functions.
    #
    # These are specific to the basis, defining the kinetic energy
    # matrix for example.

    # We need these wrappers because the state may have additional
    # indices for components etc. in front.
    def fft(self, x):
        """Perform the fft along the x axes"""
        # Makes sure that
        axis = (self.axes % len(x.shape))[0]
        return fft(x, axis=axis)

    def ifft(self, x):
        """Perform the fft along the x axes"""
        axis = (self.axes % len(x.shape))[0]
        return ifft(x, axis=axis)

    def _get_K(self, l=0):
        r"""Return `(K, r1, r2, w)`: the DVR kinetic term for the radial function
        and the appropriate factors for converting to the radial coordinates.

        This term effects the $-d^2/dr^2 - (\nu^2 - 1/4)/r^2$ term.

        Returns
        -------
        K : array
           Operates on radial wavefunctions
        r1, r2 : array
           K*r1*r2 operators on the full wavefunction (but is no longer
           Hermitian)
        w : array
           Quadrature integration weights.
        """
        nu = self.nu(l=l)
        if l == 0:
            r = self.xyz[1].ravel()
        else:
            r = self._r(self.Nxr[1], l=l)
        z = self._kmax * r
        n = np.arange(len(z))
        i1 = (slice(None), None)
        i2 = (None, slice(None))

        # Quadrature weights
        w = 2.0 / (self._kmax * z * bessel.J(nu=nu, d=1)(z) ** 2)

        # DVR kinetic term for radial function:
        K = np.ma.divide(
            (-1.0) ** (n[i1] - n[i2]) * 8.0 * z[i1] * z[i2],
            (z[i1] ** 2 - z[i2] ** 2) ** 2,
        ).filled(0)
        K[n, n] = 1.0 / 3.0 * (1.0 + 2.0 * (nu ** 2 - 1.0) / z ** 2)
        K *= self._kmax ** 2

        # Here we convert from the wavefunction Psi(r) to the radial
        # function u(r) = sqrt(r)*Psi(r) and back with factors of
        # sqrt(wr).  This includes the integration weights (since K is
        # defined acting on the basis functions).
        # Note: this makes the matrix non-hermitian, so don't do this if you
        # want to diagonalize.
        _tmp = np.sqrt(w * r)
        r2 = _tmp[i2]
        r1 = 1.0 / _tmp[i1]

        return K, r1, r2, w

    def nu(self, l):
        """Return `nu = l + d/2 - 1` for the centrifugal term.

        Arguments
        ---------
        l : int
           Angular quantum number.
        """
        nu = l + self._d / 2 - 1
        return nu

    def _r(self, N, l=0):
        r"""Return the abscissa."""
        # l=0 cylindrical: nu = l + d/2 - 1
        return bessel.j_root(nu=self.nu(l=l), N=N) / self._kmax

    def _F(self, n, r, d=0):
        r"""Return the dth derivative of the n'th basis function."""
        nu = 0.0  # l=0 cylindrical: nu = l + d/2 - 1
        rn = self.xyz[1].ravel()[n]
        zn = self._kmax * rn
        z = self._kmax * r
        H = bessel.J_sqrt_pole(nu=nu, zn=zn, d=0)
        coeff = math.sqrt(2.0 * self._kmax) * (-1.0) ** (n + 1) / (1.0 + r / rn)
        if 0 == d:
            return coeff * H(z)
        elif 1 == d:
            dH = bessel.J_sqrt_pole(nu=nu, zn=zn, d=1)
            return coeff * (dH(z) - H(z) / (z + zn)) * self._kmax
        else:
            raise NotImplementedError

    def get_F(self, r):
        """Return a function that can extrapolate a radial
        wavefunction to a new set of abscissa (x, r)."""
        x, r0 = self.xyz
        n = np.arange(r0.size)[:, None]

        # Here is the transform matrix
        _F = self._F(n, r) / self._F(n, r0.T)

        def F(u):
            return np.dot(u, _F)

        return F

    def F(self, u, xr):
        r"""Return u evaluated on the new abscissa (Assumes x does not
        change for now)"""
        x0, r0 = self.xyz
        x, r = xr
        assert np.allclose(x, x0)

        return self.get_F(r)(u)

    def get_Psi(self, r, return_matrix=False):
        r"""Return a function that can extrapolate a wavefunction to a
        new set of abscissa (x, r).

