threept.cpp

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// Triumvirate: Three-Point Clustering Measurements in LSS
//
// Copyright (C) 2023 Mike S Wang & Naonori S Sugiyama [GPL-3.0-or-later]
//
// This file is part of the Triumvirate program. See the COPYRIGHT
// and LICENCE files at the top-level directory of this distribution
// for details of copyright and licensing.
//
// This program is free software: you can redistribute it and/or modify it
// under the terms of the GNU General Public License as published by
// the Free Software Foundation, either version 3 of the License, or
// (at your option) any later version.
//
// This program is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
// See the GNU General Public License for more details.
//
// You should have received a copy of the GNU General Public License
// along with this program. If not, see <https://www.gnu.org/licenses/>.

 
#include "threept.hpp"
 
namespace trvs = trv::sys;
namespace trvm = trv::maths;
 
namespace trv {
 
// ***********************************************************************
// Spherical orders
// ***********************************************************************
 
SphericalOrderTriplet SphericalOrderTriplet::operator+(
  const SphericalOrderTriplet& other
) {
  SphericalOrderTriplet sph_order_triplet = {
    this->m1 + other.m1, this->m2 + other.m2, this->M + other.M
  };
  return sph_order_triplet;
}
 
bool SphericalOrderTriplet::is_zeros() {
  return (this->m1 == 0) && (this->m2 == 0) && (this->M == 0);
}
 
bool SphericalOrderTriplet::is_inverse(const SphericalOrderTriplet& other) {
  return
    (this->m1 + other.m1 == 0) &&
    (this->m2 + other.m2 == 0) &&
    (this->M + other.M == 0);
}
 
 
// ***********************************************************************
// Coupling coefficients
// ***********************************************************************
 
double calc_coupling_coeff_3pt(
  int ell1, int ell2, int ELL, int m1, int m2, int M
) {
  return double(2*ell1 + 1) * double(2*ell2 + 1) * double(2*ELL + 1)
    * trvm::wigner_3j(ell1, ell2, ELL, 0, 0, 0)
    * trvm::wigner_3j(ell1, ell2, ELL, m1, m2, M);
}
 
void validate_multipole_coupling(trv::ParameterSet& params) {
  double coupling_ = trvm::wigner_3j(
    params.ell1, params.ell2, params.ELL, 0, 0, 0
  );
  if (std::fabs(coupling_) < trvm::eps_coupling) {
    if (trvs::currTask == 0) {
      trvs::logger.error(
        "Specified three-point correlator multipole "
        "vanishes identically owing to zero-valued Wigner 3-j symbol."
      );
    }
    throw trvs::InvalidParameterError(
      "Specified three-point correlator multipole "
      "vanishes identically owing to zero-valued Wigner 3-j symbol."
    );
  }
}
 
 
// ***********************************************************************
// Normalisation
// ***********************************************************************
 
double calc_bispec_normalisation_from_particles(
  ParticleCatalogue& particles, double alpha
) {
  if (particles.pdata == nullptr) {
    if (trvs::currTask == 0) {
      trvs::logger.error("Particle data are uninitialised.");
    }
    throw trvs::InvalidDataError("Particle data are uninitialised.");
  }
 
  double norm = 0.;  // I₃
 
#ifdef TRV_USE_OMP
#if defined(__GNUC__) && !defined(__clang__)
#pragma omp parallel for simd reduction(+:norm)
#else  // !__GNUC__ || __clang__
#pragma omp parallel for reduction(+:norm)
#endif  // __GNUC__ && !__clang__
#endif  // TRV_USE_OMP
  for (int pid = 0; pid < particles.ntotal; pid++) {
    norm += particles[pid].ws
      * std::pow(particles[pid].nz, 2) * std::pow(particles[pid].wc, 3);
  }
 
  if (norm == 0.) {
    if (trvs::currTask == 0) {
      trvs::logger.error(
        "Particle 'nz' values appear to be all zeros. "
        "Check the input catalogue contains valid 'nz' field."
      );
    }
    throw trvs::InvalidDataError(
      "Particle 'nz' values appear to be all zeros. "
      "Check the input catalogue contains valid 'nz' field."
    );
  }
 
  double norm_factor = 1. / (alpha * norm);  // 1/I₃
 
  return norm_factor;
}
 
double calc_bispec_normalisation_from_mesh(
  ParticleCatalogue& particles, trv::ParameterSet& params, double alpha
) {
  MeshField catalogue_mesh(params, false, "`catalogue_mesh`");
 
  double norm_factor =
    catalogue_mesh.calc_grid_based_powlaw_norm(particles, 3);
 
  norm_factor /= std::pow(alpha, 3);
 
  return norm_factor;
}
 
 
// ***********************************************************************
// Shot noise
// ***********************************************************************
 
std::complex<double> calc_ylm_wgtd_shotnoise_amp_for_bispec(
  ParticleCatalogue& particles_data, ParticleCatalogue& particles_rand,
  LineOfSight* los_data, LineOfSight* los_rand,
  double alpha, int ell, int m
) {
  double sn_data_real = 0., sn_data_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:sn_data_real, sn_data_imag)
#endif
  for (int pid = 0; pid < particles_data.ntotal; pid++) {
    double los_[3] = {
      los_data[pid].pos[0], los_data[pid].pos[1], los_data[pid].pos[2]
    };
 
    std::complex<double> ylm = trvm::SphericalHarmonicCalculator::
      calc_reduced_spherical_harmonic(ell, m, los_);
 
    std::complex<double> sn_part = ylm * std::pow(particles_data[pid].w, 3);
    double sn_part_real = sn_part.real();
    double sn_part_imag = sn_part.imag();
 
    sn_data_real += sn_part_real;
    sn_data_imag += sn_part_imag;
  }
 
  std::complex<double> sn_data(sn_data_real, sn_data_imag);
 
  double sn_rand_real = 0., sn_rand_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:sn_rand_real, sn_rand_imag)
#endif
  for (int pid = 0; pid < particles_rand.ntotal; pid++) {
    double los_[3] = {
      los_rand[pid].pos[0], los_rand[pid].pos[1], los_rand[pid].pos[2]
    };
 
    std::complex<double> ylm = trvm::SphericalHarmonicCalculator::
      calc_reduced_spherical_harmonic(ell, m, los_);
 
    std::complex<double> sn_part = ylm * std::pow(particles_rand[pid].w, 3);
    double sn_part_real = sn_part.real();
    double sn_part_imag = sn_part.imag();
 
    sn_rand_real += sn_part_real;
    sn_rand_imag += sn_part_imag;
  }
 
  std::complex<double> sn_rand(sn_rand_real, sn_rand_imag);
 
  return sn_data + std::pow(alpha, 3) * sn_rand;
}
 
std::complex<double> calc_ylm_wgtd_shotnoise_amp_for_bispec(
  ParticleCatalogue& particles, LineOfSight* los,
  double alpha, int ell, int m
) {
  double sn_real = 0., sn_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:sn_real, sn_imag)
#endif
  for (int pid = 0; pid < particles.ntotal; pid++) {
    double los_[3] = {los[pid].pos[0], los[pid].pos[1], los[pid].pos[2]};
 
    std::complex<double> ylm = trvm::SphericalHarmonicCalculator::
      calc_reduced_spherical_harmonic(ell, m, los_);
 
    std::complex<double> sn_part = ylm * std::pow(particles[pid].w, 3);
    double sn_part_real = sn_part.real();
    double sn_part_imag = sn_part.imag();
 
    sn_real += sn_part_real;
    sn_imag += sn_part_imag;
  }
 
  std::complex<double> sn(sn_real, sn_imag);
 
  return std::pow(alpha, 3) * sn;
}
 
 
// ***********************************************************************
// Full statistics
// ***********************************************************************
 
// STYLE: Standard naming convention is not always followed for
// intermediary quantities in the functions below.
 
// Hereafter 'the Paper' refers to Sugiyama et al. (2019) [1803.02132].
 
trv::BispecMeasurements compute_bispec(
  ParticleCatalogue& catalogue_data, ParticleCatalogue& catalogue_rand,
  LineOfSight* los_data, LineOfSight* los_rand,
  trv::ParameterSet& params, trv::Binning& kbinning,
  double norm_factor
) {
  trvs::logger.reset_level(params.verbose);
 
  if (trvs::currTask == 0) {
    trvs::logger.stat(
      "Computing bispectrum from paired survey-type catalogues..."
    );
  }
 
  // ---------------------------------------------------------------------
  // Set-up
  // ---------------------------------------------------------------------
 
  // Set up/check input.
  validate_multipole_coupling(params);
 
  double alpha = catalogue_data.wstotal / catalogue_rand.wstotal;
 
  std::complex<double> factor_phase =
    std::pow(trvm::M_I, params.ell1 + params.ell2);
 
  // Set up output.
  int dv_dim = 0;  // data vector dimension
  if (params.shape == "diag" || params.shape == "row" ) {
    dv_dim = kbinning.num_bins;
  } else
  if (params.shape == "off-diag") {
    dv_dim = kbinning.num_bins - std::abs(params.idx_bin);
  } else
  if (params.shape == "full") {
    dv_dim = kbinning.num_bins * kbinning.num_bins;
  } else
  if (params.shape == "triu") {
    dv_dim = kbinning.num_bins * (kbinning.num_bins + 1) / 2;
  } else {
    if (trvs::currTask == 0) {
      trvs::logger.error(
        "Three-point statistic form is not recognised: `form` = '%s'.",
        params.form.c_str()
      );
    }
    throw trvs::InvalidParameterError(
      "Three-point statistic form is not recognised: `form` = '%s'.",
      params.form.c_str()
    );
  }
 
  int* nmodes1_dv = new int[dv_dim]();
  int* nmodes2_dv = new int[dv_dim]();
  double* k1bin_dv = new double[dv_dim]();
  double* k2bin_dv = new double[dv_dim]();
  double* k1eff_dv = new double[dv_dim]();
  double* k2eff_dv = new double[dv_dim]();
  std::complex<double>* bk_dv = new std::complex<double>[dv_dim];
  std::complex<double>* sn_dv = new std::complex<double>[dv_dim];
 
  // ---------------------------------------------------------------------
  // Measurement
  // ---------------------------------------------------------------------
 
#if defined(TRV_USE_OMP) && defined(TRV_USE_FFTWOMP)
  if (!trvs::is_gpu_enabled()) {
    fftw_init_threads();
  }
#endif  // TRV_USE_OMP && TRV_USE_FFTWOMP
 
  // Compute common field quantities.
  MeshField dn_00(params, true, "`dn_00`");  // δn_00(k)
  dn_00.compute_ylm_wgtd_field(
    catalogue_data, catalogue_rand, los_data, los_rand, alpha, 0, 0
  );
  dn_00.fourier_transform();
 
  MeshField& dn_00_for_sn = dn_00;  // δn_00(k) (for shot noise)
 
  double vol_cell = dn_00.vol_cell;
 
  MeshField N_00(params, true, "`N_00`");  // N_00(k)
  N_00.compute_ylm_wgtd_quad_field(
    catalogue_data, catalogue_rand, los_data, los_rand, alpha, 0, 0
  );
  N_00.fourier_transform();
 
  trvm::SphericalBesselCalculator sj_a(params.ell1);  // j_l_a
  trvm::SphericalBesselCalculator sj_b(params.ell2);  // j_l_b
 
  FieldStats stats_sn(params);
 
  // Initialise reduced-spherical-harmonic weights on mesh grids.
  std::vector< std::complex<double> > ylm_k_a(params.nmesh);
  std::vector< std::complex<double> > ylm_k_b(params.nmesh);
  std::vector< std::complex<double> > ylm_r_a(params.nmesh);
  std::vector< std::complex<double> > ylm_r_b(params.nmesh);
  trvs::count_cgrid += 4;
  trvs::count_grid += 4;
  trvs::update_maxcntgrid();
  trvs::gbytesMem +=
    trvs::size_in_gb< std::complex<double> >(4*params.nmesh);
  trvs::update_maxmem();
 
  // Compute bispectrum terms including shot noise.
  std::vector<trv::SphericalOrderTriplet> sph_orders_computed;
  int count_terms = 0;
  for (int m1_ = - params.ell1; m1_ <= params.ell1; m1_++) {
    for (int m2_ = - params.ell2; m2_ <= params.ell2; m2_++) {
      // Check for if all Wigner-3j symbols are zero.
      std::string flag_vanishing = "true";
      for (int M_ = - params.ELL; M_ <= params.ELL; M_++) {
        double coupling = trv::calc_coupling_coeff_3pt(
          params.ell1, params.ell2, params.ELL, m1_, m2_, M_
        );
        if (std::fabs(coupling) > trvm::eps_coupling) {
          flag_vanishing = "false";
          break;
        }
      }
      if (flag_vanishing == "true") {continue;}
 
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_fourier_space(
          params.ell1, m1_, params.boxsize, params.ngrid, ylm_k_a
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_fourier_space(
          params.ell2, m2_, params.boxsize, params.ngrid, ylm_k_b
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_config_space(
          params.ell1, m1_, params.boxsize, params.ngrid, ylm_r_a
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_config_space(
          params.ell2, m2_, params.boxsize, params.ngrid, ylm_r_b
        );
 
      for (int M_ = - params.ELL; M_ <= params.ELL; M_++) {
        // Check if the current spherical order triplet is redundant.
        SphericalOrderTriplet sph_order_current = {m1_, m2_, M_};
 
        std::string flag_redundant = "false";
        for (SphericalOrderTriplet sph_order : sph_orders_computed) {
          if (sph_order_current.is_inverse(sph_order)) {
            flag_redundant = "true";
          }
        }
        if (flag_redundant == "true") {continue;}
 
        // Calculate the coupling coefficient.
        double coupling = trv::calc_coupling_coeff_3pt(
          params.ell1, params.ell2, params.ELL, m1_, m2_, M_
        );  // Wigner 3-j's
        if (std::fabs(coupling) < trvm::eps_coupling) {continue;}
 
        // The factors consist of products of
        // ```
        // {std::pow(-1., params.ell1 + params.ell2 + params.ELL),
        //  1},
        // {std::pow(-1., m1_ + m2_ + M_),
        //  std::pow(-1., 2*M_)} ≡ 1,
        // {std::pow(-1., params.ell1 + params.ell2),
        //  std::pow(-1., params.ELL)} ≡ std::pow(-1., params.ELL)}
        // ```
        // which, by parity symmetry & Wigner 3-j properties, simplify to:
        double factor_mirror = sph_order_current.is_zeros() ?
          0. : std::pow(-1., params.ELL);
        double factor_mirror_w3j = sph_order_current.is_zeros() ?
          0. : 1.;  // std::pow(-1., params.ell1 + params.ell2 + params.ELL);
 
        // Set up FFT counter and progress bar.
        int progbar_task_count = 4 + 3 * dv_dim;
        std::stringstream progbar_name_ss;
        progbar_name_ss
          << "component orders (" << m1_ << ", " << m2_ << ", " << M_ << ")";
        trvs::ProgressBar progbar(progbar_task_count, progbar_name_ss.str());
 
        std::vector<float> progbar_nodes;
        if (params.progbar != "false" && params.progbar != "true") {
          progbar_nodes = trvs::set_nodes_by_str(params.progbar);
          if (progbar_nodes.size() > 0) {
            progbar.set_nodes(progbar_nodes);
          }
        }
 
        int count_fft_init = trvs::count_fft + trvs::count_ifft;
        auto update_progbar = [&]() {
          if (params.progbar != "false") {
            if (trvs::currTask == 0) {
              progbar.update(
                trvs::count_fft + trvs::count_ifft - count_fft_init
              );
            }
          }
        };
 
        // ·······························································
        // Raw bispectrum
        // ·······························································
 
        // Compute bispectrum components in eqs. (41) & (42) in the Paper.
        MeshField G_LM(params, true, "`G_LM`");  // G_LM
        G_LM.compute_ylm_wgtd_field(
          catalogue_data, catalogue_rand, los_data, los_rand, alpha,
          params.ELL, M_
        );
        G_LM.fourier_transform();
        G_LM.apply_assignment_compensation();
        G_LM.inv_fourier_transform();
        update_progbar();
 
        MeshField F_lm_a(params, true, "`F_lm_a`");  // F_lm_a
        MeshField F_lm_b(params, true, "`F_lm_b`");  // F_lm_b
 
        if (params.shape == "diag") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin = idx_dv;
 
            double k_lower = kbinning.bin_edges[ibin];
            double k_upper = kbinning.bin_edges[ibin + 1];
 
            double k_eff_a_, k_eff_b_;
            int nmodes_a_, nmodes_b_;
 
            F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_00, ylm_k_a, k_lower, k_upper, k_eff_a_, nmodes_a_
            );
            F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_00, ylm_k_b, k_lower, k_upper, k_eff_b_, nmodes_b_
            );
            update_progbar();
 
            if (count_terms == 0) {
              k1bin_dv[idx_dv] = kbinning.bin_centres[ibin];
              k2bin_dv[idx_dv] = kbinning.bin_centres[ibin];
              k1eff_dv[idx_dv] = k_eff_a_;
              k2eff_dv[idx_dv] = k_eff_b_;
              nmodes1_dv[idx_dv] = nmodes_a_;
              nmodes2_dv[idx_dv] = nmodes_b_;
            }
 
