import jax
import jax.numpy as jnp
import jax_cosmo as jc
from jax_cosmo import Cosmology

from jaxpm.growth import growth_factor, growth_rate, dGfa, growth_factor_second, growth_rate_second, dGf2a
from jaxpm.kernels import PGD_kernel, fftk, gradient_kernel, invlaplace_kernel, longrange_kernel
from jaxpm.painting import cic_paint, cic_read



def pm_forces(positions, mesh_shape, delta=None, r_split=0):
    """
    Computes gravitational forces on particles using a PM scheme
    """
    if delta is None:
        delta_k = jnp.fft.rfftn(cic_paint(jnp.zeros(mesh_shape), positions))
    elif jnp.isrealobj(delta):
        delta_k = jnp.fft.rfftn(delta)
    else:
        delta_k = delta

    # Computes gravitational potential
    kvec = fftk(mesh_shape)
    pot_k = delta_k * invlaplace_kernel(kvec) * longrange_kernel(kvec, r_split=r_split)
    # Computes gravitational forces
    return jnp.stack([cic_read(jnp.fft.irfftn(- gradient_kernel(kvec, i) * pot_k), positions) 
                      for i in range(3)], axis=-1)


def lpt(cosmo:Cosmology, init_mesh, positions, a, order=1):
    """
    Computes first and second order LPT displacement and momentum, 
    e.g. Eq. 2 and 3 [Jenkins2010](https://arxiv.org/pdf/0910.0258)
    """
    a = jnp.atleast_1d(a)
    E = jnp.sqrt(jc.background.Esqr(cosmo, a)) 
    delta_k = jnp.fft.rfftn(init_mesh) # TODO: pass the modes directly to save one or two fft?
    mesh_shape = init_mesh.shape

    init_force = pm_forces(positions, mesh_shape, delta=delta_k)
    dx = growth_factor(cosmo, a) * init_force
    p = a**2 * growth_rate(cosmo, a) * E * dx
    f = a**2 * E * dGfa(cosmo, a) * init_force

    if order == 2:
        kvec = fftk(mesh_shape)
        pot_k = delta_k * invlaplace_kernel(kvec)

        delta2 = 0
        shear_acc = 0
        # for i, ki in enumerate(kvec):
        for i in range(3):
            # Add products of diagonal terms = 0 + s11*s00 + s22*(s11+s00)...
            # shear_ii = jnp.fft.irfftn(- ki**2 * pot_k)
            nabla_i_nabla_i = gradient_kernel(kvec, i)**2
            shear_ii = jnp.fft.irfftn(nabla_i_nabla_i * pot_k)
            delta2 += shear_ii * shear_acc 
            shear_acc += shear_ii

            # for kj in kvec[i+1:]:
            for j in range(i+1, 3):
                # Substract squared strict-up-triangle terms
                # delta2 -= jnp.fft.irfftn(- ki * kj * pot_k)**2
                nabla_i_nabla_j = gradient_kernel(kvec, i) * gradient_kernel(kvec, j)
                delta2 -= jnp.fft.irfftn(nabla_i_nabla_j * pot_k)**2

        init_force2 = pm_forces(positions, mesh_shape, delta=jnp.fft.rfftn(delta2))
        # NOTE: growth_factor_second is renormalized: - D2 = 3/7 * growth_factor_second
        dx2 = 3/7 * growth_factor_second(cosmo, a) * init_force2
        p2 = a**2 * growth_rate_second(cosmo, a) * E * dx2
        f2 = a**2 * E * dGf2a(cosmo, a) * init_force2

        dx += dx2
        p  += p2
        f  += f2

    return dx, p, f


def linear_field(mesh_shape, box_size, pk, seed):
    """
    Generate initial conditions.
    """
    kvec = fftk(mesh_shape)
    kmesh = sum((kk / box_size[i] * mesh_shape[i])**2
                for i, kk in enumerate(kvec))**0.5
    pkmesh = pk(kmesh) * (mesh_shape[0] * mesh_shape[1] * mesh_shape[2]) / (
        box_size[0] * box_size[1] * box_size[2])