        This includes the factor of $\sqrt{r}$ that converts the
        wavefunction to the radial function, then uses the basis to
        extrapolate the radial function.

        Arguments
        ---------
        r : array
           The new abscissa in the radial direction (the $x$ values
           stay the same.)
        return_matrix : bool
           If True, then return the extrapolation matrix F so that
           ``Psi = np.dot(psi, F)``
        """
        x, r0 = self.xyz
        n = np.arange(r0.size)[:, None]

        # Here is the transform matrix
        _F = (np.sqrt(r) * self._F(n, r)) / (np.sqrt(r0.T) * self._F(n, r0.T))

        if return_matrix:
            return _F

        def Psi(psi):
            return np.dot(psi, _F)

        return Psi

    def Psi(self, psi, xr):
        r"""Return psi evaluated on the new abscissa (Assumes x does not
        change for now)"""
        x0, r0 = self.xyz
        x, r = xr
        assert np.allclose(x, x0)

        return self.get_Psi(r)(psi)

    def integrate1(self, n):
        """Return the integral of n over y and z."""
        n = np.asarray(n)
        x, r = self.xyz
        x_axis, r_axis = self.axes
        bcast = [None] * len(n.shape)
        bcast[x_axis] = slice(None)
        bcast[r_axis] = slice(None)
        return ((2 * np.pi * r * self.weights)[tuple(bcast)] * n).sum(axis=r_axis)

    def integrate2(self, n, y=None, Nz=100):
        """Return the integral of n over z (line-of-sight integral) at y.

        This is an Abel transform, and is used to compute the 1D
        line-of-sight integral as would be seen by a photographic
        image through an axial cloud.

        Arguments
        ---------
        n : array
           (Nx, Nr) array of the function to be integrated tabulated
           on the abscissa.  Note: the extrapolation assumes that `n =
           abs(psi)**2` where `psi` is well represented in the basis.
        y : array, None
           Ny points at which the resulting integral should be
           returned.  If not provided, then the function will be
           tabulated at the radial abscissa.
        Nz : int
           Number of points to use in z integral.
        """
        n = np.asarray(n)
        x, r = self.xyz
        if y is None:
            y = r

        y = y.ravel()
        Ny = len(y)

        x_axis, r_axis = self.axes
        y_axis = r_axis
        bcast_y = [None] * len(n.shape)
        bcast_z = [None] * len(n.shape)
        bcast_y[y_axis] = slice(None)
        bcast_y.append(None)
        bcast_z.append(slice(None))

        bcast_y, bcast_z = tuple(bcast_y), tuple(bcast_z)

        z = np.linspace(0, r.max(), Nz)
        shape_xyz = n.shape[:-1] + (Ny, Nz)
        rs = np.sqrt(y.ravel()[bcast_y] ** 2 + z[bcast_z] ** 2)
        n_xyz = (abs(self.Psi(np.sqrt(n), (x, rs.ravel()))) ** 2).reshape(shape_xyz)
        n_2D = 2 * np.trapz(n_xyz, z, axis=-1)
        return n_2D
'''