            // B_{l₁ l₂ L}^{m₁ m₂ M}
            double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
            for (long long gid = 0; gid < params.nmesh; gid++) {
              std::complex<double> F_lm_a_gridpt(
                F_lm_a[gid][0], F_lm_a[gid][1]
              );
              std::complex<double> F_lm_b_gridpt(
                F_lm_b[gid][0], F_lm_b[gid][1]
              );
              std::complex<double> G_LM_gridpt(G_LM[gid][0], G_LM[gid][1]);
              std::complex<double> bk_gridpt =
                F_lm_a_gridpt * F_lm_b_gridpt * G_LM_gridpt;
 
              bk_comp_real += bk_gridpt.real();
              bk_comp_imag += bk_gridpt.imag();
            }
 
            std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
            bk_dv[idx_dv] += coupling * vol_cell * (
              bk_component + factor_mirror * std::conj(bk_component)
            );
          }
        }
 
        if (params.shape == "off-diag") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin_row, ibin_col;
            if (params.idx_bin >= 0) {
              ibin_row = idx_dv;
              ibin_col = idx_dv + std::abs(params.idx_bin);
            } else {
              ibin_row = idx_dv + std::abs(params.idx_bin);
              ibin_col = idx_dv;
            }
 
            double k_lower_a = kbinning.bin_edges[ibin_row];
            double k_upper_a = kbinning.bin_edges[ibin_row + 1];
            double k_lower_b = kbinning.bin_edges[ibin_col];
            double k_upper_b = kbinning.bin_edges[ibin_col + 1];
 
            double k_eff_a_, k_eff_b_;
            int nmodes_a_, nmodes_b_;
 
            F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_00, ylm_k_a, k_lower_a, k_upper_a, k_eff_a_, nmodes_a_
            );
            F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_00, ylm_k_b, k_lower_b, k_upper_b, k_eff_b_, nmodes_b_
            );
            update_progbar();
 
            if (count_terms == 0) {
              k1bin_dv[idx_dv] = kbinning.bin_centres[ibin_row];
              k2bin_dv[idx_dv] = kbinning.bin_centres[ibin_col];
              k1eff_dv[idx_dv] = k_eff_a_;
              k2eff_dv[idx_dv] = k_eff_b_;
              nmodes1_dv[idx_dv] = nmodes_a_;
              nmodes2_dv[idx_dv] = nmodes_b_;
            }
 
            // B_{l₁ l₂ L}^{m₁ m₂ M}
            double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
            for (long long gid = 0; gid < params.nmesh; gid++) {
              std::complex<double> F_lm_a_gridpt(
                F_lm_a[gid][0], F_lm_a[gid][1]
              );
              std::complex<double> F_lm_b_gridpt(
                F_lm_b[gid][0], F_lm_b[gid][1]
              );
              std::complex<double> G_LM_gridpt(G_LM[gid][0], G_LM[gid][1]);
              std::complex<double> bk_gridpt =
                F_lm_a_gridpt * F_lm_b_gridpt * G_LM_gridpt;
 
              bk_comp_real += bk_gridpt.real();
              bk_comp_imag += bk_gridpt.imag();
            }
 
            std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
            bk_dv[idx_dv] += coupling * vol_cell * (
              bk_component + factor_mirror * std::conj(bk_component)
            );
          }
        }
 
        if (params.shape == "row") {
          int ibin_row = params.idx_bin;
 
          double k_lower_a = kbinning.bin_edges[ibin_row];
          double k_upper_a = kbinning.bin_edges[ibin_row + 1];
 
          double k_eff_a_;
          int nmodes_a_;
 
          F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
            dn_00, ylm_k_a, k_lower_a, k_upper_a, k_eff_a_, nmodes_a_
          );
 
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin_col = idx_dv;
 
            double k_lower_b = kbinning.bin_edges[ibin_col];
            double k_upper_b = kbinning.bin_edges[ibin_col + 1];
 
            double k_eff_b_;
            int nmodes_b_;
 
            F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_00, ylm_k_b, k_lower_b, k_upper_b, k_eff_b_, nmodes_b_
            );
            update_progbar();
 
            if (count_terms == 0) {
              k1bin_dv[idx_dv] = kbinning.bin_centres[ibin_row];
              k2bin_dv[idx_dv] = kbinning.bin_centres[ibin_col];
              k1eff_dv[idx_dv] = k_eff_a_;
              k2eff_dv[idx_dv] = k_eff_b_;
              nmodes1_dv[idx_dv] = nmodes_a_;
              nmodes2_dv[idx_dv] = nmodes_b_;
            }
 
            // B_{l₁ l₂ L}^{m₁ m₂ M}
            double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
            for (long long gid = 0; gid < params.nmesh; gid++) {
              std::complex<double> F_lm_a_gridpt(
                F_lm_a[gid][0], F_lm_a[gid][1]
              );
              std::complex<double> F_lm_b_gridpt(
                F_lm_b[gid][0], F_lm_b[gid][1]
              );
              std::complex<double> G_LM_gridpt(G_LM[gid][0], G_LM[gid][1]);
              std::complex<double> bk_gridpt =
                F_lm_a_gridpt * F_lm_b_gridpt * G_LM_gridpt;
 
              bk_comp_real += bk_gridpt.real();
              bk_comp_imag += bk_gridpt.imag();
            }
 
            std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
            bk_dv[idx_dv] += coupling * vol_cell * (
              bk_component + factor_mirror * std::conj(bk_component)
            );
          }
        }
 
        if (params.shape == "full") {
          for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
            for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
              int idx_dv = idx_row * params.num_bins + idx_col;
 
              double k_lower_a = kbinning.bin_edges[idx_row];
              double k_upper_a = kbinning.bin_edges[idx_row + 1];
              double k_lower_b = kbinning.bin_edges[idx_col];
              double k_upper_b = kbinning.bin_edges[idx_col + 1];
 
              double k_eff_a_, k_eff_b_;
              int nmodes_a_, nmodes_b_;
 
              F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
                dn_00, ylm_k_a, k_lower_a, k_upper_a, k_eff_a_, nmodes_a_
              );
              F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
                dn_00, ylm_k_b, k_lower_b, k_upper_b, k_eff_b_, nmodes_b_
              );
              update_progbar();
 
              if (count_terms == 0) {
                k1bin_dv[idx_dv] = kbinning.bin_centres[idx_row];
                k2bin_dv[idx_dv] = kbinning.bin_centres[idx_col];
                k1eff_dv[idx_dv] = k_eff_a_;
                k2eff_dv[idx_dv] = k_eff_b_;
                nmodes1_dv[idx_dv] = nmodes_a_;
                nmodes2_dv[idx_dv] = nmodes_b_;
              }
 
              // B_{l₁ l₂ L}^{m₁ m₂ M}
              double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
              for (long long gid = 0; gid < params.nmesh; gid++) {
                std::complex<double> F_lm_a_gridpt(
                  F_lm_a[gid][0], F_lm_a[gid][1]
                );
                std::complex<double> F_lm_b_gridpt(
                  F_lm_b[gid][0], F_lm_b[gid][1]
                );
                std::complex<double> G_LM_gridpt(G_LM[gid][0], G_LM[gid][1]);
                std::complex<double> bk_gridpt =
                  F_lm_a_gridpt * F_lm_b_gridpt * G_LM_gridpt;
 
                bk_comp_real += bk_gridpt.real();
                bk_comp_imag += bk_gridpt.imag();
              }
 
              std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
              bk_dv[idx_dv] += coupling * vol_cell * (
                bk_component + factor_mirror * std::conj(bk_component)
              );
            }
          }
        }
 
        if (params.shape == "triu") {
          for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
            for (int idx_col = idx_row; idx_col < params.num_bins; idx_col++) {
              int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2
                + (idx_col - idx_row);
 
              double k_lower_a = kbinning.bin_edges[idx_row];
              double k_upper_a = kbinning.bin_edges[idx_row + 1];
              double k_lower_b = kbinning.bin_edges[idx_col];
              double k_upper_b = kbinning.bin_edges[idx_col + 1];
 
              double k_eff_a_, k_eff_b_;
              int nmodes_a_, nmodes_b_;
 
              F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
                dn_00, ylm_k_a, k_lower_a, k_upper_a, k_eff_a_, nmodes_a_
              );
              F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
                dn_00, ylm_k_b, k_lower_b, k_upper_b, k_eff_b_, nmodes_b_
              );
              update_progbar();
 
              if (count_terms == 0) {
                k1bin_dv[idx_dv] = kbinning.bin_centres[idx_row];
                k2bin_dv[idx_dv] = kbinning.bin_centres[idx_col];
                k1eff_dv[idx_dv] = k_eff_a_;
                k2eff_dv[idx_dv] = k_eff_b_;
                nmodes1_dv[idx_dv] = nmodes_a_;
                nmodes2_dv[idx_dv] = nmodes_b_;
              }
 
              // B_{l₁ l₂ L}^{m₁ m₂ M}
              double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
              for (long long gid = 0; gid < params.nmesh; gid++) {
                std::complex<double> F_lm_a_gridpt(
                  F_lm_a[gid][0], F_lm_a[gid][1]
                );
                std::complex<double> F_lm_b_gridpt(
                  F_lm_b[gid][0], F_lm_b[gid][1]
                );
                std::complex<double> G_LM_gridpt(G_LM[gid][0], G_LM[gid][1]);
                std::complex<double> bk_gridpt =
                  F_lm_a_gridpt * F_lm_b_gridpt * G_LM_gridpt;
 
                bk_comp_real += bk_gridpt.real();
                bk_comp_imag += bk_gridpt.imag();
              }
 
              std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
              bk_dv[idx_dv] += coupling * vol_cell * (
                bk_component + factor_mirror * std::conj(bk_component)
              );
            }
          }
        }
 
        // ·······························································
        // Shot noise
        // ·······························································
 
        // Compute shot noise components in eqs. (45) & (46) in the Paper.
        MeshField dn_LM_for_sn(params, true, "`dn_LM_for_sn`");  // δn_LM(k)
                                                                 // (for
                                                                 // shot noise)
        dn_LM_for_sn.compute_ylm_wgtd_field(
          catalogue_data, catalogue_rand, los_data, los_rand, alpha,
          params.ELL, M_
        );
        dn_LM_for_sn.fourier_transform();
 
        MeshField N_LM(params, true, "`N_LM`");  // N_LM(k)
        N_LM.compute_ylm_wgtd_quad_field(
          catalogue_data, catalogue_rand, los_data, los_rand, alpha,
          params.ELL, M_
        );
        N_LM.fourier_transform();
        update_progbar();
 
        std::complex<double> Sbar_LM = calc_ylm_wgtd_shotnoise_amp_for_bispec(
          catalogue_data, catalogue_rand, los_data, los_rand, alpha,
          params.ELL, M_
        );  // \bar{S}_LM
 
        if (params.ell1 == 0 && params.ell2 == 0) {
          // When l₁ = l₂ = 0, the Wigner 3-j symbol enforces L = 0
          // and the pre-factors involving degrees and orders become 1.
          std::complex<double> S_ijk = coupling * Sbar_LM;  // S|{i = j = k}
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            sn_dv[idx_dv] += S_ijk;
          }
        }
 
        if (params.ell2 == 0) {  // S|{i ≠ j = k}
          // When l₂ = 0, the Wigner 3-j symbol enforces L = l₁.
          stats_sn.compute_ylm_wgtd_2pt_stats_in_fourier(
            dn_00_for_sn, N_LM, Sbar_LM, params.ell1, m1_, kbinning
          );
 
          if (params.shape == "diag") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin = idx_dv;
              std::complex<double> S_i_jk = coupling * (
                stats_sn.pk[ibin] - stats_sn.sn[ibin]
              );
              sn_dv[idx_dv] += S_i_jk + factor_mirror * std::conj(S_i_jk);
            }
          }
 
          if (params.shape == "off-diag") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin_row;
              if (params.idx_bin >= 0) {
                ibin_row = idx_dv;
              } else {
                ibin_row = idx_dv + std::abs(params.idx_bin);
              }
              std::complex<double> S_i_jk = coupling * (
                stats_sn.pk[ibin_row] - stats_sn.sn[ibin_row]
              );
              sn_dv[idx_dv] += S_i_jk + factor_mirror * std::conj(S_i_jk);
            }
          }
 
          if (params.shape == "row") {
            std::complex<double> S_i_jk_row_ = coupling * (
              stats_sn.pk[params.idx_bin] - stats_sn.sn[params.idx_bin]
            );
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              sn_dv[idx_dv] +=
                S_i_jk_row_ + factor_mirror * std::conj(S_i_jk_row_);
            }
          }
 
          if (params.shape == "full") {
            for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
              std::complex<double> S_i_jk_row_ = coupling * (
                stats_sn.pk[idx_row] - stats_sn.sn[idx_row]
              );
              for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
                int idx_dv = idx_row * params.num_bins + idx_col;
                sn_dv[idx_dv] +=
                  S_i_jk_row_ + factor_mirror * std::conj(S_i_jk_row_);
              }
            }
          }
 
          if (params.shape == "triu") {
            for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
              std::complex<double> S_i_jk_row_ = coupling * (
                stats_sn.pk[idx_row] - stats_sn.sn[idx_row]
              );
              for (int idx_col = idx_row; idx_col < params.num_bins; idx_col++)
              {
                int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2
                  + (idx_col - idx_row);
                sn_dv[idx_dv] +=
                  S_i_jk_row_ + factor_mirror * std::conj(S_i_jk_row_);
              }
            }
          }
        }
 
        if (params.ell1 == 0) {  // S|{j ≠ i = k}
          // When l₁ = 0, the Wigner 3-j symbol enforces L = l₂.
          stats_sn.compute_ylm_wgtd_2pt_stats_in_fourier(
            dn_00_for_sn, N_LM, Sbar_LM, params.ell2, m2_, kbinning
          );
 
          if (params.shape == "diag") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin = idx_dv;
              std::complex<double> S_ik_j_ki = coupling * (
                stats_sn.pk[ibin] - stats_sn.sn[ibin]
              );
              sn_dv[idx_dv] +=
                S_ik_j_ki + factor_mirror * std::conj(S_ik_j_ki);
            }
          }
 
          if (params.shape == "off-diag") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin_col;
              if (params.idx_bin >= 0) {
                ibin_col = idx_dv + params.idx_bin;
              } else {
                ibin_col = idx_dv;
              }
              std::complex<double> S_ik_j_ki = coupling * (
                stats_sn.pk[ibin_col] - stats_sn.sn[ibin_col]
              );
              sn_dv[idx_dv] +=
                S_ik_j_ki + factor_mirror * std::conj(S_ik_j_ki);
            }
          }
 
          if (params.shape == "row") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin_col = idx_dv;
              std::complex<double> S_ik_j_ki = coupling * (
                stats_sn.pk[ibin_col] - stats_sn.sn[ibin_col]
              );
              sn_dv[idx_dv] +=
                S_ik_j_ki + factor_mirror * std::conj(S_ik_j_ki);
            }
          }
 
          if (params.shape == "full") {
            for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
              std::complex<double> S_ik_j_ki_col_ = coupling * (
                stats_sn.pk[idx_col] - stats_sn.sn[idx_col]
              );
              for (int idx_row = 0; idx_row <= params.num_bins; idx_row++) {
                int idx_dv = idx_row * params.num_bins + idx_col;
                sn_dv[idx_dv] +=
                  S_ik_j_ki_col_ + factor_mirror * std::conj(S_ik_j_ki_col_);
              }
            }
          }
 
          if (params.shape == "triu") {
            for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
              std::complex<double> S_ik_j_ki_col_ = coupling * (
                stats_sn.pk[idx_col] - stats_sn.sn[idx_col]
              );
              for (int idx_row = 0; idx_row <= idx_col; idx_row++) {
                int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2
                  + (idx_col - idx_row);
                sn_dv[idx_dv] +=
                  S_ik_j_ki_col_ + factor_mirror * std::conj(S_ik_j_ki_col_);
              }
            }
          }
        }
 
        for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
          double k_a = k1eff_dv[idx_dv];
          double k_b = k2eff_dv[idx_dv];
 
          std::complex<double> S_ij_k = coupling
            * stats_sn.compute_uncoupled_shotnoise_for_bispec_per_bin(
              dn_LM_for_sn, N_00, ylm_r_a, ylm_r_b, sj_a, sj_b,
              Sbar_LM, k_a, k_b
            );  // S|{i = j ≠ k}
          update_progbar();
 
          sn_dv[idx_dv] += factor_phase * (
            S_ij_k + factor_mirror_w3j * std::conj(S_ij_k)
          );
        }
 
        sph_orders_computed.push_back(sph_order_current);
        count_terms++;
        if (trvs::currTask == 0) {
          trvs::logger.stat(
            "Bispectrum term computed at orders (m₁, m₂, M) = ±(%d, %d, %d).",
            m1_, m2_, M_
          );
        }
      }
    }
  }
 