    field = jax.random.normal(seed, mesh_shape)
    field = jnp.fft.rfftn(field) * pkmesh**0.5
    field = jnp.fft.irfftn(field)
    return field


def make_ode_fn(mesh_shape):

    def nbody_ode(state, a, cosmo):
        """
        state is a tuple (position, velocities)
        """
        pos, vel = state

        forces = pm_forces(pos, mesh_shape=mesh_shape) * 1.5 * cosmo.Omega_m

        # Computes the update of position (drift)
        dpos = 1. / (a**3 * jnp.sqrt(jc.background.Esqr(cosmo, a))) * vel

        # Computes the update of velocity (kick)
        dvel = 1. / (a**2 * jnp.sqrt(jc.background.Esqr(cosmo, a))) * forces

        return dpos, dvel

    return nbody_ode

def get_ode_fn(cosmo:Cosmology, mesh_shape):

    def nbody_ode(a, state, args):
        """
        State is an array [position, velocities]

        Compatible with [Diffrax API](https://docs.kidger.site/diffrax/)
        """
        pos, vel = state
        forces = pm_forces(pos, mesh_shape) * 1.5 * cosmo.Omega_m

        # Computes the update of position (drift)
        dpos = 1. / (a**3 * jnp.sqrt(jc.background.Esqr(cosmo, a))) * vel
        
        # Computes the update of velocity (kick)
        dvel = 1. / (a**2 * jnp.sqrt(jc.background.Esqr(cosmo, a))) * forces

        return jnp.stack([dpos, dvel])

    return nbody_ode


def pgd_correction(pos, mesh_shape, params):
    """
    improve the short-range interactions of PM-Nbody simulations with potential gradient descent method, 
    based on https://arxiv.org/abs/1804.00671

    args:
      pos: particle positions [npart, 3]
      params: [alpha, kl, ks] pgd parameters
    """
    kvec = fftk(mesh_shape)
    delta = cic_paint(jnp.zeros(mesh_shape), pos)
    alpha, kl, ks = params
    delta_k = jnp.fft.rfftn(delta)
    PGD_range=PGD_kernel(kvec, kl, ks)
    
    pot_k_pgd=(delta_k * invlaplace_kernel(kvec))*PGD_range

    forces_pgd= jnp.stack([cic_read(jnp.fft.irfftn(- gradient_kernel(kvec, i)*pot_k_pgd), pos) 
                      for i in range(3)],axis=-1)
    
    dpos_pgd = forces_pgd*alpha
   
    return dpos_pgd


def make_neural_ode_fn(model, mesh_shape):
    def neural_nbody_ode(state, a, cosmo:Cosmology, params):
        """
        state is a tuple (position, velocities)
        """
        pos, vel = state
        kvec = fftk(mesh_shape)

        delta = cic_paint(jnp.zeros(mesh_shape), pos)

        delta_k = jnp.fft.rfftn(delta)

        # Computes gravitational potential
        pot_k = delta_k * invlaplace_kernel(kvec) * longrange_kernel(kvec, r_split=0)

        # Apply a correction filter
        kk = jnp.sqrt(sum((ki/jnp.pi)**2 for ki in kvec))
        pot_k = pot_k *(1. + model.apply(params, kk, jnp.atleast_1d(a)))

        # Computes gravitational forces
        forces = jnp.stack([cic_read(jnp.fft.irfftn(- gradient_kernel(kvec, i)*pot_k), pos) 
                          for i in range(3)],axis=-1)

        forces = forces * 1.5 * cosmo.Omega_m

        # Computes the update of position (drift)
        dpos = 1. / (a**3 * jnp.sqrt(jc.background.Esqr(cosmo, a))) * vel

        # Computes the update of velocity (kick)
        dvel = 1. / (a**2 * jnp.sqrt(jc.background.Esqr(cosmo, a))) * forces

        return dpos, dvel
    return neural_nbody_ode