1-dimensional basis for radial problems.      We represent exactly `N` positive abscissa, excluding the origin and use     the discrete sine transform.  We represent the square-root of the     wavefunctions here so that a factor of `r` is required to convert these     into the radial functions.  Unlike the DVR techniques, this approach allows     us to compute the Coulomb interaction for example. Return the laplacian of `y` times `factor` or the         exponential of this.          Arguments         ---------         factor : float            Additional factor (mostly used with `exp=True`).  The            implementation must be careful to allow the factor to            broadcast across the components.         exp : bool            If `True`, then compute the exponential of the laplacian.            This is used for split evolvers. Form for the truncated Coulomb kernel. Modified Coulomb convolution to include form-factors (if provided).          This version implemented a 3D spherically symmetric convolution. # Padded arrays with trailing _ Return the periodic convolution `int(C(x-r)*y(r),r)`.          Note: this is the 3D convolution. dim-dimensional periodic bases.      Parameters     ----------     Nxyz : (Nx, Ny, ...)        Number of lattice points in basis.     Lxyz : (Lx, Ly, ...)        Size of each dimension (length of box and radius)     symmetric_lattice: bool        If True, then shift the lattice so that it is symmetric about        the origin.  The default is to ensure that there is a lattice        point at the origin which will make the lattice asymmetric for        even Nxyz.     axes : (int, int, ...)        Axes in array y which correspond to the x, y, ... axes here.        This is required for cases where y has additional dimensions.        The default is the last dim axes (best for performance).     boost_pxyz : float        Momentum of moving frame.  Momenta are shifted by this, which        corresponds to working in a boosted frame with velocity `vx = px/m`.     smoothing_cutoff : float        Fraction of maximum momentum used in the function smooth(). # Select operations are performed using self.xp instead of numpy. # This can be replaced cupy to provide gpu support with minimal # code changes.  Similarly with the fft functions and a generic # function to convert an array into a numpy array on the host. # Convert to numpy array # Zero out odd highest frequency component. # Add boosts # np.exp(-p2_pc2**4) # self._smoothing_factor = 1.0 # Memoize momentum sums for speed Return the laplacian of `y` times `factor` or the exponential of this.          Arguments         ---------         factor : float            Additional factor (mostly used with `exp=True`).  The            implementation must be careful to allow the factor to            broadcast across the components.         exp : bool            If `True`, then compute the exponential of the laplacian.            This is used for split evolvers. Only allowed to be `True`            if `kwz2 == 0`.         kx2 : array, optional            Replacement for the default `kx2=kx**2` used when computing the            "laplacian".  This would allow you, for example, to implement a            modified dispersion relationship like ``1-cos(kx)`` rather than            ``kx**2``.         kwz2 : None, float            Angular velocity of the frame expressed as `kwz2 = m*omega_z/hbar`.         k2 : array, optional            Replacement for `k2 = kx**2 + ky**2 + kz**2`.         twist_phase_x : array, optional            To implement twisted boundary conditions, one needs to remove an            overall phase from the wavefunction rendering it periodic for use            the the FFT.  This the the phase that should be removed.  Note: to            compensate, the momenta should be shifted as well::                -factor * twist_phase_x*ifft((k+k_twist)**2*fft(y/twist_phase_x) Apply `Lz/hbar` to `y`. # We need these wrappers because the state may have additional # indices for components etc. in front. Perform the fft along self.axes[axis] Perform the ifft along self.axes[axis] Perform the fft along spatial axes Perform the ifft along spatial axes Smooth the state by multiplying by form factor. # TODO: Check this for the highest momentum issue. # TODO: Check this for the highest momentum issue. Use this to broadcast a 1D array along the n'th of N dimensions Form for the Coulomb kernel.          The normalization here is that the k=0 component is set to         zero.  This means that the charge distribution has an overall         constant background removed so that the net charge in the unit         cell is zero. Periodic convolution with the Coulomb kernel. # This broadcasts to the appropriate size if there are # multiple components. # dim = len(np.asarray(self.Lxyz)) # N = np.asarray(y.shape) # b_cast = [None] * (dim - len(N)) + [slice(None)]*dim Return the periodic convolution `int(C(x-r)*y(r),r)`.          