  // ---------------------------------------------------------------------
  // Results
  // ---------------------------------------------------------------------
 
  trv::BispecMeasurements bispec_out;
  for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
    bispec_out.k1_bin.push_back(k1bin_dv[idx_dv]);
    bispec_out.k1_eff.push_back(k1eff_dv[idx_dv]);
    bispec_out.nmodes_1.push_back(nmodes1_dv[idx_dv]);
    bispec_out.k2_bin.push_back(k2bin_dv[idx_dv]);
    bispec_out.k2_eff.push_back(k2eff_dv[idx_dv]);
    bispec_out.nmodes_2.push_back(nmodes2_dv[idx_dv]);
    bispec_out.bk_raw.push_back(norm_factor * bk_dv[idx_dv]);
    bispec_out.bk_shot.push_back(norm_factor * sn_dv[idx_dv]);
  }
  bispec_out.dim = dv_dim;
 
  delete[] nmodes1_dv; delete[] nmodes2_dv;
  delete[] k1bin_dv; delete[] k2bin_dv;
  delete[] k1eff_dv; delete[] k2eff_dv;
  delete[] bk_dv; delete[] sn_dv;
 
  trvs::count_cgrid -= 4;
  trvs::count_grid -= 4;
  trvs::gbytesMem -=
    trvs::size_in_gb< std::complex<double> >(4*params.nmesh);
 
  if (trvs::currTask == 0) {
    trvs::logger.stat(
      "... computed bispectrum from paired survey-type catalogues."
    );
  }
 
  return bispec_out;
}
 
trv::ThreePCFMeasurements compute_3pcf(
  ParticleCatalogue& catalogue_data, ParticleCatalogue& catalogue_rand,
  LineOfSight* los_data, LineOfSight* los_rand,
  trv::ParameterSet& params, trv::Binning& rbinning,
  double norm_factor
) {
  trvs::logger.reset_level(params.verbose);
 
  if (trvs::currTask == 0) {
    trvs::logger.stat(
      "Computing three-point correlation function "
      "from paired survey-type catalogues..."
    );
  }
 
  // ---------------------------------------------------------------------
  // Set-up
  // ---------------------------------------------------------------------
 
  // Set up/check input.
  validate_multipole_coupling(params);
 
  double alpha = catalogue_data.wstotal / catalogue_rand.wstotal;
 
  std::complex<double> factor_phase =
    std::pow(trvm::M_I, params.ell1 + params.ell2);
  double factor_parity = std::pow(-1., params.ell1 + params.ell2);
 
  // Set up output.
  int dv_dim = 0;  // data vector dimension
  if (params.shape == "diag" || params.shape == "row" ) {
    dv_dim = rbinning.num_bins;
  } else
  if (params.shape == "off-diag") {
    dv_dim = rbinning.num_bins - std::abs(params.idx_bin);
  } else
  if (params.shape == "full") {
    dv_dim = rbinning.num_bins * rbinning.num_bins;
  } else
  if (params.shape == "triu") {
    dv_dim = rbinning.num_bins * (rbinning.num_bins + 1) / 2;
  } else {
    if (trvs::currTask == 0) {
      trvs::logger.error(
        "Three-point statistic form is not recognised: `form` = '%s'.",
        params.form.c_str()
      );
    }
    throw trvs::InvalidParameterError(
      "Three-point statistic form is not recognised: `form` = '%s'.",
      params.form.c_str()
    );
  }
 
  int* npairs1_dv = new int[dv_dim]();
  int* npairs2_dv = new int[dv_dim]();
  double* r1bin_dv = new double[dv_dim]();
  double* r2bin_dv = new double[dv_dim]();
  double* r1eff_dv = new double[dv_dim]();
  double* r2eff_dv = new double[dv_dim]();
  std::complex<double>* zeta_dv = new std::complex<double>[dv_dim];
  std::complex<double>* sn_dv = new std::complex<double>[dv_dim];
 
  // ---------------------------------------------------------------------
  // Measurement
  // ---------------------------------------------------------------------
 
#if defined(TRV_USE_OMP) && defined(TRV_USE_FFTWOMP)
  if (!trvs::is_gpu_enabled()) {
    fftw_init_threads();
  }
#endif  // TRV_USE_OMP && TRV_USE_FFTWOMP
 
  // Compute common field quantities.
  MeshField dn_00(params, true, "`dn_00`");  // δn_00(k)
  dn_00.compute_ylm_wgtd_field(
    catalogue_data, catalogue_rand, los_data, los_rand, alpha, 0, 0
  );
  dn_00.fourier_transform();
 
  double vol_cell = dn_00.vol_cell;
 
  MeshField N_00(params, true, "`N_00`");  // N_00(k)
  N_00.compute_ylm_wgtd_quad_field(
    catalogue_data, catalogue_rand, los_data, los_rand, alpha, 0, 0
  );
  N_00.fourier_transform();
 
  trvm::SphericalBesselCalculator sj_a(params.ell1);  // j_l_a
  trvm::SphericalBesselCalculator sj_b(params.ell2);  // j_l_b
 
  FieldStats stats_sn(params);
 
  // Initialise reduced-spherical-harmonic weights on mesh grids.
  std::vector< std::complex<double> > ylm_r_a(params.nmesh);
  std::vector< std::complex<double> > ylm_r_b(params.nmesh);
  std::vector< std::complex<double> > ylm_k_a(params.nmesh);
  std::vector< std::complex<double> > ylm_k_b(params.nmesh);
  trvs::count_cgrid += 4;
  trvs::count_grid += 4;
  trvs::update_maxcntgrid();
  trvs::gbytesMem +=
    trvs::size_in_gb< std::complex<double> >(4*params.nmesh);
  trvs::update_maxmem();
 
  // Compute 3PCF terms including shot noise.
  std::vector<trv::SphericalOrderTriplet> sph_orders_computed;
  int count_terms = 0;
  for (int m1_ = - params.ell1; m1_ <= params.ell1; m1_++) {
    for (int m2_ = - params.ell2; m2_ <= params.ell2; m2_++) {
      // Check for if all Wigner-3j symbols are zero.
      std::string flag_vanishing = "true";
      for (int M_ = - params.ELL; M_ <= params.ELL; M_++) {
        double coupling = trv::calc_coupling_coeff_3pt(
          params.ell1, params.ell2, params.ELL, m1_, m2_, M_
        );
        if (std::fabs(coupling) > trvm::eps_coupling) {
          flag_vanishing = "false";
          break;
        }
      }
      if (flag_vanishing == "true") {continue;}
 
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_config_space(
          params.ell1, m1_, params.boxsize, params.ngrid, ylm_r_a
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_config_space(
          params.ell2, m2_, params.boxsize, params.ngrid, ylm_r_b
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_fourier_space(
          params.ell1, m1_, params.boxsize, params.ngrid, ylm_k_a
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_fourier_space(
          params.ell2, m2_, params.boxsize, params.ngrid, ylm_k_b
        );
 
      for (int M_ = - params.ELL; M_ <= params.ELL; M_++) {
        // Check if the current spherical order triplet is redundant.
        SphericalOrderTriplet sph_order_current = {m1_, m2_, M_};
 
        std::string flag_redundant = "false";
        for (SphericalOrderTriplet sph_order : sph_orders_computed) {
          if (sph_order_current.is_inverse(sph_order)) {
            flag_redundant = "true";
          }
        }
        if (flag_redundant == "true") {continue;}
 
        // Calculate the coupling coefficient.
        double coupling = trv::calc_coupling_coeff_3pt(
          params.ell1, params.ell2, params.ELL, m1_, m2_, M_
        );  // Wigner 3-j's
        if (std::fabs(coupling) < trvm::eps_coupling) {continue;}
 
        // The factor should be
        // ```
        // std::pow(-1., params.ell1 + params.ell2 + params.ELL) *
        // std::pow(-1., m1_ + m2_ + M_) *
        // {std::pow(-1., params.ell1 + params.ell2), 1};
        // ```
        // which, by parity symmetry and Wigner 3-j properties,
        // simplifies to:
        double factor_mirror = sph_order_current.is_zeros() ?
          0. : std::pow(-1., params.ELL);
        double factor_mirror_w3j = sph_order_current.is_zeros() ?
          0. : 1.;  // std::pow(-1., params.ell1 + params.ell2 + params.ELL);
 
        // Set up FFT counter and progress bar.
        int progbar_task_count = 4 + 2 * dv_dim;
        std::stringstream progbar_name_ss;
        progbar_name_ss
          << "component orders (" << m1_ << ", " << m2_ << ", " << M_ << ")";
        trvs::ProgressBar progbar(progbar_task_count, progbar_name_ss.str());
 
        std::vector<float> progbar_nodes;
        if (params.progbar != "false" && params.progbar != "true") {
          progbar_nodes = trvs::set_nodes_by_str(params.progbar);
          if (progbar_nodes.size() > 0) {
            progbar.set_nodes(progbar_nodes);
          }
        }
 
        int count_fft_init = trvs::count_fft + trvs::count_ifft;
        auto update_progbar = [&]() {
          if (params.progbar != "false") {
            if (trvs::currTask == 0) {
              progbar.update(
                trvs::count_fft + trvs::count_ifft - count_fft_init
              );
            }
          }
        };
 
        // ·······························································
        // Shot noise
        // ·······························································
 
        // Compute shot noise components in eq. (51) in the Paper.
        MeshField dn_LM_for_sn(params, true, "`dn_LM_for_sn`");  // δn_LM(k)
                                                                 // (for
                                                                 // shot noise)
        dn_LM_for_sn.compute_ylm_wgtd_field(
          catalogue_data, catalogue_rand, los_data, los_rand, alpha,
          params.ELL, M_
        );
        dn_LM_for_sn.fourier_transform();
        update_progbar();
 
        std::complex<double> Sbar_LM = calc_ylm_wgtd_shotnoise_amp_for_bispec(
          catalogue_data, catalogue_rand, los_data, los_rand, alpha,
          params.ELL, M_
        );  // \bar{S}_LM
 
        stats_sn.compute_uncoupled_shotnoise_for_3pcf(
          dn_LM_for_sn, N_00, ylm_r_a, ylm_r_b, Sbar_LM, rbinning
        );  // S|{i = j ≠ k}
        update_progbar();
 
        // Enforce the Kronecker delta in eq. (51) in the Paper.
        if (params.shape == "diag") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin = idx_dv;
            sn_dv[idx_dv] += coupling * factor_parity * (
              stats_sn.xi[ibin]
              + factor_mirror_w3j * std::conj(stats_sn.xi[ibin])
            );
          }
        }
 
        if (params.shape == "off-diag" && params.idx_bin == 0) {
          // Note that ``idx_col == idx_row``.
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin = idx_dv;
            sn_dv[idx_dv] += coupling * factor_parity * (
              stats_sn.xi[ibin]
              + factor_mirror_w3j * std::conj(stats_sn.xi[ibin])
            );
          }
        }
 
        if (params.shape == "row") {
          sn_dv[params.idx_bin] += coupling * factor_parity * (
            stats_sn.xi[params.idx_bin]
            + factor_mirror_w3j * std::conj(stats_sn.xi[params.idx_bin])
          );
        }
 
        if (params.shape == "full") {
          for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
            // Note that ``idx_col == idx_row``.
            int idx_dv = idx_row * params.num_bins + idx_row;
            sn_dv[idx_dv] += coupling * factor_parity * (
              stats_sn.xi[idx_row]
              + factor_mirror_w3j * std::conj(stats_sn.xi[idx_row])
            );
          }
        }
 
        if (params.shape == "triu") {
          for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
            // Note that ``idx_col == idx_row``.
            int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2;
            sn_dv[idx_dv] += coupling * factor_parity * (
              stats_sn.xi[idx_row]
              + factor_mirror_w3j * std::conj(stats_sn.xi[idx_row])
            );
          }
        }
 
        // Only record the binned coordinates and counts once.
        if (count_terms == 0) {
          if (params.shape == "diag") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin = idx_dv;
 
              r1bin_dv[idx_dv] = rbinning.bin_centres[ibin];
              r2bin_dv[idx_dv] = rbinning.bin_centres[ibin];
              r1eff_dv[idx_dv] = stats_sn.r[ibin];
              r2eff_dv[idx_dv] = stats_sn.r[ibin];
              npairs1_dv[idx_dv] = stats_sn.npairs[ibin];
              npairs2_dv[idx_dv] = stats_sn.npairs[ibin];
            }
          }
 
          if (params.shape == "off-diag") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin_row, ibin_col;
              if (params.idx_bin >= 0) {
                ibin_row = idx_dv;
                ibin_col = idx_dv + std::abs(params.idx_bin);
              } else {
                ibin_row = idx_dv + std::abs(params.idx_bin);
                ibin_col = idx_dv;
              }
 
              r1bin_dv[idx_dv] = rbinning.bin_centres[ibin_row];
              r2bin_dv[idx_dv] = rbinning.bin_centres[ibin_col];
              r1eff_dv[idx_dv] = stats_sn.r[ibin_row];
              r2eff_dv[idx_dv] = stats_sn.r[ibin_col];
              npairs1_dv[idx_dv] = stats_sn.npairs[ibin_row];
              npairs2_dv[idx_dv] = stats_sn.npairs[ibin_col];
            }
          }
 
          if (params.shape == "row") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin_row = params.idx_bin;
              int ibin_col = idx_dv;
 
              r1bin_dv[idx_dv] = rbinning.bin_centres[ibin_row];
              r2bin_dv[idx_dv] = rbinning.bin_centres[ibin_col];
              r1eff_dv[idx_dv] = stats_sn.r[ibin_row];
              r2eff_dv[idx_dv] = stats_sn.r[ibin_col];
              npairs1_dv[idx_dv] = stats_sn.npairs[ibin_row];
              npairs2_dv[idx_dv] = stats_sn.npairs[ibin_col];
            }
          }
 
          if (params.shape == "full") {
            for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
              for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
                int idx_dv = idx_row * params.num_bins + idx_col;
 
                r1bin_dv[idx_dv] = rbinning.bin_centres[idx_row];
                r2bin_dv[idx_dv] = rbinning.bin_centres[idx_col];
                r1eff_dv[idx_dv] = stats_sn.r[idx_row];
                r2eff_dv[idx_dv] = stats_sn.r[idx_col];
                npairs1_dv[idx_dv] = stats_sn.npairs[idx_row];
                npairs2_dv[idx_dv] = stats_sn.npairs[idx_col];
              }
            }
          }
 
          if (params.shape == "triu") {
            for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
              for (int idx_col = idx_row; idx_col < params.num_bins; idx_col++)
              {
                int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2
                  + (idx_col - idx_row);
 
                r1bin_dv[idx_dv] = rbinning.bin_centres[idx_row];
                r2bin_dv[idx_dv] = rbinning.bin_centres[idx_col];
                r1eff_dv[idx_dv] = stats_sn.r[idx_row];
                r2eff_dv[idx_dv] = stats_sn.r[idx_col];
                npairs1_dv[idx_dv] = stats_sn.npairs[idx_row];
                npairs2_dv[idx_dv] = stats_sn.npairs[idx_col];
              }
            }
          }
        }
 
        // ·······························································
        // Raw 3PCF
        // ·······························································
 
        // Compute 3PCF components in eqs. (42), (48) & (49) in the Paper.
        MeshField G_LM(params, true, "`G_LM`");  // G_LM
        G_LM.compute_ylm_wgtd_field(
          catalogue_data, catalogue_rand, los_data, los_rand, alpha,
          params.ELL, M_
        );
        G_LM.fourier_transform();
        G_LM.apply_assignment_compensation();
        G_LM.inv_fourier_transform();
        update_progbar();
 
        MeshField F_lm_a(params, true, "`F_lm_a`");  // F_lm_a
        MeshField F_lm_b(params, true, "`F_lm_b`");  // F_lm_b
 
        for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
          double r_a = r1eff_dv[idx_dv];
          F_lm_a.inv_fourier_transform_sjl_ylm_wgtd_field(
            dn_00, ylm_k_a, sj_a, r_a
          );
 
          double r_b = r2eff_dv[idx_dv];
          F_lm_b.inv_fourier_transform_sjl_ylm_wgtd_field(
            dn_00, ylm_k_b, sj_b, r_b
          );
          update_progbar();
 
          // ζ_{l₁ l₂ L}^{m₁ m₂ M}
          double zeta_comp_real = 0., zeta_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:zeta_comp_real, zeta_comp_imag)
#endif  // TRV_USE_OMP
          for (long long gid = 0; gid < params.nmesh; gid++) {
            std::complex<double> F_lm_a_gridpt(F_lm_a[gid][0], F_lm_a[gid][1]);
            std::complex<double> F_lm_b_gridpt(F_lm_b[gid][0], F_lm_b[gid][1]);
            std::complex<double> G_LM_gridpt(G_LM[gid][0], G_LM[gid][1]);
            std::complex<double> zeta_gridpt =
              F_lm_a_gridpt * F_lm_b_gridpt * G_LM_gridpt;
 
            zeta_comp_real += zeta_gridpt.real();
            zeta_comp_imag += zeta_gridpt.imag();
          }
 
          std::complex<double> zeta_component(zeta_comp_real, zeta_comp_imag);
 
          zeta_dv[idx_dv] += coupling * vol_cell * factor_phase * (
            zeta_component + factor_mirror * std::conj(zeta_component)
          );
        }
 
        sph_orders_computed.push_back(sph_order_current);
        count_terms++;
        if (trvs::currTask == 0) {
          trvs::logger.stat(
            "Three-point correlation function term computed at orders "
            "(m₁, m₂, M) = ±(%d, %d, %d).",
            m1_, m2_, M_
          );
        }
      }
    }
  }
 