Arguments         ---------         y : array            Usually the density, but can be any array         C : array            Convolution kernel. The convolution will be performed using the FFT.         Ck : function (optional)            If provided, then this function will be used instead directly in            momentum space.  Assumed to be spherically symmetric (will be passed            only the magnitude `k`) N-dimensional periodic bases but with Coulomb convolution that does not     use periodic images.  Use this for nuclei in free space.      Parameters     ----------     Nxyz : (Nx, Ny, ...)        Number of lattice points in basis.     Lxyz : (Lx, Ly, ...)        Size of each dimension (length of box and radius)     symmetric_lattice: bool        If True, then shift the lattice so that it is symmetric about        the origin.  The default is to ensure that there is a lattice        point at the origin which will make the lattice asymmetric for        even Nxyz.     axes : (int, int, ...)        Axes in array y which correspond to the x, y, ... axes here.        This is required for cases where y has additional dimensions.        The default is the last dim axes (best for performance).     fast_coulomb : bool        If `True`, use the fast Coulomb algorithm which is slightly less        accurate but much faster. Return the approximate convolution `int(C(x-r)*y(r),r)` where          .. math::            C(r) = 1/r          is the Coulomb potential (without charges etc.)          Arguments         ---------         y : array            Usually the density, but can be any array         correct : bool            If `True`, then include the high frequency components via the            periodic convolution.          Notes         -----         This version uses the Truncated Kernel Expansion method which uses the         Truncated Kernel          .. math::            4\pi(1-\cos\sqrt{3}Lk)/k^2          on a padded array to remove the images, approximating the linear         convolution without the highest frequency modes.  By choosing the         smaller lattice to be at least 3 times smaller we can guarantee that         the padded array will fit into memory.  This can be augmented by the         periodic convolution to fill in the higher modes.          There are two sources of error here:          * We presently use the same periodic ``resample`` method to interpolate           the linear convolution to the larger grid.  This assumes that the           function is periodic -- which it is not -- and can introduce some           aliasing artifacts.  Some preliminary experimentation shows, however,           that these are generally small.  Perhaps cubic spline interpolation           could be used to improve the interpolation, but this is not clear           yet.         * The contribution from the higher modes are computed from the periodic           convolution which could in principle be contaminated by images.           However, for smooth functions, there should be little amplitude here,           and it should consist only of higher multipoles, so the contamination           should be small. Return the convolution `int(C(x-r)*y(r),r)` where          .. math::            C(r) = 1/r          is the Coulomb potential (without charges etc.)          Arguments         ---------         y : array            Usually the density, but can be any array         method : 'sum', 'pad'            Either zero-pad the array (takes extra memory but can use multiple            cores) or sum over the 27 small transforms (slow).          This function is designed for computing the Coulomb potential of a         charge distribution.  In this case, one would have the kernel:          .. math::            4\pi/k^2          or in the case of non-periodic convolution to remove the images          .. math::            4\pi(1-\cos\sqrt{3}Lk)/k^2 # Diameter of cell # Sum with a loop.  Minimizes the memory usage, but will not # use multiple cores. # This broadcasts to the appropriate size 2D basis for Cylindrical coordinates via a DVR basis.      This represents 3-dimensional problems with axial symmetry, but only has     two dimensions (x, r).      Parameters     ----------     Nxr : (Nx, Nr)        Number of lattice points in basis.     Lxr : (L, R)        Size of each dimension (length of box and radius)     twist : float        Twist (angle) in periodic dimension.  This adds a constant offset to the        momenta allowing one to study Bloch waves.     boost_px : float        Momentum of moving frame (along the x axis).  Momenta are shifted by        this, which corresponds to working in a boosted frame with velocity        `vx = boost_px/m`.     axes : (int, int)        Axes in array y which correspond to the x and r axes here.        This is required for cases where y has additional dimensions.        The default is the last two axes (best for performance). # Dimension of spherical part (see nu()) # For large n, the roots of the bessel function are approximately # z[n] = (n + 0.75)*pi, so R = r_max = z_max/k_max = (N-0.25)*pi/kmax # This self._kmax defines the DVR basis, not self.k_max # This is just the maximum momentum for diagnostics, # determining cutoffs etc. # Do this after setting _kmax # Get the DVR kinetic piece for radial component # We did not apply the sqrt(r) factors so at this point, K is still # Hermitian and we can diagonalize for later exponentiation. # K = np.dot(V*d, V.T) # Here we convert from the wavefunction Psi(r) to the radial # function u(r) = sqrt(r)*Psi(r) and back with factors of sqrt(r). # For use when exponentiating # And factor for x. # Cache for K_data from apply_exp_K. ###################################################################### # IBasisMinimal: Required methods Return the laplacian of y.          Arguments         ---------         factor : float            Additional factor (mostly used with `exp=True`).  The            implementation must be careful to allow the factor to            broadcast across the components.         exp : bool            If `True`, then compute the exponential of the laplacian.            This is used for split evolvers. Only allowed to be `True`            if `kwz2 == 0`.         kx2 : array, optional            Replacement for the default `kx2=kx**2` used when computing the            "laplacian".  This would allow you, for example, to implement a            modified dispersion relationship like ``1-cos(kx)`` rather than            ``kx**2``.         kwz2 : float            Angular velocity of the frame expressed as `kwz2 = m*omega_z/hbar`.         twist_phase_x : array, optional            To implement twisted boundary conditions, one needs to remove an            overall phase from the wavefunction rendering it periodic for use            the the FFT.  This the the phase that should be removed.  Note: to            compensate, the momenta should be shifted as well::                -factor * twist_phase_x*ifft((k+k_twist)**2*fft(y/twist_phase_x) ###################################################################### Returns the gradient along the x axis. Apply the Pz operator to y without any px.          Requires :attr:`_pxyz` to be defined. Return `exp(K*factor)*y` or return precomputed data if         `K_data` is `None`. # Reduce storage Return `K*y` where `K = k**2/2` # Here is how the indices work: # C <- alpha*B*A + beta*C    A = A^T  zSYMM or zHYMM but not supported # maybe cvxopt.blas?  Actually, A is not symmetric... so be careful! ###################################################################### # FFT and DVR Helper functions. # # These are specific to the basis, defining the kinetic energy # matrix for example. # We need these wrappers because the state may have additional # indices for components etc. in front. Perform the fft along the x axes # Makes sure that Perform the fft along the x axes Return `(K, r1, r2, w)`: the DVR kinetic term for the radial function         and the appropriate factors for converting to the radial coordinates.          This term effects the $-d^2/dr^2 - (\nu^2 - 1/4)/r^2$ term.          Returns         -------         K : array            Operates on radial wavefunctions         r1, r2 : array            K*r1*r2 operators on the full wavefunction (but is no longer            Hermitian)         w : array            Quadrature integration weights. # Quadrature weights # DVR kinetic term for radial function: # Here we convert from the wavefunction Psi(r) to the radial # function u(r) = sqrt(r)*Psi(r) and back with factors of # sqrt(wr).  This includes the integration weights (since K is # defined acting on the basis functions). # Note: this makes the matrix non-hermitian, so don't do this if you # want to diagonalize. Return `nu = l + d/2 - 1` for the centrifugal term.          Arguments         ---------         l : int            Angular quantum number. Return the abscissa. # l=0 cylindrical: nu = l + d/2 - 1 Return the dth derivative of the n'th basis function. # l=0 cylindrical: nu = l + d/2 - 1 Return a function that can extrapolate a radial         wavefunction to a new set of abscissa (x, r). # Here is the transform matrix Return u evaluated on the new abscissa (Assumes x does not         change for now) Return a function that can extrapolate a wavefunction to a         new set of abscissa (x, r).          This includes the factor of $\sqrt{r}$ that converts the         wavefunction to the radial function, then uses the basis to         extrapolate the radial function.          