  // ---------------------------------------------------------------------
  // Results
  // ---------------------------------------------------------------------
 
  trv::ThreePCFMeasurements threepcf_out;
  for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
    threepcf_out.r1_bin.push_back(r1bin_dv[idx_dv]);
    threepcf_out.r1_eff.push_back(r1eff_dv[idx_dv]);
    threepcf_out.npairs_1.push_back(npairs1_dv[idx_dv]);
    threepcf_out.r2_bin.push_back(r2bin_dv[idx_dv]);
    threepcf_out.r2_eff.push_back(r2eff_dv[idx_dv]);
    threepcf_out.npairs_2.push_back(npairs2_dv[idx_dv]);
    threepcf_out.zeta_raw.push_back(norm_factor * zeta_dv[idx_dv]);
    threepcf_out.zeta_shot.push_back(norm_factor * sn_dv[idx_dv]);
  }
  threepcf_out.dim = dv_dim;
 
  delete[] npairs1_dv; delete[] npairs2_dv;
  delete[] r1bin_dv; delete[] r2bin_dv;
  delete[] r1eff_dv; delete[] r2eff_dv;
  delete[] zeta_dv; delete[] sn_dv;
 
  trvs::count_cgrid -= 4;
  trvs::count_grid -= 4;
  trvs::gbytesMem -=
    trvs::size_in_gb< std::complex<double> >(4*params.nmesh);
 
  if (trvs::currTask == 0) {
    trvs::logger.stat(
      "... computed three-point correlation function "
      "from paired survey-type catalogues."
    );
  }
 
  return threepcf_out;
}
 
trv::BispecMeasurements compute_bispec_in_gpp_box(
  ParticleCatalogue& catalogue_data,
  trv::ParameterSet& params, trv::Binning kbinning,
  double norm_factor
) {
  trvs::logger.reset_level(params.verbose);
 
  if (trvs::currTask == 0) {
    trvs::logger.stat(
      "Computing bispectrum from a periodic-box simulation-type catalogue "
      "in the global plane-parallel approximation..."
    );
  }
 
  // ---------------------------------------------------------------------
  // Set-up
  // ---------------------------------------------------------------------
 
  // Set up/check input.
  validate_multipole_coupling(params);
 
  std::complex<double> factor_phase =
    std::pow(trvm::M_I, params.ell1 + params.ell2);
 
  // Set up output.
  int dv_dim = 0;  // data vector dimension
  if (params.shape == "diag" || params.shape == "row" ) {
    dv_dim = kbinning.num_bins;
  } else
  if (params.shape == "off-diag") {
    dv_dim = kbinning.num_bins - std::abs(params.idx_bin);
  } else
  if (params.shape == "full") {
    dv_dim = kbinning.num_bins * kbinning.num_bins;
  } else
  if (params.shape == "triu") {
    dv_dim = kbinning.num_bins * (kbinning.num_bins + 1) / 2;
  } else {
    if (trvs::currTask == 0) {
      trvs::logger.error(
        "Three-point statistic form is not recognised: `form` = '%s'.",
        params.form.c_str()
      );
    }
    throw trvs::InvalidParameterError(
      "Three-point statistic form is not recognised: `form` = '%s'.",
      params.form.c_str()
    );
  }
 
  int* nmodes1_dv = new int[dv_dim]();
  int* nmodes2_dv = new int[dv_dim]();
  double* k1bin_dv = new double[dv_dim]();
  double* k2bin_dv = new double[dv_dim]();
  double* k1eff_dv = new double[dv_dim]();
  double* k2eff_dv = new double[dv_dim]();
  std::complex<double>* bk_dv = new std::complex<double>[dv_dim];
  std::complex<double>* sn_dv = new std::complex<double>[dv_dim];
 
  // ---------------------------------------------------------------------
  // Measurement
  // ---------------------------------------------------------------------
 
#if defined(TRV_USE_OMP) && defined(TRV_USE_FFTWOMP)
  if (!trvs::is_gpu_enabled()) {
    fftw_init_threads();
  }
#endif  // TRV_USE_OMP && TRV_USE_FFTWOMP
 
  // Compute common field quantities.
  MeshField dn_00(params, true, "`dn_00`");  // δn_00(k)
  dn_00.compute_unweighted_field_fluctuations_insitu(catalogue_data);
  dn_00.fourier_transform();
 
  MeshField& dn_00_for_sn = dn_00;  // δn_00(k) (for shot noise)
  MeshField& dn_L0_for_sn = dn_00;  // δn_L0(k) (for shot noise)
 
  double vol_cell = dn_00.vol_cell;
 
  // Under the global plane-parallel approximation, y_{LM} = δᴰ_{M0}
  // (L-invariant) for the line-of-sight spherical harmonic.
  MeshField N_L0(params, true, "`N_L0`");  // N_L0(k)
  N_L0.compute_unweighted_field(catalogue_data);
  N_L0.fourier_transform();
 
  MeshField& N_00 = N_L0;  // N_00(k)
 
  trvm::SphericalBesselCalculator sj_a(params.ell1);  // j_l_a
  trvm::SphericalBesselCalculator sj_b(params.ell2);  // j_l_b
 
  FieldStats stats_sn(params);
 
  // Initialise/reset spherical harmonic mesh grids.
  std::vector< std::complex<double> > ylm_k_a(params.nmesh);
  std::vector< std::complex<double> > ylm_k_b(params.nmesh);
  std::vector< std::complex<double> > ylm_r_a(params.nmesh);
  std::vector< std::complex<double> > ylm_r_b(params.nmesh);
  trvs::count_cgrid += 4;
  trvs::count_grid += 4;
  trvs::update_maxcntgrid();
  trvs::gbytesMem +=
    trvs::size_in_gb< std::complex<double> >(4*params.nmesh);
  trvs::update_maxmem();
 
  // Compute bispectrum terms including shot noise.
  std::vector<trv::SphericalOrderTriplet> sph_orders_computed;
  int count_terms = 0;
  for (int m1_ = - params.ell1; m1_ <= params.ell1; m1_++) {
    for (int m2_ = - params.ell2; m2_ <= params.ell2; m2_++) {
      // Under the global plane-parallel approximation, δᴰ_{M0} enforces
      // M = 0 for any spherical-harmonic-weighted field fluctuations.
      int M_ = 0;
 
      // Check if the current spherical order triplet is redundant.
      SphericalOrderTriplet sph_order_current = {m1_, m2_, M_};
 
      std::string flag_redundant = "false";
      for (SphericalOrderTriplet sph_order : sph_orders_computed) {
        if (sph_order_current.is_inverse(sph_order)) {
          flag_redundant = "true";
        }
      }
      if (flag_redundant == "true") {continue;}
 
      // Calculate the coupling coefficient.
      double coupling = trv::calc_coupling_coeff_3pt(
        params.ell1, params.ell2, params.ELL, m1_, m2_, M_
      );  // Wigner 3-j's
      if (std::fabs(coupling) < trvm::eps_coupling) {continue;}
 
      // The factor consist of products of
      // ```
      // {std::pow(-1., params.ell1 + params.ell2 + params.ELL),
      //  1},
      // {std::pow(-1., m1_ + m2_ + M_),
      //  std::pow(-1., 2*M_)} ≡ 1,
      // {std::pow(-1., params.ell1 + params.ell2),
      //  std::pow(-1., params.ELL)} ≡ std::pow(-1., params.ELL)}
      // ```
      // which, by parity symmetry & Wigner 3-j properties, simplify to:
      double factor_mirror = sph_order_current.is_zeros() ?
        0. : std::pow(-1., params.ELL);
      double factor_mirror_w3j = sph_order_current.is_zeros() ?
        0. : 1.;  // std::pow(-1., params.ell1 + params.ell2 + params.ELL);
 
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_fourier_space(
          params.ell1, m1_, params.boxsize, params.ngrid, ylm_k_a
        );
      trvm::SphericalHarmonicCalculator
        ::store_reduced_spherical_harmonic_in_fourier_space(
          params.ell2, m2_, params.boxsize, params.ngrid, ylm_k_b
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_config_space(
          params.ell1, m1_, params.boxsize, params.ngrid, ylm_r_a
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_config_space(
          params.ell2, m2_, params.boxsize, params.ngrid, ylm_r_b
        );
 
      // Set up FFT counter and progress bar.
      int progbar_task_count = 2 + 3 * dv_dim;
      std::stringstream progbar_name_ss;
      progbar_name_ss
        << "component orders (" << m1_ << ", " << m2_ << ", " << M_ << ")";
      trvs::ProgressBar progbar(progbar_task_count, progbar_name_ss.str());
 
      std::vector<float> progbar_nodes;
      if (params.progbar != "false" && params.progbar != "true") {
        progbar_nodes = trvs::set_nodes_by_str(params.progbar);
        if (progbar_nodes.size() > 0) {
          progbar.set_nodes(progbar_nodes);
        }
      }
 
      int count_fft_init = trvs::count_fft + trvs::count_ifft;
      auto update_progbar = [&]() {
        if (params.progbar != "false") {
          if (trvs::currTask == 0) {
            progbar.update(
              trvs::count_fft + trvs::count_ifft - count_fft_init
            );
          }
        }
      };
 
      // ·································································
      // Raw bispectrum
      // ·································································
 
      MeshField G_00(params, true, "`G_00`");  // G_00
      G_00.compute_unweighted_field_fluctuations_insitu(catalogue_data);
      G_00.fourier_transform();
      G_00.apply_assignment_compensation();
      G_00.inv_fourier_transform();
      update_progbar();
 
      MeshField F_lm_a(params, true, "`F_lm_a`");  // F_lm_a
      MeshField F_lm_b(params, true, "`F_lm_b`");  // F_lm_b
 
      if (params.shape == "diag") {
        for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
          int ibin = idx_dv;
 
          double k_lower = kbinning.bin_edges[ibin];
          double k_upper = kbinning.bin_edges[ibin + 1];
 
          double k_eff_a_, k_eff_b_;
          int nmodes_a_, nmodes_b_;
 
          F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
            dn_00, ylm_k_a, k_lower, k_upper, k_eff_a_, nmodes_a_
          );
          F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
            dn_00, ylm_k_b, k_lower, k_upper, k_eff_b_, nmodes_b_
          );
          update_progbar();
 
          if (count_terms == 0) {
            k1bin_dv[idx_dv] = kbinning.bin_centres[ibin];
            k2bin_dv[idx_dv] = kbinning.bin_centres[ibin];
            k1eff_dv[idx_dv] = k_eff_a_;
            k2eff_dv[idx_dv] = k_eff_b_;
            nmodes1_dv[idx_dv] = nmodes_a_;
            nmodes2_dv[idx_dv] = nmodes_b_;
          }
 
          // B_{l₁ l₂ L}^{m₁ m₂ M}
          double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
          for (long long gid = 0; gid < params.nmesh; gid++) {
            std::complex<double> F_lm_a_gridpt(F_lm_a[gid][0], F_lm_a[gid][1]);
            std::complex<double> F_lm_b_gridpt(F_lm_b[gid][0], F_lm_b[gid][1]);
            std::complex<double> G_00_gridpt(G_00[gid][0], G_00[gid][1]);
            std::complex<double> bk_gridpt =
              F_lm_a_gridpt * F_lm_b_gridpt * G_00_gridpt;
 
            bk_comp_real += bk_gridpt.real();
            bk_comp_imag += bk_gridpt.imag();
          }
 
          std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
          bk_dv[idx_dv] += coupling * vol_cell * (
            bk_component + factor_mirror * std::conj(bk_component)
          );
        }
      }
 
      if (params.shape == "off-diag") {
        for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
          int ibin_row, ibin_col;
          if (params.idx_bin >= 0) {
            ibin_row = idx_dv;
            ibin_col = idx_dv + std::abs(params.idx_bin);
          } else {
            ibin_row = idx_dv + std::abs(params.idx_bin);
            ibin_col = idx_dv;
          }
 
          double k_lower_a = kbinning.bin_edges[ibin_row];
          double k_upper_a = kbinning.bin_edges[ibin_row + 1];
          double k_lower_b = kbinning.bin_edges[ibin_col];
          double k_upper_b = kbinning.bin_edges[ibin_col + 1];
 
          double k_eff_a_, k_eff_b_;
          int nmodes_a_, nmodes_b_;
 
          F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
            dn_00, ylm_k_a, k_lower_a, k_upper_a, k_eff_a_, nmodes_a_
          );
          F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
            dn_00, ylm_k_b, k_lower_b, k_upper_b, k_eff_b_, nmodes_b_
          );
          update_progbar();
 
          if (count_terms == 0) {
            k1bin_dv[idx_dv] = kbinning.bin_centres[ibin_row];
            k2bin_dv[idx_dv] = kbinning.bin_centres[ibin_col];
            k1eff_dv[idx_dv] = k_eff_a_;
            k2eff_dv[idx_dv] = k_eff_b_;
            nmodes1_dv[idx_dv] = nmodes_a_;
            nmodes2_dv[idx_dv] = nmodes_b_;
          }
 
          // B_{l₁ l₂ L}^{m₁ m₂ M}
          double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
          for (long long gid = 0; gid < params.nmesh; gid++) {
            std::complex<double> F_lm_a_gridpt(F_lm_a[gid][0], F_lm_a[gid][1]);
            std::complex<double> F_lm_b_gridpt(F_lm_b[gid][0], F_lm_b[gid][1]);
            std::complex<double> G_00_gridpt(G_00[gid][0], G_00[gid][1]);
            std::complex<double> bk_gridpt =
              F_lm_a_gridpt * F_lm_b_gridpt * G_00_gridpt;
 
            bk_comp_real += bk_gridpt.real();
            bk_comp_imag += bk_gridpt.imag();
          }
 
          std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
          bk_dv[idx_dv] += coupling * vol_cell * (
            bk_component + factor_mirror * std::conj(bk_component)
          );
        }
      }
 
      if (params.shape == "row") {
        int ibin_row = params.idx_bin;
 
        double k_lower_a = kbinning.bin_edges[ibin_row];
        double k_upper_a = kbinning.bin_edges[ibin_row + 1];
 
        double k_eff_a_;
        int nmodes_a_;
 
        F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
          dn_00, ylm_k_a, k_lower_a, k_upper_a, k_eff_a_, nmodes_a_
        );
 
        for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
          int ibin_col = idx_dv;
 
          double k_lower_b = kbinning.bin_edges[ibin_col];
          double k_upper_b = kbinning.bin_edges[ibin_col + 1];
 
          double k_eff_b_;
          int nmodes_b_;
 
          F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
            dn_00, ylm_k_b, k_lower_b, k_upper_b, k_eff_b_, nmodes_b_
          );
          update_progbar();
 
          if (count_terms == 0) {
            k1bin_dv[idx_dv] = kbinning.bin_centres[ibin_row];
            k2bin_dv[idx_dv] = kbinning.bin_centres[ibin_col];
            k1eff_dv[idx_dv] = k_eff_a_;
            k2eff_dv[idx_dv] = k_eff_b_;
            nmodes1_dv[idx_dv] = nmodes_a_;
            nmodes2_dv[idx_dv] = nmodes_b_;
          }
 
          // B_{l₁ l₂ L}^{m₁ m₂ M}
          double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
          for (long long gid = 0; gid < params.nmesh; gid++) {
            std::complex<double> F_lm_a_gridpt(F_lm_a[gid][0], F_lm_a[gid][1]);
            std::complex<double> F_lm_b_gridpt(F_lm_b[gid][0], F_lm_b[gid][1]);
            std::complex<double> G_00_gridpt(G_00[gid][0], G_00[gid][1]);
            std::complex<double> bk_gridpt =
              F_lm_a_gridpt * F_lm_b_gridpt * G_00_gridpt;
 
            bk_comp_real += bk_gridpt.real();
            bk_comp_imag += bk_gridpt.imag();
          }
 
          std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
          bk_dv[idx_dv] += coupling * vol_cell * (
            bk_component + factor_mirror * std::conj(bk_component)
          );
        }
      }
 
      if (params.shape == "full") {
        for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
          for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
            int idx_dv = idx_row * params.num_bins + idx_col;
 
            double k_lower_a = kbinning.bin_edges[idx_row];
            double k_upper_a = kbinning.bin_edges[idx_row + 1];
            double k_lower_b = kbinning.bin_edges[idx_col];
            double k_upper_b = kbinning.bin_edges[idx_col + 1];
 
            double k_eff_a_, k_eff_b_;
            int nmodes_a_, nmodes_b_;
 