Arguments         ---------         r : array            The new abscissa in the radial direction (the $x$ values            stay the same.)         return_matrix : bool            If True, then return the extrapolation matrix F so that            ``Psi = np.dot(psi, F)`` # Here is the transform matrix Return psi evaluated on the new abscissa (Assumes x does not         change for now) Return the integral of n over y and z. Return the integral of n over z (line-of-sight integral) at y.          This is an Abel transform, and is used to compute the 1D         line-of-sight integral as would be seen by a photographic         image through an axial cloud.          Arguments         ---------         n : array            (Nx, Nr) array of the function to be integrated tabulated            on the abscissa.  Note: the extrapolation assumes that `n =            abs(psi)**2` where `psi` is well represented in the basis.         y : array, None            Ny points at which the resulting integral should be            returned.  If not provided, then the function will be            tabulated at the radial abscissa.         Nz : int            Number of points to use in z integral. Incomplete! @implementer(IBasis, IBasisKx) class SphericalDVRBasis(ObjectBase, BasisMixin):     r"""3D Spherical DVR basis.      This represents 3-dimensional problems with spherical symmetry.      Parameters     ----------     N : int        Number of lattice points in basis.     R : float        Radius of the basis.     """     _d = 3  # Dimension of spherical part (see nu())      def __init__(self, N, R):         self.N = N         self.R = R         super().__init__()      def init(self):         R = self.R         N = self.N          # For large n, the roots of the bessel function are approximately         # z[n] = (n + 0.75)*pi, so R = r_max = z_max/k_max = (N-0.25)*pi/kmax         # This self._kmax defines the DVR basis, not self.k_max         self._kmax = (Nr - 0.25) * np.pi / R          # This is just the maximum momentum for diagnostics,         # determining cutoffs etc.         self.k_max = np.array([abs(self.kx).max(), self._kmax])          nr = np.arange(Nr)[None, :]         r = self._r(Nr)[None, :]  # Do this after setting _kmax         self.xyz = [x, r]          _lambda = np.asarray(             [1.0 / (self._F(_nr, _r)) ** 2 for _nr, _r in zip(nr.ravel(), r.ravel())]         )[None, :]         self.metric = 2 * np.pi * r * _lambda * (Lx / Nx)         self.metric.setflags(write=False)         # Get the DVR kinetic piece for radial component         K, r1, r2, w = self._get_K()          # We did not apply the sqrt(r) factors so at this point, K is still         # Hermitian and we can diagonalize for later exponentiation.         d, V = sp.linalg.eigh(K)  # K = np.dot(V*d, V.T)          # Here we convert from the wavefunction Psi(r) to the radial         # function u(r) = sqrt(r)*Psi(r) and back with factors of sqrt(r).         K *= r1         K *= r2          self.weights = w         self._Kr = K         self._Kr_diag = (r1, r2, V, d)  # For use when exponentiating          # And factor for x.         self._Kx = self._kx2          # Cache for K_data from apply_exp_K.         self._K_data = []      @property     def Lx(self):         return self.Lxr[0]      @property     def Nx(self):         return self.Nxr[0]      ######################################################################     # IBasisMinimal: Required methods     def laplacian(self, y, factor=1.0, exp=False, kx2=None, twist_phase_x=None):         r"""Return the laplacian of y.          Arguments         ---------         factor : float            Additional factor (mostly used with `exp=True`).  The            implementation must be careful to allow the factor to            broadcast across the components.         exp : bool            If `True`, then compute the exponential of the laplacian.            This is used for split evolvers. Only allowed to be `True`            if `kwz2 == 0`.         kx2 : array, optional            Replacement for the default `kx2=kx**2` used when computing the            "laplacian".  This would allow you, for example, to implement a            modified dispersion relationship like ``1-cos(kx)`` rather than            ``kx**2``.         kwz2 : float            Angular velocity of the frame expressed as `kwz2 = m*omega_z/hbar`.         twist_phase_x : array, optional            To implement twisted boundary conditions, one needs to remove an            overall phase from the wavefunction rendering it periodic for use            the the FFT.  This the the phase that should be removed.  