            F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_00, ylm_k_a, k_lower_a, k_upper_a, k_eff_a_, nmodes_a_
            );
            F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_00, ylm_k_b, k_lower_b, k_upper_b, k_eff_b_, nmodes_b_
            );
            update_progbar();
 
            if (count_terms == 0) {
              k1bin_dv[idx_dv] = kbinning.bin_centres[idx_row];
              k2bin_dv[idx_dv] = kbinning.bin_centres[idx_col];
              k1eff_dv[idx_dv] = k_eff_a_;
              k2eff_dv[idx_dv] = k_eff_b_;
              nmodes1_dv[idx_dv] = nmodes_a_;
              nmodes2_dv[idx_dv] = nmodes_b_;
            }
 
            // B_{l₁ l₂ L}^{m₁ m₂ M}
            double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
            for (long long gid = 0; gid < params.nmesh; gid++) {
              std::complex<double> F_lm_a_gridpt(
                F_lm_a[gid][0], F_lm_a[gid][1]
              );
              std::complex<double> F_lm_b_gridpt(
                F_lm_b[gid][0], F_lm_b[gid][1]
              );
              std::complex<double> G_00_gridpt(G_00[gid][0], G_00[gid][1]);
              std::complex<double> bk_gridpt =
                F_lm_a_gridpt * F_lm_b_gridpt * G_00_gridpt;
 
              bk_comp_real += bk_gridpt.real();
              bk_comp_imag += bk_gridpt.imag();
            }
 
            std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
            bk_dv[idx_dv] += coupling * vol_cell * (
              bk_component + factor_mirror * std::conj(bk_component)
            );
          }
        }
      }
 
      if (params.shape == "triu") {
        for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
          for (int idx_col = idx_row; idx_col < params.num_bins; idx_col++) {
            int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2
              + (idx_col - idx_row);
 
            double k_lower_a = kbinning.bin_edges[idx_row];
            double k_upper_a = kbinning.bin_edges[idx_row + 1];
            double k_lower_b = kbinning.bin_edges[idx_col];
            double k_upper_b = kbinning.bin_edges[idx_col + 1];
 
            double k_eff_a_, k_eff_b_;
            int nmodes_a_, nmodes_b_;
 
            F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_00, ylm_k_a, k_lower_a, k_upper_a, k_eff_a_, nmodes_a_
            );
            F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_00, ylm_k_b, k_lower_b, k_upper_b, k_eff_b_, nmodes_b_
            );
            update_progbar();
 
            if (count_terms == 0) {
              k1bin_dv[idx_dv] = kbinning.bin_centres[idx_row];
              k2bin_dv[idx_dv] = kbinning.bin_centres[idx_col];
              k1eff_dv[idx_dv] = k_eff_a_;
              k2eff_dv[idx_dv] = k_eff_b_;
              nmodes1_dv[idx_dv] = nmodes_a_;
              nmodes2_dv[idx_dv] = nmodes_b_;
            }
 
            // B_{l₁ l₂ L}^{m₁ m₂ M}
            double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
            for (long long gid = 0; gid < params.nmesh; gid++) {
              std::complex<double> F_lm_a_gridpt(
                F_lm_a[gid][0], F_lm_a[gid][1]
              );
              std::complex<double> F_lm_b_gridpt(
                F_lm_b[gid][0], F_lm_b[gid][1]
              );
              std::complex<double> G_00_gridpt(G_00[gid][0], G_00[gid][1]);
              std::complex<double> bk_gridpt =
                F_lm_a_gridpt * F_lm_b_gridpt * G_00_gridpt;
 
              bk_comp_real += bk_gridpt.real();
              bk_comp_imag += bk_gridpt.imag();
            }
 
            std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
            bk_dv[idx_dv] += coupling * vol_cell * (
              bk_component + factor_mirror * std::conj(bk_component)
            );
          }
        }
      }
 
      // ·································································
      // Shot noise
      // ·································································
 
      // Under the global plane-parallel approximation, y_{LM} = δᴰ_{M0}
      // (L-invariant) for the line-of-sight spherical harmonic.
      // Also note the field is unweighted from simulation sources.
      std::complex<double> Sbar_LM =
        double(catalogue_data.ntotal);  // \bar{S}_LM
      std::complex<double> Sbar_L0 = Sbar_LM;  // \bar{S}_L0
 
      if (params.ell1 == 0 && params.ell2 == 0) {
        std::complex<double> S_ijk = coupling * Sbar_LM;  // S|{i = j = k}
        for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
          sn_dv[idx_dv] += S_ijk;
        }
      }
 
      if (params.ell2 == 0) {  // S|{i ≠ j = k}
        // When l₂ = 0, the Wigner 3-j symbol enforces L = l₁.
        stats_sn.compute_ylm_wgtd_2pt_stats_in_fourier(
          dn_00_for_sn, N_L0, Sbar_LM, params.ell1, m1_, kbinning
        );
 
        if (params.shape == "diag") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin = idx_dv;
            std::complex<double> S_i_jk = coupling * (
              stats_sn.pk[ibin] - stats_sn.sn[ibin]
            );
            sn_dv[idx_dv] += S_i_jk + factor_mirror * std::conj(S_i_jk);
          }
        }
 
        if (params.shape == "off-diag") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin_row;
            if (params.idx_bin >= 0) {
              ibin_row = idx_dv;
            } else {
              ibin_row = idx_dv + std::abs(params.idx_bin);
            }
            std::complex<double> S_i_jk = coupling * (
              stats_sn.pk[ibin_row] - stats_sn.sn[ibin_row]
            );
            sn_dv[idx_dv] += S_i_jk + factor_mirror * std::conj(S_i_jk);
          }
        }
 
        if (params.shape == "row") {
          std::complex<double> S_i_jk_row_ = coupling * (
            stats_sn.pk[params.idx_bin] - stats_sn.sn[params.idx_bin]
          );
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            sn_dv[idx_dv] +=
              S_i_jk_row_ + factor_mirror * std::conj(S_i_jk_row_);
          }
        }
 
        if (params.shape == "full") {
          for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
            std::complex<double> S_i_jk_row_ = coupling * (
              stats_sn.pk[idx_row] - stats_sn.sn[idx_row]
            );
            for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
              int idx_dv = idx_row * params.num_bins + idx_col;
              sn_dv[idx_dv] +=
                S_i_jk_row_ + factor_mirror * std::conj(S_i_jk_row_);
            }
          }
        }
 
        if (params.shape == "triu") {
          for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
            std::complex<double> S_i_jk_row_ = coupling * (
              stats_sn.pk[idx_row] - stats_sn.sn[idx_row]
            );
            for (int idx_col = idx_row; idx_col < params.num_bins; idx_col++) {
              int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2
                + (idx_col - idx_row);
              sn_dv[idx_dv] +=
                S_i_jk_row_ + factor_mirror * std::conj(S_i_jk_row_);
            }
          }
        }
      }
 
      if (params.ell1 == 0) {  // S|{j ≠ i = k}
        // When l₁ = 0, the Wigner 3-j symbol enforces L = l₂.
        stats_sn.compute_ylm_wgtd_2pt_stats_in_fourier(
          dn_00_for_sn, N_L0, Sbar_LM, params.ell2, m2_, kbinning
        );
 
        if (params.shape == "diag") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin = idx_dv;
            std::complex<double> S_ik_j_ki = coupling * (
              stats_sn.pk[ibin] - stats_sn.sn[ibin]
            );
            sn_dv[idx_dv] += S_ik_j_ki + factor_mirror * std::conj(S_ik_j_ki);
          }
        }
 
        if (params.shape == "off-diag") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin_col;
            if (params.idx_bin >= 0) {
              ibin_col = idx_dv + params.idx_bin;
            } else {
              ibin_col = idx_dv;
            }
            std::complex<double> S_ik_j_ki = coupling * (
              stats_sn.pk[ibin_col] - stats_sn.sn[ibin_col]
            );
            sn_dv[idx_dv] += S_ik_j_ki + factor_mirror * std::conj(S_ik_j_ki);
          }
        }
 
        if (params.shape == "row") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin_col = idx_dv;
            std::complex<double> S_ik_j_ki = coupling * (
              stats_sn.pk[ibin_col] - stats_sn.sn[ibin_col]
            );
            sn_dv[idx_dv] += S_ik_j_ki + factor_mirror * std::conj(S_ik_j_ki);
          }
        }
 
        if (params.shape == "full") {
          for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
            std::complex<double> S_ik_j_ki_col_ = coupling * (
              stats_sn.pk[idx_col] - stats_sn.sn[idx_col]
            );
            for (int idx_row = 0; idx_row <= params.num_bins; idx_row++) {
              int idx_dv = idx_row * params.num_bins + idx_col;
              sn_dv[idx_dv] +=
                S_ik_j_ki_col_ + factor_mirror * std::conj(S_ik_j_ki_col_);
            }
          }
        }
 
        if (params.shape == "triu") {
          for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
            std::complex<double> S_ik_j_ki_col_ = coupling * (
              stats_sn.pk[idx_col] - stats_sn.sn[idx_col]
            );
            for (int idx_row = 0; idx_row <= idx_col; idx_row++) {
              int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2
                + (idx_col - idx_row);
              sn_dv[idx_dv] +=
                S_ik_j_ki_col_ + factor_mirror * std::conj(S_ik_j_ki_col_);
            }
          }
        }
      }
 
      for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
        double k_a = k1eff_dv[idx_dv];
        double k_b = k2eff_dv[idx_dv];
 
        std::complex<double> S_ij_k = coupling
          * stats_sn.compute_uncoupled_shotnoise_for_bispec_per_bin(
            dn_L0_for_sn, N_00, ylm_r_a, ylm_r_b, sj_a, sj_b,
            Sbar_L0, k_a, k_b
          );  // S|{i = j ≠ k}
        update_progbar();
 
        sn_dv[idx_dv] += factor_phase * (
          S_ij_k + factor_mirror_w3j * std::conj(S_ij_k)
        );
      }
 
      sph_orders_computed.push_back(sph_order_current);
      count_terms++;
      if (trvs::currTask == 0) {
        trvs::logger.stat(
          "Bispectrum term computed at orders (m₁, m₂, M) = ±(%d, %d, 0).",
          m1_, m2_
        );
      }
    }
  }
 
  // ---------------------------------------------------------------------
  // Results
  // ---------------------------------------------------------------------
 
  trv::BispecMeasurements bispec_out;
  for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
    bispec_out.k1_bin.push_back(k1bin_dv[idx_dv]);
    bispec_out.k1_eff.push_back(k1eff_dv[idx_dv]);
    bispec_out.nmodes_1.push_back(nmodes1_dv[idx_dv]);
    bispec_out.k2_bin.push_back(k2bin_dv[idx_dv]);
    bispec_out.k2_eff.push_back(k2eff_dv[idx_dv]);
    bispec_out.nmodes_2.push_back(nmodes2_dv[idx_dv]);
    bispec_out.bk_raw.push_back(norm_factor * bk_dv[idx_dv]);
    bispec_out.bk_shot.push_back(norm_factor * sn_dv[idx_dv]);
  }
  bispec_out.dim = dv_dim;
 
  delete[] nmodes1_dv; delete[] nmodes2_dv;
  delete[] k1bin_dv; delete[] k2bin_dv;
  delete[] k1eff_dv; delete[] k2eff_dv;
  delete[] bk_dv; delete[] sn_dv;
 
  trvs::count_cgrid -= 4;
  trvs::count_grid -= 4;
  trvs::gbytesMem -=
    trvs::size_in_gb< std::complex<double> >(4*params.nmesh);
 
  if (trvs::currTask == 0) {
    trvs::logger.stat(
      "... computed bispectrum from a periodic-box simulation-type catalogue "
      "in the global plane-parallel approximation."
    );
  }
 
  return bispec_out;
}
 
trv::ThreePCFMeasurements compute_3pcf_in_gpp_box(
  ParticleCatalogue& catalogue_data,
  trv::ParameterSet& params, trv::Binning& rbinning,
  double norm_factor
) {
  trvs::logger.reset_level(params.verbose);
 
  if (trvs::currTask == 0) {
    trvs::logger.stat(
      "Computing three-point correlation function "
      "from a periodic-box simulation-type catalogue "
      "in the global plane-parallel approximation..."
    );
  }
 
  // ---------------------------------------------------------------------
  // Set-up
  // ---------------------------------------------------------------------
 
  // Set up/check input.
  validate_multipole_coupling(params);
 
  std::complex<double> factor_phase =
    std::pow(trvm::M_I, params.ell1 + params.ell2);
  double factor_parity = std::pow(-1., params.ell1 + params.ell2);
 
  // Set up output.
  int dv_dim = 0;  // data vector dimension
  if (params.shape == "diag" || params.shape == "row" ) {
    dv_dim = rbinning.num_bins;
  } else
  if (params.shape == "off-diag") {
    dv_dim = rbinning.num_bins - std::abs(params.idx_bin);
  } else
  if (params.shape == "full") {
    dv_dim = rbinning.num_bins * rbinning.num_bins;
  } else
  if (params.shape == "triu") {
    dv_dim = rbinning.num_bins * (rbinning.num_bins + 1) / 2;
  } else {
    if (trvs::currTask == 0) {
      trvs::logger.error(
        "Three-point statistic form is not recognised: `form` = '%s'.",
        params.form.c_str()
      );
    }
    throw trvs::InvalidParameterError(
      "Three-point statistic form is not recognised: `form` = '%s'.",
      params.form.c_str()
    );
  }
 
  int* npairs1_dv = new int[dv_dim]();
  int* npairs2_dv = new int[dv_dim]();
  double* r1bin_dv = new double[dv_dim]();
  double* r2bin_dv = new double[dv_dim]();
  double* r1eff_dv = new double[dv_dim]();
  double* r2eff_dv = new double[dv_dim]();
  std::complex<double>* zeta_dv = new std::complex<double>[dv_dim];
  std::complex<double>* sn_dv = new std::complex<double>[dv_dim];
 
  // ---------------------------------------------------------------------
  // Measurement
  // ---------------------------------------------------------------------
 
#if defined(TRV_USE_OMP) && defined(TRV_USE_FFTWOMP)
  if (!trvs::is_gpu_enabled()) {
    fftw_init_threads();
  }
#endif  // TRV_USE_OMP && TRV_USE_FFTWOMP
 
  // Compute common field quantities.
  MeshField dn_00(params, true, "`dn_00`");  // δn_00(k)
  dn_00.compute_unweighted_field_fluctuations_insitu(catalogue_data);
  dn_00.fourier_transform();
 
  MeshField& dn_L0_for_sn = dn_00;  // δn_L0(k)
 
  double vol_cell = dn_00.vol_cell;
 
  MeshField N_00(params, true, "`N_00`");  // N_00(k)
  N_00.compute_unweighted_field(catalogue_data);
  N_00.fourier_transform();
 
  trvm::SphericalBesselCalculator sj_a(params.ell1);  // j_l_a
  trvm::SphericalBesselCalculator sj_b(params.ell2);  // j_l_b
 
  FieldStats stats_sn(params);
 
  // Initialise/reset spherical harmonic mesh grids.
  std::vector< std::complex<double> > ylm_r_a(params.nmesh);
  std::vector< std::complex<double> > ylm_r_b(params.nmesh);
  std::vector< std::complex<double> > ylm_k_a(params.nmesh);
  std::vector< std::complex<double> > ylm_k_b(params.nmesh);
  trvs::count_cgrid += 4;
  trvs::count_grid += 4;
  trvs::update_maxcntgrid();
  trvs::gbytesMem +=
    trvs::size_in_gb< std::complex<double> >(4*params.nmesh);
  trvs::update_maxmem();
 
  // Compute 3PCF terms including shot noise.
  std::vector<trv::SphericalOrderTriplet> sph_orders_computed;
  int count_terms = 0;
  for (int m1_ = - params.ell1; m1_ <= params.ell1; m1_++) {
    for (int m2_ = - params.ell2; m2_ <= params.ell2; m2_++) {
      // Under the global plane-parallel approximation, δᴰ_{M0} enforces
      // M = 0 for any spherical-harmonic-weighted field fluctuations.
      int M_ = 0;
 
      // Check if the current spherical order triplet is redundant.
      SphericalOrderTriplet sph_order_current = {m1_, m2_, M_};
 
      std::string flag_redundant = "false";
      for (SphericalOrderTriplet sph_order : sph_orders_computed) {
        if (sph_order_current.is_inverse(sph_order)) {
          flag_redundant = "true";
        }
      }
      if (flag_redundant == "true") {continue;}
 
      // Calculate the coupling coefficient.
      double coupling = trv::calc_coupling_coeff_3pt(
        params.ell1, params.ell2, params.ELL, m1_, m2_, M_
      );  // Wigner 3-j's
      if (std::fabs(coupling) < trvm::eps_coupling) {continue;}
 
      // The factor should be
      // ```
      // std::pow(-1., params.ell1 + params.ell2 + params.ELL) *
      // std::pow(-1., m1_ + m2_ + M_) *
      // {std::pow(-1., params.ell1 + params.ell2), 1};
      // ```
      // which, by parity symmetry and Wigner 3-j properties,
      // simplifies to:
      double factor_mirror = sph_order_current.is_zeros() ?
        0. : std::pow(-1., params.ELL);
      double factor_mirror_w3j = sph_order_current.is_zeros() ?
        0. : 1.;  // std::pow(-1., params.ell1 + params.ell2 + params.ELL);
 
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_config_space(
          params.ell1, m1_, params.boxsize, params.ngrid, ylm_r_a
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_config_space(
          params.ell2, m2_, params.boxsize, params.ngrid, ylm_r_b
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_fourier_space(
          params.ell1, m1_, params.boxsize, params.ngrid, ylm_k_a
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_fourier_space(
          params.ell2, m2_, params.boxsize, params.ngrid, ylm_k_b
        );
 
      // Set up FFT counter and progress bar.
      int progbar_task_count = 3 + 2 * dv_dim;
      std::stringstream progbar_name_ss;
      progbar_name_ss
        << "component orders (" << m1_ << ", " << m2_ << ", " << M_ << ")";
      trvs::ProgressBar progbar(progbar_task_count, progbar_name_ss.str());
 
      std::vector<float> progbar_nodes;
      if (params.progbar != "false" && params.progbar != "true") {
        progbar_nodes = trvs::set_nodes_by_str(params.progbar);
        if (progbar_nodes.size() > 0) {
          progbar.set_nodes(progbar_nodes);
        }
      }
 
      int count_fft_init = trvs::count_fft + trvs::count_ifft;
      auto update_progbar = [&]() {
        if (params.progbar != "false") {
          if (trvs::currTask == 0) {
            progbar.update(
              trvs::count_fft + trvs::count_ifft - count_fft_init
            );
          }
        }
      };
 
      // ·································································
      // Shot noise
      // ·································································
 
      // Under the global plane-parallel approximation, y_{LM} = δᴰ_{M0}
      // (L-invariant) for the line-of-sight spherical harmonic.
      // Also note the field is unweighted from simulation sources.
      std::complex<double> Sbar_L0 =
        double(catalogue_data.ntotal);  // \bar{S}_L0
 
      stats_sn.compute_uncoupled_shotnoise_for_3pcf(
        dn_L0_for_sn, N_00, ylm_r_a, ylm_r_b, Sbar_L0, rbinning
      );  // S|{i = j ≠ k}
      update_progbar();
 
      // Enforce the Kronecker delta in eq. (51) in the Paper.
      if (params.shape == "diag") {
        for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
          int ibin = idx_dv;
          sn_dv[idx_dv] += coupling * factor_parity * (
            stats_sn.xi[ibin]
            + factor_mirror_w3j * std::conj(stats_sn.xi[ibin])
          );
        }
      }
 
      if (params.shape == "off-diag" && params.idx_bin == 0) {
        // Note that ``idx_col == idx_row``.
        for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
          int ibin = idx_dv;
          sn_dv[idx_dv] += coupling * factor_parity * (
            stats_sn.xi[ibin]
            + factor_mirror_w3j * std::conj(stats_sn.xi[ibin])
          );
        }
      }
 
      if (params.shape == "row") {
        sn_dv[params.idx_bin] += coupling * factor_parity * (
          stats_sn.xi[params.idx_bin]
          + factor_mirror_w3j * std::conj(stats_sn.xi[params.idx_bin])
        );
      }
 
      if (params.shape == "full") {
        for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
          // Note that ``idx_col == idx_row``.
          int idx_dv = idx_row * params.num_bins + idx_row;
          sn_dv[idx_dv] += coupling * factor_parity * (
            stats_sn.xi[idx_row]
            + factor_mirror_w3j * std::conj(stats_sn.xi[idx_row])
          );
        }
      }
 
      if (params.shape == "triu") {
        for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
          // Note that ``idx_col == idx_row``.
          int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2;
          sn_dv[idx_dv] += coupling * factor_parity * (
            stats_sn.xi[idx_row]
            + factor_mirror_w3j * std::conj(stats_sn.xi[idx_row])
          );
        }
      }
 
      // Only record the binned coordinates and counts once.
      if (count_terms == 0) {
        if (params.shape == "diag") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin = idx_dv;
 
            r1bin_dv[idx_dv] = rbinning.bin_centres[ibin];
            r2bin_dv[idx_dv] = rbinning.bin_centres[ibin];
            r1eff_dv[idx_dv] = stats_sn.r[ibin];
            r2eff_dv[idx_dv] = stats_sn.r[ibin];
            npairs1_dv[idx_dv] = stats_sn.npairs[ibin];
            npairs2_dv[idx_dv] = stats_sn.npairs[ibin];
          }
        }
 
        if (params.shape == "off-diag") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin_row, ibin_col;
            if (params.idx_bin >= 0) {
              ibin_row = idx_dv;
              ibin_col = idx_dv + std::abs(params.idx_bin);
            } else {
              ibin_row = idx_dv + std::abs(params.idx_bin);
              ibin_col = idx_dv;
            }
 
            r1bin_dv[idx_dv] = rbinning.bin_centres[ibin_row];
            r2bin_dv[idx_dv] = rbinning.bin_centres[ibin_col];
            r1eff_dv[idx_dv] = stats_sn.r[ibin_row];
            r2eff_dv[idx_dv] = stats_sn.r[ibin_col];
            npairs1_dv[idx_dv] = stats_sn.npairs[ibin_row];
            npairs2_dv[idx_dv] = stats_sn.npairs[ibin_col];
          }
        }
 
        if (params.shape == "row") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin_row = params.idx_bin;
            int ibin_col = idx_dv;
 
            r1bin_dv[idx_dv] = rbinning.bin_centres[ibin_row];
            r2bin_dv[idx_dv] = rbinning.bin_centres[ibin_col];
            r1eff_dv[idx_dv] = stats_sn.r[ibin_row];
            r2eff_dv[idx_dv] = stats_sn.r[ibin_col];
            npairs1_dv[idx_dv] = stats_sn.npairs[ibin_row];
            npairs2_dv[idx_dv] = stats_sn.npairs[ibin_col];
          }
        }
 
        if (params.shape == "full") {
          for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
            for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
              int idx_dv = idx_row * params.num_bins + idx_col;
 
              r1bin_dv[idx_dv] = rbinning.bin_centres[idx_row];
              r2bin_dv[idx_dv] = rbinning.bin_centres[idx_col];
              r1eff_dv[idx_dv] = stats_sn.r[idx_row];
              r2eff_dv[idx_dv] = stats_sn.r[idx_col];
              npairs1_dv[idx_dv] = stats_sn.npairs[idx_row];
              npairs2_dv[idx_dv] = stats_sn.npairs[idx_col];
            }
          }
        }
 
        if (params.shape == "triu") {
          for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
            for (int idx_col = idx_row; idx_col < params.num_bins; idx_col++) {
              int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2
                + (idx_col - idx_row);
 
              r1bin_dv[idx_dv] = rbinning.bin_centres[idx_row];
              r2bin_dv[idx_dv] = rbinning.bin_centres[idx_col];
              r1eff_dv[idx_dv] = stats_sn.r[idx_row];
              r2eff_dv[idx_dv] = stats_sn.r[idx_col];
              npairs1_dv[idx_dv] = stats_sn.npairs[idx_row];
              npairs2_dv[idx_dv] = stats_sn.npairs[idx_col];
            }
          }
        }
      }
 
      // ·································································
      // Raw 3PCF
      // ·································································
 
      // Compute 3PCF components in eqs. (42), (48) & (49) in the Paper.
      MeshField G_00(params, true, "`G_00`");  // G_00
      G_00.compute_unweighted_field_fluctuations_insitu(catalogue_data);
      G_00.fourier_transform();
      G_00.apply_assignment_compensation();
      G_00.inv_fourier_transform();
      update_progbar();
 
      MeshField F_lm_a(params, true, "`F_lm_a`");  // F_lm_a
      MeshField F_lm_b(params, true, "`F_lm_b`");  // F_lm_b
 
      for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
        double r_a = r1eff_dv[idx_dv];
        F_lm_a.inv_fourier_transform_sjl_ylm_wgtd_field(
          dn_00, ylm_k_a, sj_a, r_a
        );
 
        double r_b = r2eff_dv[idx_dv];
        F_lm_b.inv_fourier_transform_sjl_ylm_wgtd_field(
          dn_00, ylm_k_b, sj_b, r_b
        );
        update_progbar();
 
        // ζ_{l₁ l₂ L}^{m₁ m₂ M}
        double zeta_comp_real = 0., zeta_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:zeta_comp_real, zeta_comp_imag)
#endif  // TRV_USE_OMP
        for (long long gid = 0; gid < params.nmesh; gid++) {
          std::complex<double> F_lm_a_gridpt(F_lm_a[gid][0], F_lm_a[gid][1]);
          std::complex<double> F_lm_b_gridpt(F_lm_b[gid][0], F_lm_b[gid][1]);
          std::complex<double> G_00_gridpt(G_00[gid][0], G_00[gid][1]);
          std::complex<double> zeta_gridpt =
            F_lm_a_gridpt * F_lm_b_gridpt * G_00_gridpt;
 
          zeta_comp_real += zeta_gridpt.real();
          zeta_comp_imag += zeta_gridpt.imag();
        }
 
        std::complex<double> zeta_component(zeta_comp_real, zeta_comp_imag);
 
        zeta_dv[idx_dv] += coupling * vol_cell * factor_phase * (
          zeta_component + factor_mirror * std::conj(zeta_component)
        );
      }
 
      sph_orders_computed.push_back(sph_order_current);
      count_terms++;
      if (trvs::currTask == 0) {
        trvs::logger.stat(
          "Three-point correlation function term computed at orders "
          "(m₁, m₂, M) = ±(%d, %d, 0).",
          m1_, m2_
        );
      }
    }
  }
 
  // ---------------------------------------------------------------------
  // Results
  // ---------------------------------------------------------------------
 
  trv::ThreePCFMeasurements threepcf_out;
  for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
    threepcf_out.r1_bin.push_back(r1bin_dv[idx_dv]);
    threepcf_out.r1_eff.push_back(r1eff_dv[idx_dv]);
    threepcf_out.npairs_1.push_back(npairs1_dv[idx_dv]);
    threepcf_out.r2_bin.push_back(r2bin_dv[idx_dv]);
    threepcf_out.r2_eff.push_back(r2eff_dv[idx_dv]);
    threepcf_out.npairs_2.push_back(npairs2_dv[idx_dv]);
    threepcf_out.zeta_raw.push_back(norm_factor * zeta_dv[idx_dv]);
    threepcf_out.zeta_shot.push_back(norm_factor * sn_dv[idx_dv]);
  }
  threepcf_out.dim = dv_dim;
 
  delete[] npairs1_dv; delete[] npairs2_dv;
  delete[] r1bin_dv; delete[] r2bin_dv;
  delete[] r1eff_dv; delete[] r2eff_dv;
  delete[] zeta_dv; delete[] sn_dv;
 
  trvs::count_cgrid -= 4;
  trvs::count_grid -= 4;
  trvs::gbytesMem -=
    trvs::size_in_gb< std::complex<double> >(4*params.nmesh);
 
  if (trvs::currTask == 0) {
    trvs::logger.stat(
      "... computed three-point correlation function "
      "from a periodic-box simulation-type catalogue "
      "in the global plane-parallel approximation."
    );
  }
 
  return threepcf_out;
}
 
trv::ThreePCFWindowMeasurements compute_3pcf_window(
  ParticleCatalogue& catalogue_rand, LineOfSight* los_rand,
  trv::ParameterSet& params, trv::Binning& rbinning,
  double alpha, double norm_factor, bool wide_angle
) {
  std::string msg_tag = wide_angle ? "wide-angle corrections " : "";
 
  trvs::logger.reset_level(params.verbose);
 
  if (trvs::currTask == 0) {
    trvs::logger.stat(
      "Computing three-point correlation function window %s"
      "from random catalogue...",
      msg_tag.c_str()
    );
  }
 
  // ---------------------------------------------------------------------
  // Set-up
  // ---------------------------------------------------------------------
 
  // Set up/check input.
  validate_multipole_coupling(params);
 
  std::complex<double> factor_phase =
    std::pow(trvm::M_I, params.ell1 + params.ell2);
  double factor_parity = std::pow(-1., params.ell1 + params.ell2);
 
  // Set up output.
  int dv_dim = 0;  // data vector dimension
  if (params.shape == "diag" || params.shape == "row" ) {
    dv_dim = rbinning.num_bins;
  } else
  if (params.shape == "off-diag") {
    dv_dim = rbinning.num_bins - std::abs(params.idx_bin);
  } else
  if (params.shape == "full") {
    dv_dim = rbinning.num_bins * rbinning.num_bins;
  } else
  if (params.shape == "triu") {
    dv_dim = rbinning.num_bins * (rbinning.num_bins + 1) / 2;
  } else {
    if (trvs::currTask == 0) {
      trvs::logger.error(
        "Three-point statistic form is not recognised: `form` = '%s'.",
        params.form.c_str()
      );
    }
    throw trvs::InvalidParameterError(
      "Three-point statistic form is not recognised: `form` = '%s'.",
      params.form.c_str()
    );
  }
 
  int* npairs1_dv = new int[dv_dim]();
  int* npairs2_dv = new int[dv_dim]();
  double* r1bin_dv = new double[dv_dim]();
  double* r2bin_dv = new double[dv_dim]();
  double* r1eff_dv = new double[dv_dim]();
  double* r2eff_dv = new double[dv_dim]();
  std::complex<double>* zeta_dv = new std::complex<double>[dv_dim];
  std::complex<double>* sn_dv = new std::complex<double>[dv_dim];
 
  // ---------------------------------------------------------------------
  // Measurement
  // ---------------------------------------------------------------------
 
#if defined(TRV_USE_OMP) && defined(TRV_USE_FFTWOMP)
  if (!trvs::is_gpu_enabled()) {
    fftw_init_threads();
  }
#endif  // TRV_USE_OMP && TRV_USE_FFTWOMP
 
  // Compute common field quantities.
  MeshField n_00(params, true, "`n_00`");  // n_00(k)
  n_00.compute_ylm_wgtd_field(catalogue_rand, los_rand, alpha, 0, 0);
  n_00.fourier_transform();
 
  double vol_cell = n_00.vol_cell;
 
  MeshField N_00(params, true, "`N_00`");  // N_00(k)
  N_00.compute_ylm_wgtd_quad_field(catalogue_rand, los_rand, alpha, 0, 0);
  N_00.fourier_transform();
 
  trvm::SphericalBesselCalculator sj_a(params.ell1);  // j_l_a
  trvm::SphericalBesselCalculator sj_b(params.ell2);  // j_l_b
 
  FieldStats stats_sn(params);
 
  // Initialise reduced-spherical-harmonic weights on mesh grids.
  std::vector< std::complex<double> > ylm_r_a(params.nmesh);
  std::vector< std::complex<double> > ylm_r_b(params.nmesh);
  std::vector< std::complex<double> > ylm_k_a(params.nmesh);
  std::vector< std::complex<double> > ylm_k_b(params.nmesh);
  trvs::count_cgrid += 4;
  trvs::count_grid += 4;
  trvs::update_maxcntgrid();
  trvs::gbytesMem +=
    trvs::size_in_gb< std::complex<double> >(4*params.nmesh);
  trvs::update_maxmem();
 
  // Compute 3PCF window terms including shot noise.
  std::vector<trv::SphericalOrderTriplet> sph_orders_computed;
  int count_terms = 0;
  for (int m1_ = - params.ell1; m1_ <= params.ell1; m1_++) {
    for (int m2_ = - params.ell2; m2_ <= params.ell2; m2_++) {
      // Check for if all Wigner-3j symbols are zero.
      std::string flag_vanishing = "true";
      for (int M_ = - params.ELL; M_ <= params.ELL; M_++) {
        double coupling = trv::calc_coupling_coeff_3pt(
          params.ell1, params.ell2, params.ELL, m1_, m2_, M_
        );
        if (std::fabs(coupling) > trvm::eps_coupling) {
          flag_vanishing = "false";
          break;
        }
      }
      if (flag_vanishing == "true") {continue;}
 
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_config_space(
          params.ell1, m1_, params.boxsize, params.ngrid, ylm_r_a
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_config_space(
          params.ell2, m2_, params.boxsize, params.ngrid, ylm_r_b
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_fourier_space(
          params.ell1, m1_, params.boxsize, params.ngrid, ylm_k_a
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_fourier_space(
          params.ell2, m2_, params.boxsize, params.ngrid, ylm_k_b
        );
 
      for (int M_ = - params.ELL; M_ <= params.ELL; M_++) {
        // Check if the current spherical order triplet is redundant.
        SphericalOrderTriplet sph_order_current = {m1_, m2_, M_};
 
        std::string flag_redundant = "false";
        for (SphericalOrderTriplet sph_order : sph_orders_computed) {
          if (sph_order_current.is_inverse(sph_order)) {
            flag_redundant = "true";
          }
        }
        if (flag_redundant == "true") {continue;}
 
        // Calculate the coupling coefficient.
        double coupling = trv::calc_coupling_coeff_3pt(
          params.ell1, params.ell2, params.ELL, m1_, m2_, M_
        );  // Wigner 3-j's
        if (std::fabs(coupling) < trvm::eps_coupling) {continue;}
 