Note: to            compensate, the momenta should be shifted as well::                -factor * twist_phase_x*ifft((k+k_twist)**2*fft(y/twist_phase_x)         """         if not exp:             return self.apply_K(y=y, kx2=kx2, twist_phase_x=twist_phase_x) * (-factor)         else:             return self.apply_exp_K(                 y=y, factor=-factor, kx2=kx2, twist_phase_x=twist_phase_x             )      ######################################################################      def get_gradient(self, y):         """Returns the gradient along the x axis."""         kx = self.kx         return [self.ifft(1j * kx * self.fft(y)), NotImplemented]      def apply_Lz(self, y, hermitian=False):         raise NotImplementedError      def apply_Px(self, y, hermitian=False):         r"""Apply the Pz operator to y without any px.          Requires :attr:`_pxyz` to be defined.         """         return self.y_twist * self.ifft(self._kx0 * self.fft(y / self.y_twist))      def apply_exp_K(self, y, factor, kx2=None, twist_phase_x=None):         r"""Return `exp(K*factor)*y` or return precomputed data if         `K_data` is `None`.         """         if kx2 is None:             kx2 = self._Kx         _K_data_max_len = 3         ind = None         for _i, (_f, _d) in enumerate(self._K_data):             if np.allclose(factor, _f):                 ind = _i         if ind is None:             _r1, _r2, V, d = self._Kr_diag             exp_K_r = _r1 * np.dot(V * np.exp(factor * d), V.T) * _r2             exp_K_x = np.exp(factor * kx2)             K_data = (exp_K_r, exp_K_x)             self._K_data.append((factor, K_data))             ind = -1             while len(self._K_data) > _K_data_max_len:                 # Reduce storage                 self._K_data.pop(0)          K_data = self._K_data[ind][1]         exp_K_r, exp_K_x = K_data         if twist_phase_x is None or self.twist == 0:             tmp = self.ifft(exp_K_x * self.fft(y))         else:             if twist_phase_x is None:                 twist_phase_x = self.y_twist             tmp = twist_phase_x * self.ifft(exp_K_x * self.fft(y / twist_phase_x))         return np.einsum("...ij,...yj->...yi", exp_K_r, tmp)      def apply_K(self, y, kx2=None, twist_phase_x=None):         r"""Return `K*y` where `K = k**2/2`"""         # Here is how the indices work:         if kx2 is None:             kx2 = self._Kx          if twist_phase_x is None or self.twist == 0:             yt = self.fft(y)             yt *= kx2             yt = self.ifft(yt)         else:             if twist_phase_x is None:                 twist_phase_x = self.y_twist             yt = self.fft(y / twist_phase_x)             yt *= kx2             yt = self.ifft(yt)             yt *= twist_phase_x          # C <- alpha*B*A + beta*C    A = A^T  zSYMM or zHYMM but not supported         # maybe cvxopt.blas?  Actually, A is not symmetric... so be careful!         yt += np.dot(y, self._Kr.T)         return yt      ######################################################################     # FFT and DVR Helper functions.     #     # These are specific to the basis, defining the kinetic energy     # matrix for example.      # We need these wrappers because the state may have additional     # indices for components etc. in front.     def fft(self, x):         """Perform the fft along the x axes"""         # Makes sure that         axis = (self.axes % len(x.shape))[0]         return fft(x, axis=axis)      def ifft(self, x):         """Perform the fft along the x axes"""         axis = (self.axes % len(x.shape))[0]         return ifft(x, axis=axis)      def _get_K(self, l=0):         r"""Return `(K, r1, r2, w)`: the DVR kinetic term for the radial function         and the appropriate factors for converting to the radial coordinates.          This term effects the $-d^2/dr^2 - (\nu^2 - 1/4)/r^2$ term.          Returns         -------         K : array            Operates on radial wavefunctions         r1, r2 : array            K*r1*r2 operators on the full wavefunction (but is no longer            Hermitian)         w : array            Quadrature integration weights.         """         nu = self.nu(l=l)         if l == 0:             r = self.xyz[1].ravel()         else:             r = self._r(self.Nxr[1], l=l)         z = self._kmax * r         n = np.arange(len(z))         i1 = (slice(None), None)         i2 = (None, slice(None))          # Quadrature weights         w = 2.0 / (self._kmax * z * bessel.J(nu=nu, d=1)(z) ** 2)          # DVR kinetic term for radial function:         K = np.ma.divide(             (-1.0) ** (n[i1] - n[i2]) * 8.0 * z[i1] * z[i2],             (z[i1] ** 2 - z[i2] ** 2) ** 2,         ).filled(0)         K[n, n] = 1.0 / 3.0 * (1.0 + 2.0 * (nu ** 2 - 1.0) / z ** 2)         K *= self._kmax ** 2          # Here we convert from the wavefunction Psi(r) to the radial         # function u(r) = sqrt(r)*Psi(r) and back with factors of         # sqrt(wr).  This includes the integration weights (since K is         # defined acting on the basis functions).         # Note: this makes the matrix non-hermitian, so don't do this if you         # want to diagonalize.         _tmp = np.sqrt(w * r)         r2 = _tmp[i2]         r1 = 1.0 / _tmp[i1]          return K, r1, r2, w      def nu(self, l):         """Return `nu = l + d/2 - 1` for the centrifugal term.          Arguments         ---------         l : int            Angular quantum number.         """         nu = l + self._d / 2 - 1         return nu      def _r(self, N, l=0):         r"""Return the abscissa."""         # l=0 cylindrical: nu = l + d/2 - 1         return bessel.j_root(nu=self.nu(l=l), N=N) / self._kmax      def _F(self, n, r, d=0):         r"""Return the dth derivative of the n'th basis function."""         nu = 0.0  # l=0 cylindrical: nu = l + d/2 - 1         rn = self.xyz[1].ravel()[n]         zn = self._kmax * rn         z = self._kmax * r         H = bessel.J_sqrt_pole(nu=nu, zn=zn, d=0)         coeff = math.sqrt(2.0 * self._kmax) * (-1.0) ** (n + 1) / (1.0 + r / rn)         if 0 == d:             return coeff * H(z)         elif 1 == d:             dH = bessel.J_sqrt_pole(nu=nu, zn=zn, d=1)             return coeff * (dH(z) - H(z) / (z + zn)) * self._kmax         else:             raise NotImplementedError      def get_F(self, r):         """Return a function that can extrapolate a radial         wavefunction to a new set of abscissa (x, r)."""         x, r0 = self.xyz         n = np.arange(r0.size)[:, None]          # Here is the transform matrix         _F = self._F(n, r) / self._F(n, r0.T)          def F(u):             return np.dot(u, _F)          return F      def F(self, u, xr):         r"""Return u evaluated on the new abscissa (Assumes x does not         change for now)"""         x0, r0 = self.xyz         x, r = xr         assert np.allclose(x, x0)          return self.get_F(r)(u)      def get_Psi(self, r, return_matrix=False):         r"""Return a function that can extrapolate a wavefunction to a         new set of abscissa (x, r).          This includes the factor of $\sqrt{r}$ that converts the         wavefunction to the radial function, then uses the basis to         extrapolate the radial function.          Arguments         ---------         r : array            The new abscissa in the radial direction (the $x$ values            stay the same.)         return_matrix : bool            If True, then return the extrapolation matrix F so that            ``Psi = np.dot(psi, F)``         """         x, r0 = self.xyz         n = np.arange(r0.size)[:, None]          # Here is the transform matrix         _F = (np.sqrt(r) * self._F(n, r)) / (np.sqrt(r0.T) * self._F(n, r0.T))          if return_matrix:             return _F          def Psi(psi):             return np.dot(psi, _F)          return Psi      def Psi(self, psi, xr):         r"""Return psi evaluated on the new abscissa (Assumes x does not         change for now)"""         x0, r0 = self.xyz         x, r = xr         assert np.allclose(x, x0)          return self.get_Psi(r)(psi)      def integrate1(self, n):         """Return the integral of n over y and z."""         n = np.asarray(n)         x, r = self.xyz         x_axis, r_axis = self.axes         bcast = [None] * len(n.shape)         bcast[x_axis] = slice(None)         bcast[r_axis] = slice(None)         return ((2 * np.pi * r * self.weights)[tuple(bcast)] * n).sum(axis=r_axis)      def integrate2(self, n, y=None, Nz=100):         """Return the integral of n over z (line-of-sight integral) at y.          This is an Abel transform, and is used to compute the 1D         line-of-sight integral as would be seen by a photographic         image through an axial cloud.          Arguments         ---------         n : array            (Nx, Nr) array of the function to be integrated tabulated            on the abscissa.  Note: the extrapolation assumes that `n =            abs(psi)**2` where `psi` is well represented in the basis.         y : array, None            Ny points at which the resulting integral should be            returned.  If not provided, then the function will be            tabulated at the radial abscissa.         Nz : int            Number of points to use in z integral.         """         n = np.asarray(n)         x, r = self.xyz         if y is None:             y = r          y = y.ravel()         Ny = len(y)          x_axis, r_axis = self.axes         y_axis = r_axis         bcast_y = [None] * len(n.shape)         bcast_z = [None] * len(n.shape)         bcast_y[y_axis] = slice(None)         bcast_y.append(None)         bcast_z.append(slice(None))          bcast_y, bcast_z = tuple(bcast_y), tuple(bcast_z)          z = np.linspace(0, r.max(), Nz)         shape_xyz = n.shape[:-1] + (Ny, Nz)         rs = np.sqrt(y.ravel()[bcast_y] ** 2 + z[bcast_z] ** 2)         n_xyz = (abs(self.Psi(np.sqrt(n), (x, rs.ravel()))) ** 2).reshape(shape_xyz)         n_2D = 2 * np.trapz(n_xyz, z, axis=-1)         return n_2D