        // The factor should be
        // ```
        // std::pow(-1., params.ell1 + params.ell2 + params.ELL) *
        // std::pow(-1., m1_ + m2_ + M_) *
        // {std::pow(-1., params.ell1 + params.ell2), 1};
        // ```
        // which, by parity symmetry and Wigner 3-j properties,
        // simplifies to:
        double factor_mirror = sph_order_current.is_zeros() ?
          0. : std::pow(-1., params.ELL);
        double factor_mirror_w3j = sph_order_current.is_zeros() ?
          0. : 1.;  // std::pow(-1., params.ell1 + params.ell2 + params.ELL);
 
        // Set up FFT counter and progress bar.
        int progbar_task_count = 4 + 2 * dv_dim;
        std::stringstream progbar_name_ss;
        progbar_name_ss
          << "component orders (" << m1_ << ", " << m2_ << ", " << M_ << ")";
        trvs::ProgressBar progbar(progbar_task_count, progbar_name_ss.str());
 
        std::vector<float> progbar_nodes;
        if (params.progbar != "false" && params.progbar != "true") {
          progbar_nodes = trvs::set_nodes_by_str(params.progbar);
          if (progbar_nodes.size() > 0) {
            progbar.set_nodes(progbar_nodes);
          }
        }
 
        int count_fft_init = trvs::count_fft + trvs::count_ifft;
        auto update_progbar = [&]() {
          if (params.progbar != "false") {
            if (trvs::currTask == 0) {
              progbar.update(
                trvs::count_fft + trvs::count_ifft - count_fft_init
              );
            }
          }
        };
 
        // ·······························································
        // Shot noise
        // ·······························································
 
        // Compute shot noise components in eq. (51) in the Paper.
        MeshField n_LM_for_sn(params, true, "`n_LM_for_sn`");  // δn_LM(k)
                                                               // (for
                                                               // shot noise)
        n_LM_for_sn.compute_ylm_wgtd_field(
          catalogue_rand, los_rand, alpha, params.ELL, M_
        );
        n_LM_for_sn.fourier_transform();
        update_progbar();
 
        std::complex<double> Sbar_LM = calc_ylm_wgtd_shotnoise_amp_for_bispec(
          catalogue_rand, los_rand, alpha, params.ELL, M_
        );  // \bar{S}_LM
 
        stats_sn.compute_uncoupled_shotnoise_for_3pcf(
          n_LM_for_sn, N_00, ylm_r_a, ylm_r_b, Sbar_LM, rbinning
        );  // S|{i = j ≠ k}
        update_progbar();
 
        // Enforce the Kronecker delta in eq. (51) in the Paper.
        if (params.shape == "diag") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin = idx_dv;
            sn_dv[idx_dv] += coupling * factor_parity * (
              stats_sn.xi[ibin]
              + factor_mirror_w3j * std::conj(stats_sn.xi[ibin])
            );
          }
        }
 
        if (params.shape == "off-diag" && params.idx_bin == 0) {
          // Note that ``idx_col == idx_row``.
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin = idx_dv;
            sn_dv[idx_dv] += coupling * factor_parity * (
              stats_sn.xi[ibin]
              + factor_mirror_w3j * std::conj(stats_sn.xi[ibin])
            );
          }
        }
 
        if (params.shape == "row") {
          sn_dv[params.idx_bin] += coupling * factor_parity * (
            stats_sn.xi[params.idx_bin]
            + factor_mirror_w3j * std::conj(stats_sn.xi[params.idx_bin])
          );
        }
 
        if (params.shape == "full") {
          for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
            // Note that ``idx_col == idx_row``.
            int idx_dv = idx_row * params.num_bins + idx_row;
            sn_dv[idx_dv] += coupling * factor_parity * (
              stats_sn.xi[idx_row]
              + factor_mirror_w3j * std::conj(stats_sn.xi[idx_row])
            );
          }
        }
 
        if (params.shape == "triu") {
          for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
            // Note that ``idx_col == idx_row``.
            int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2;
            sn_dv[idx_dv] += coupling * factor_parity * (
              stats_sn.xi[idx_row]
              + factor_mirror_w3j * std::conj(stats_sn.xi[idx_row])
            );
          }
        }
 
        if (count_terms == 0) {
          if (params.shape == "diag") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin = idx_dv;
 
              r1bin_dv[idx_dv] = rbinning.bin_centres[ibin];
              r2bin_dv[idx_dv] = rbinning.bin_centres[ibin];
              r1eff_dv[idx_dv] = stats_sn.r[ibin];
              r2eff_dv[idx_dv] = stats_sn.r[ibin];
              npairs1_dv[idx_dv] = stats_sn.npairs[ibin];
              npairs2_dv[idx_dv] = stats_sn.npairs[ibin];
            }
          }
 
          if (params.shape == "off-diag") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin_row, ibin_col;
              if (params.idx_bin >= 0) {
                ibin_row = idx_dv;
                ibin_col = idx_dv + std::abs(params.idx_bin);
              } else {
                ibin_row = idx_dv + std::abs(params.idx_bin);
                ibin_col = idx_dv;
              }
 
              r1bin_dv[idx_dv] = rbinning.bin_centres[ibin_row];
              r2bin_dv[idx_dv] = rbinning.bin_centres[ibin_col];
              r1eff_dv[idx_dv] = stats_sn.r[ibin_row];
              r2eff_dv[idx_dv] = stats_sn.r[ibin_col];
              npairs1_dv[idx_dv] = stats_sn.npairs[ibin_row];
              npairs2_dv[idx_dv] = stats_sn.npairs[ibin_col];
            }
          }
 
          if (params.shape == "row") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin_row = params.idx_bin;
              int ibin_col = idx_dv;
 
              r1bin_dv[idx_dv] = rbinning.bin_centres[ibin_row];
              r2bin_dv[idx_dv] = rbinning.bin_centres[ibin_col];
              r1eff_dv[idx_dv] = stats_sn.r[ibin_row];
              r2eff_dv[idx_dv] = stats_sn.r[ibin_col];
              npairs1_dv[idx_dv] = stats_sn.npairs[ibin_row];
              npairs2_dv[idx_dv] = stats_sn.npairs[ibin_col];
            }
          }
 
          if (params.shape == "full") {
            for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
              for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
                int idx_dv = idx_row * params.num_bins + idx_col;
 
                r1bin_dv[idx_dv] = rbinning.bin_centres[idx_row];
                r2bin_dv[idx_dv] = rbinning.bin_centres[idx_col];
                r1eff_dv[idx_dv] = stats_sn.r[idx_row];
                r2eff_dv[idx_dv] = stats_sn.r[idx_col];
                npairs1_dv[idx_dv] = stats_sn.npairs[idx_row];
                npairs2_dv[idx_dv] = stats_sn.npairs[idx_col];
              }
            }
          }
 
          if (params.shape == "triu") {
            for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
              for (int idx_col = idx_row; idx_col < params.num_bins; idx_col++)
              {
                int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2
                  + (idx_col - idx_row);
 
                r1bin_dv[idx_dv] = rbinning.bin_centres[idx_row];
                r2bin_dv[idx_dv] = rbinning.bin_centres[idx_col];
                r1eff_dv[idx_dv] = stats_sn.r[idx_row];
                r2eff_dv[idx_dv] = stats_sn.r[idx_col];
                npairs1_dv[idx_dv] = stats_sn.npairs[idx_row];
                npairs2_dv[idx_dv] = stats_sn.npairs[idx_col];
              }
            }
          }
        }
 
        // ·······························································
        // Raw 3PCF
        // ·······························································
 
        // Compute 3PCF components in eqs. (42), (48) & (49) in the Paper.
        MeshField G_LM(params, true, "`G_LM`");  // G_LM
        G_LM.compute_ylm_wgtd_field(
          catalogue_rand, los_rand, alpha, params.ELL, M_
        );
        G_LM.fourier_transform();
        G_LM.apply_assignment_compensation();
        G_LM.inv_fourier_transform();
        update_progbar();
 
        // Perform wide-angle corrections if required.
        if (wide_angle) {
          G_LM.apply_wide_angle_pow_law_kernel();
        }
 
        MeshField F_lm_a(params, true, "`F_lm_a`");  // F_lm_a
        MeshField F_lm_b(params, true, "`F_lm_b`");  // F_lm_b
 
        for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
          double r_a = r1eff_dv[idx_dv];
          F_lm_a.inv_fourier_transform_sjl_ylm_wgtd_field(
            n_00, ylm_k_a, sj_a, r_a
          );
 
          double r_b = r2eff_dv[idx_dv];
          F_lm_b.inv_fourier_transform_sjl_ylm_wgtd_field(
            n_00, ylm_k_b, sj_b, r_b
          );
 
          update_progbar();
 
          // ζ_{l₁ l₂ L}^{m₁ m₂ M}
          double zeta_comp_real = 0., zeta_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:zeta_comp_real, zeta_comp_imag)
#endif  // TRV_USE_OMP
          for (long long gid = 0; gid < params.nmesh; gid++) {
            std::complex<double> F_lm_a_gridpt(F_lm_a[gid][0], F_lm_a[gid][1]);
            std::complex<double> F_lm_b_gridpt(F_lm_b[gid][0], F_lm_b[gid][1]);
            std::complex<double> G_LM_gridpt(G_LM[gid][0], G_LM[gid][1]);
            std::complex<double> zeta_gridpt =
              F_lm_a_gridpt * F_lm_b_gridpt * G_LM_gridpt;
 
            zeta_comp_real += zeta_gridpt.real();
            zeta_comp_imag += zeta_gridpt.imag();
          }
 
          std::complex<double> zeta_component(zeta_comp_real, zeta_comp_imag);
 
          zeta_dv[idx_dv] += coupling * vol_cell * factor_phase * (
            zeta_component + factor_mirror * std::conj(zeta_component)
          );
        }
 
        sph_orders_computed.push_back(sph_order_current);
        count_terms++;
        if (trvs::currTask == 0) {
          trvs::logger.stat(
            "Three-point correlation function window term computed at orders "
            "(m₁, m₂, M) = ±(%d, %d, %d).",
            m1_, m2_, M_
          );
        }
      }
    }
  }
 
  // ---------------------------------------------------------------------
  // Results
  // ---------------------------------------------------------------------
 
  trv::ThreePCFWindowMeasurements threepcfwin_out;
  for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
    threepcfwin_out.r1_bin.push_back(r1bin_dv[idx_dv]);
    threepcfwin_out.r1_eff.push_back(r1eff_dv[idx_dv]);
    threepcfwin_out.npairs_1.push_back(npairs1_dv[idx_dv]);
    threepcfwin_out.r2_bin.push_back(r2bin_dv[idx_dv]);
    threepcfwin_out.r2_eff.push_back(r2eff_dv[idx_dv]);
    threepcfwin_out.npairs_2.push_back(npairs2_dv[idx_dv]);
    threepcfwin_out.zeta_raw.push_back(norm_factor * zeta_dv[idx_dv]);
    threepcfwin_out.zeta_shot.push_back(norm_factor * sn_dv[idx_dv]);
  }
  threepcfwin_out.dim = dv_dim;
 
  delete[] npairs1_dv; delete[] npairs2_dv;
  delete[] r1bin_dv; delete[] r2bin_dv;
  delete[] r1eff_dv; delete[] r2eff_dv;
  delete[] zeta_dv; delete[] sn_dv;
 
  trvs::count_cgrid -= 4;
  trvs::count_grid -= 4;
  trvs::gbytesMem -=
    trvs::size_in_gb< std::complex<double> >(4*params.nmesh);
 
  if (trvs::currTask == 0) {
    trvs::logger.stat(
      "... computed three-point correlation function window %s"
      "from random catalogue.",
      msg_tag.c_str()
    );
  }
 
  return threepcfwin_out;
}
 
#ifdef TRV_USE_LEGACY_CODE
trv::BispecMeasurements compute_bispec_for_los_choice(
  ParticleCatalogue& catalogue_data, ParticleCatalogue& catalogue_rand,
  LineOfSight* los_data, LineOfSight* los_rand,
  int los_choice,
  trv::ParameterSet& params, trv::Binning& kbinning,
  double norm_factor
) {
  trvs::logger.reset_level(params.verbose);
 
  if (trvs::currTask == 0) {
    trvs::logger.stat(
      "Computing bispectrum from paired survey-type catalogues "
      "for line-of-sight choice %d...",
      los_choice
    );
  }
 
  // ---------------------------------------------------------------------
  // Set-up
  // ---------------------------------------------------------------------
 
  // Set up/check input.
  validate_multipole_coupling(params);
 
  double alpha = catalogue_data.wstotal / catalogue_rand.wstotal;
 
  std::complex<double> factor_phase =
    std::pow(trvm::M_I, params.ell1 + params.ell2);
 
  // Set up output.
  int dv_dim = 0;  // data vector dimension
  if (params.shape == "diag" || params.shape == "row" ) {
    dv_dim = kbinning.num_bins;
  } else
  if (params.shape == "off-diag") {
    dv_dim = kbinning.num_bins - std::abs(params.idx_bin);
  } else
  if (params.shape == "full") {
    dv_dim = kbinning.num_bins * kbinning.num_bins;
  } else
  if (params.shape == "triu") {
    dv_dim = kbinning.num_bins * (kbinning.num_bins + 1) / 2;
  } else {
    if (trvs::currTask == 0) {
      trvs::logger.error(
        "Three-point statistic form is not recognised: `form` = '%s'.",
        params.form.c_str()
      );
    }
    throw trvs::InvalidParameterError(
      "Three-point statistic form is not recognised: `form` = '%s'.",
      params.form.c_str()
    );
  }
 
  int* nmodes1_dv = new int[dv_dim]();
  int* nmodes2_dv = new int[dv_dim]();
  double* k1bin_dv = new double[dv_dim]();
  double* k2bin_dv = new double[dv_dim]();
  double* k1eff_dv = new double[dv_dim]();
  double* k2eff_dv = new double[dv_dim]();
  std::complex<double>* bk_dv = new std::complex<double>[dv_dim];
  std::complex<double>* sn_dv = new std::complex<double>[dv_dim];
 
  // ---------------------------------------------------------------------
  // Measurement
  // ---------------------------------------------------------------------
 
#if defined(TRV_USE_OMP) && defined(TRV_USE_FFTWOMP)
  if (!trvs::is_gpu_enabled()) {
    fftw_init_threads();
  }
#endif  // TRV_USE_OMP && TRV_USE_FFTWOMP
 
  // RFE: Not adopted until copy-assignment constructor is checked.
  // ---->
  // // Compute common field quantities.
  // MeshField dn_00(params, true, "`dn_00`");  // δn_00(r)
  // dn_00.compute_ylm_wgtd_field(
  //   catalogue_data, catalogue_rand, los_data, los_rand, alpha, 0, 0
  // );
 
  // double vol_cell = dn_00.vol_cell;
 
  // MeshField N_00(params, true, "`N_00`");  // N_00(r)
  // N_00.compute_ylm_wgtd_quad_field(
  //   catalogue_data, catalogue_rand, los_data, los_rand, alpha, 0, 0
  // );
  // ----<
 
  trvm::SphericalBesselCalculator sj_a(params.ell1);  // j_l_a
  trvm::SphericalBesselCalculator sj_b(params.ell2);  // j_l_b
 
  FieldStats stats_sn(params);
 
  // Initialise reduced-spherical-harmonic weights on mesh grids.
  std::vector< std::complex<double> > ylm_k_a(params.nmesh);
  std::vector< std::complex<double> > ylm_k_b(params.nmesh);
  std::vector< std::complex<double> > ylm_r_a(params.nmesh);
  std::vector< std::complex<double> > ylm_r_b(params.nmesh);
  trvs::count_cgrid += 4;
  trvs::count_grid += 4;
  trvs::update_maxcntgrid();
  trvs::gbytesMem +=
    trvs::size_in_gb< std::complex<double> >(4*params.nmesh);
  trvs::update_maxmem();
 
  // Compute bispectrum terms.
  std::vector<trv::SphericalOrderTriplet> sph_orders_computed;
  int count_terms = 0;
  for (int m1_ = - params.ell1; m1_ <= params.ell1; m1_++) {
    for (int m2_ = - params.ell2; m2_ <= params.ell2; m2_++) {
      // Check for if all Wigner-3j symbols are zero.
      std::string flag_vanishing = "true";
      for (int M_ = - params.ELL; M_ <= params.ELL; M_++) {
        double coupling = trv::calc_coupling_coeff_3pt(
          params.ell1, params.ell2, params.ELL, m1_, m2_, M_
        );
        if (std::fabs(coupling) > trvm::eps_coupling) {
          flag_vanishing = "false";
          break;
        }
      }
      if (flag_vanishing == "true") {continue;}
 
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_fourier_space(
          params.ell1, m1_, params.boxsize, params.ngrid, ylm_k_a
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_fourier_space(
          params.ell2, m2_, params.boxsize, params.ngrid, ylm_k_b
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_config_space(
          params.ell1, m1_, params.boxsize, params.ngrid, ylm_r_a
        );
      trvm::SphericalHarmonicCalculator::
        store_reduced_spherical_harmonic_in_config_space(
          params.ell2, m2_, params.boxsize, params.ngrid, ylm_r_b
        );
 
      for (int M_ = - params.ELL; M_ <= params.ELL; M_++) {
        // Check if the current spherical order triplet is redundant.
        SphericalOrderTriplet sph_order_current = {m1_, m2_, M_};
 
        std::string flag_redundant = "false";
        for (SphericalOrderTriplet sph_order : sph_orders_computed) {
          if (sph_order_current.is_inverse(sph_order)) {
            flag_redundant = "true";
          }
        }
        if (flag_redundant == "true") {continue;}
 
        // Calculate the coupling coefficient.
        double coupling = trv::calc_coupling_coeff_3pt(
          params.ell1, params.ell2, params.ELL, m1_, m2_, M_
        );  // Wigner 3-j's
        if (std::fabs(coupling) < trvm::eps_coupling) {continue;}
 
        // The factors consist of products of
        // ```
        // {std::pow(-1., params.ell1 + params.ell2 + params.ELL),
        //  1},
        // {std::pow(-1., m1_ + m2_ + M_),
        //  std::pow(-1., 2*M_)} ≡ 1,
        // {std::pow(-1., params.ell1 + params.ell2),
        //  std::pow(-1., params.ELL)} ≡ std::pow(-1., params.ELL)}
        // ```
        // which, by parity symmetry & Wigner 3-j properties, simplify to:
        double factor_mirror = sph_order_current.is_zeros() ?
          0. : std::pow(-1., params.ELL);
        double factor_mirror_w3j = sph_order_current.is_zeros() ?
          0. : 1.;  // std::pow(-1., params.ell1 + params.ell2 + params.ELL);
 
        // Set up FFT counter and progress bar.
        int progbar_task_count = 10 + 3 * dv_dim;
        std::stringstream progbar_name_ss;
        progbar_name_ss
          << "component orders (" << m1_ << ", " << m2_ << ", " << M_ << ")";
        trvs::ProgressBar progbar(progbar_task_count, progbar_name_ss.str());
 
        std::vector<float> progbar_nodes;
        if (params.progbar != "false" && params.progbar != "true") {
          progbar_nodes = trvs::set_nodes_by_str(params.progbar);
          if (progbar_nodes.size() > 0) {
            progbar.set_nodes(progbar_nodes);
          }
        }
 
        int count_fft_init = trvs::count_fft + trvs::count_ifft;
        auto update_progbar = [&]() {
          if (params.progbar != "false") {
            if (trvs::currTask == 0) {
              progbar.update(
                trvs::count_fft + trvs::count_ifft - count_fft_init
              );
            }
          }
        };
 
        // ·······························································
        // Raw bispectrum
        // ·······························································
 
        // Compute bispectrum components in eqs. (41) & (42) in the Paper.
        MeshField dn_LM_a(params, true, "`dn_LM_a`");  // δn_LM_a
        if (los_choice == 0) {
          dn_LM_a.compute_ylm_wgtd_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            params.ELL, M_
          );
        } else {
          dn_LM_a.compute_ylm_wgtd_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            0, 0
          );
        }
        dn_LM_a.fourier_transform();
        update_progbar();
 
        MeshField dn_LM_b(params, true, "`dn_LM_b`");  // δn_LM_b
        if (los_choice == 1) {
          dn_LM_b.compute_ylm_wgtd_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            params.ELL, M_
          );
        } else {
          dn_LM_b.compute_ylm_wgtd_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            0, 0
          );
        }
        dn_LM_b.fourier_transform();
        update_progbar();
 
        MeshField G_LM(params, true, "`G_LM`");  // G_LM
        if (los_choice == 2) {
          G_LM.compute_ylm_wgtd_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            params.ELL, M_
          );
        } else {
          G_LM.compute_ylm_wgtd_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            0, 0
          );
        }
        G_LM.fourier_transform();
        G_LM.apply_assignment_compensation();
        G_LM.inv_fourier_transform();
        update_progbar();
 
        double vol_cell = G_LM.vol_cell;
 
        MeshField F_lm_a(params, true, "`F_lm_a`");  // F_lm_a
        MeshField F_lm_b(params, true, "`F_lm_b`");  // F_lm_b
 
        if (params.shape == "diag") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin = idx_dv;
 
            double k_lower = kbinning.bin_edges[ibin];
            double k_upper = kbinning.bin_edges[ibin + 1];
 
            double k_eff_a_, k_eff_b_;
            int nmodes_a_, nmodes_b_;
 
            F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_LM_a, ylm_k_a, k_lower, k_upper, k_eff_a_, nmodes_a_
            );
            F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_LM_b, ylm_k_b, k_lower, k_upper, k_eff_b_, nmodes_b_
            );
            update_progbar();
 
            if (count_terms == 0) {
              k1bin_dv[idx_dv] = kbinning.bin_centres[ibin];
              k2bin_dv[idx_dv] = kbinning.bin_centres[ibin];
              k1eff_dv[idx_dv] = k_eff_a_;
              k2eff_dv[idx_dv] = k_eff_b_;
              nmodes1_dv[idx_dv] = nmodes_a_;
              nmodes2_dv[idx_dv] = nmodes_b_;
            }
 
            // B_{l₁ l₂ L}^{m₁ m₂ M}
            double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
            for (long long gid = 0; gid < params.nmesh; gid++) {
              std::complex<double> F_lm_a_gridpt(
                F_lm_a[gid][0], F_lm_a[gid][1]
              );
              std::complex<double> F_lm_b_gridpt(
                F_lm_b[gid][0], F_lm_b[gid][1]
              );
              std::complex<double> G_LM_gridpt(G_LM[gid][0], G_LM[gid][1]);
              std::complex<double> bk_gridpt =
                F_lm_a_gridpt * F_lm_b_gridpt * G_LM_gridpt;
 
              bk_comp_real += bk_gridpt.real();
              bk_comp_imag += bk_gridpt.imag();
            }
 
            std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
            bk_dv[idx_dv] += coupling * vol_cell * (
              bk_component + factor_mirror * std::conj(bk_component)
            );
          }
        }
 
        if (params.shape == "off-diag") {
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin_row, ibin_col;
            if (params.idx_bin >= 0) {
              ibin_row = idx_dv;
              ibin_col = idx_dv + std::abs(params.idx_bin);
            } else {
              ibin_row = idx_dv + std::abs(params.idx_bin);
              ibin_col = idx_dv;
            }
 
            double k_lower_a = kbinning.bin_edges[ibin_row];
            double k_upper_a = kbinning.bin_edges[ibin_row + 1];
            double k_lower_b = kbinning.bin_edges[ibin_col];
            double k_upper_b = kbinning.bin_edges[ibin_col + 1];
 
            double k_eff_a_, k_eff_b_;
            int nmodes_a_, nmodes_b_;
 
            F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_LM_a, ylm_k_a, k_lower_a, k_upper_a, k_eff_a_, nmodes_a_
            );
            F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_LM_b, ylm_k_b, k_lower_b, k_upper_b, k_eff_b_, nmodes_b_
            );
            update_progbar();
 
            if (count_terms == 0) {
              k1bin_dv[idx_dv] = kbinning.bin_centres[ibin_row];
              k2bin_dv[idx_dv] = kbinning.bin_centres[ibin_col];
              k1eff_dv[idx_dv] = k_eff_a_;
              k2eff_dv[idx_dv] = k_eff_b_;
              nmodes1_dv[idx_dv] = nmodes_a_;
              nmodes2_dv[idx_dv] = nmodes_b_;
            }
 
            // B_{l₁ l₂ L}^{m₁ m₂ M}
            double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
            for (long long gid = 0; gid < params.nmesh; gid++) {
              std::complex<double> F_lm_a_gridpt(
                F_lm_a[gid][0], F_lm_a[gid][1]
              );
              std::complex<double> F_lm_b_gridpt(
                F_lm_b[gid][0], F_lm_b[gid][1]
              );
              std::complex<double> G_LM_gridpt(G_LM[gid][0], G_LM[gid][1]);
              std::complex<double> bk_gridpt =
                F_lm_a_gridpt * F_lm_b_gridpt * G_LM_gridpt;
 
              bk_comp_real += bk_gridpt.real();
              bk_comp_imag += bk_gridpt.imag();
            }
 
            std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
            bk_dv[idx_dv] += coupling * vol_cell * (
              bk_component + factor_mirror * std::conj(bk_component)
            );
          }
        }
 
        if (params.shape == "row") {
          int ibin_row = params.idx_bin;
 
          double k_lower_a = kbinning.bin_edges[params.idx_bin];
          double k_upper_a = kbinning.bin_edges[params.idx_bin + 1];
 
          double k_eff_a_;
          int nmodes_a_;
 
          F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
            dn_LM_a, ylm_k_a, k_lower_a, k_upper_a, k_eff_a_, nmodes_a_
          );
 
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            int ibin_col = idx_dv;
 
            double k_lower_b = kbinning.bin_edges[ibin_col];
            double k_upper_b = kbinning.bin_edges[ibin_col + 1];
 
            double k_eff_b_;
            int nmodes_b_;
 
            F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
              dn_LM_b, ylm_k_b, k_lower_b, k_upper_b, k_eff_b_, nmodes_b_
            );
            update_progbar();
 
            if (count_terms == 0) {
              k1bin_dv[idx_dv] = kbinning.bin_centres[ibin_row];
              k2bin_dv[idx_dv] = kbinning.bin_centres[ibin_col];
              k1eff_dv[idx_dv] = k_eff_a_;
              k2eff_dv[idx_dv] = k_eff_b_;
              nmodes1_dv[idx_dv] = nmodes_a_;
              nmodes2_dv[idx_dv] = nmodes_b_;
            }
 
            // B_{l₁ l₂ L}^{m₁ m₂ M}
            double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
            for (long long gid = 0; gid < params.nmesh; gid++) {
              std::complex<double> F_lm_a_gridpt(
                F_lm_a[gid][0], F_lm_a[gid][1]
              );
              std::complex<double> F_lm_b_gridpt(
                F_lm_b[gid][0], F_lm_b[gid][1]
              );
              std::complex<double> G_LM_gridpt(G_LM[gid][0], G_LM[gid][1]);
              std::complex<double> bk_gridpt =
                F_lm_a_gridpt * F_lm_b_gridpt * G_LM_gridpt;
 
              bk_comp_real += bk_gridpt.real();
              bk_comp_imag += bk_gridpt.imag();
            }
 
            std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
            bk_dv[idx_dv] += coupling * vol_cell * (
              bk_component + factor_mirror * std::conj(bk_component)
            );
          }
        }
 
        if (params.shape == "full") {
          for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
            for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
              int idx_dv = idx_row * params.num_bins + idx_col;
 
              double k_lower_a = kbinning.bin_edges[idx_row];
              double k_upper_a = kbinning.bin_edges[idx_row + 1];
              double k_lower_b = kbinning.bin_edges[idx_col];
              double k_upper_b = kbinning.bin_edges[idx_col + 1];
 
              double k_eff_a_, k_eff_b_;
              int nmodes_a_, nmodes_b_;
 
              F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
                dn_LM_a, ylm_k_a, k_lower_a, k_upper_a, k_eff_a_, nmodes_a_
              );
              F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
                dn_LM_b, ylm_k_b, k_lower_b, k_upper_b, k_eff_b_, nmodes_b_
              );
              update_progbar();
 
              if (count_terms == 0) {
                k1bin_dv[idx_dv] = kbinning.bin_centres[idx_row];
                k2bin_dv[idx_dv] = kbinning.bin_centres[idx_col];
                k1eff_dv[idx_dv] = k_eff_a_;
                k2eff_dv[idx_dv] = k_eff_b_;
                nmodes1_dv[idx_dv] = nmodes_a_;
                nmodes2_dv[idx_dv] = nmodes_b_;
              }
 
              // B_{l₁ l₂ L}^{m₁ m₂ M}
              double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
              for (long long gid = 0; gid < params.nmesh; gid++) {
                std::complex<double> F_lm_a_gridpt(
                  F_lm_a[gid][0], F_lm_a[gid][1]
                );
                std::complex<double> F_lm_b_gridpt(
                  F_lm_b[gid][0], F_lm_b[gid][1]
                );
                std::complex<double> G_LM_gridpt(G_LM[gid][0], G_LM[gid][1]);
                std::complex<double> bk_gridpt =
                  F_lm_a_gridpt * F_lm_b_gridpt * G_LM_gridpt;
 
                bk_comp_real += bk_gridpt.real();
                bk_comp_imag += bk_gridpt.imag();
              }
 
              std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
              bk_dv[idx_dv] += coupling * vol_cell * (
                bk_component + factor_mirror * std::conj(bk_component)
              );
            }
          }
        }
 
        if (params.shape == "triu") {
          for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
            for (int idx_col = idx_row; idx_col < params.num_bins; idx_col++) {
              int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2
                + (idx_col - idx_row);
 
              double k_lower_a = kbinning.bin_edges[idx_row];
              double k_upper_a = kbinning.bin_edges[idx_row + 1];
              double k_lower_b = kbinning.bin_edges[idx_col];
              double k_upper_b = kbinning.bin_edges[idx_col + 1];
 
              double k_eff_a_, k_eff_b_;
              int nmodes_a_, nmodes_b_;
 
              F_lm_a.inv_fourier_transform_ylm_wgtd_field_band_limited(
                dn_LM_a, ylm_k_a, k_lower_a, k_upper_a, k_eff_a_, nmodes_a_
              );
              F_lm_b.inv_fourier_transform_ylm_wgtd_field_band_limited(
                dn_LM_b, ylm_k_b, k_lower_b, k_upper_b, k_eff_b_, nmodes_b_
              );
              update_progbar();
 
              if (count_terms == 0) {
                k1bin_dv[idx_dv] = kbinning.bin_centres[idx_row];
                k2bin_dv[idx_dv] = kbinning.bin_centres[idx_col];
                k1eff_dv[idx_dv] = k_eff_a_;
                k2eff_dv[idx_dv] = k_eff_b_;
                nmodes1_dv[idx_dv] = nmodes_a_;
                nmodes2_dv[idx_dv] = nmodes_b_;
              }
 
              // B_{l₁ l₂ L}^{m₁ m₂ M}
              double bk_comp_real = 0., bk_comp_imag = 0.;
 
#ifdef TRV_USE_OMP
#pragma omp parallel for reduction(+:bk_comp_real, bk_comp_imag)
#endif  // TRV_USE_OMP
              for (long long gid = 0; gid < params.nmesh; gid++) {
                std::complex<double> F_lm_a_gridpt(
                  F_lm_a[gid][0], F_lm_a[gid][1]
                );
                std::complex<double> F_lm_b_gridpt(
                  F_lm_b[gid][0], F_lm_b[gid][1]
                );
                std::complex<double> G_LM_gridpt(G_LM[gid][0], G_LM[gid][1]);
                std::complex<double> bk_gridpt =
                  F_lm_a_gridpt * F_lm_b_gridpt * G_LM_gridpt;
 
                bk_comp_real += bk_gridpt.real();
                bk_comp_imag += bk_gridpt.imag();
              }
 
              std::complex<double> bk_component(bk_comp_real, bk_comp_imag);
 
              bk_dv[idx_dv] += coupling * vol_cell * (
                bk_component + factor_mirror * std::conj(bk_component)
              );
            }
          }
        }
 
        // ·······························································
        // Shot noise
        // ·······························································
 
        // Compute shot noise components in eqs. (45) & (46) in the Paper.
        MeshField dn_LM_a_for_sn(params, true, "`dn_LM_a_for_sn`");  // δn_LM_a(k)
                                                               // (for shot
                                                               // noise)
        if (los_choice == 0) {
          dn_LM_a_for_sn.compute_ylm_wgtd_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            params.ELL, M_
          );
        } else {
          dn_LM_a_for_sn.compute_ylm_wgtd_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            0, 0
          );
        }
        dn_LM_a_for_sn.fourier_transform();
        update_progbar();
 
        MeshField dn_LM_b_for_sn(params, true, "`dn_LM_b_for_sn`");  // δn_LM_b(k)
                                                               // (for shot
                                                               // noise)
        if (los_choice == 1) {
          dn_LM_b_for_sn.compute_ylm_wgtd_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            params.ELL, M_
          );
        } else {
          dn_LM_b_for_sn.compute_ylm_wgtd_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            0, 0
          );
        }
        dn_LM_b_for_sn.fourier_transform();
        update_progbar();
 
        // δn_LM_c(k) (for shot noise)
        MeshField dn_LM_c_for_sn(params, true, "`dn_LM_c_for_sn`");
        if (los_choice == 2) {
          dn_LM_c_for_sn.compute_ylm_wgtd_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            params.ELL, M_
          );
        } else {
          dn_LM_c_for_sn.compute_ylm_wgtd_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            0, 0
          );
        }
        dn_LM_c_for_sn.fourier_transform();
        update_progbar();
 
        MeshField N_LM_a(params, true, "`N_LM_a`");  // N_LM_a(k)
        if (los_choice == 0) {
          N_LM_a.compute_ylm_wgtd_quad_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            0, 0
          );
        } else {
          N_LM_a.compute_ylm_wgtd_quad_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            params.ELL, M_
          );
        }
        N_LM_a.fourier_transform();
        update_progbar();
 
        MeshField N_LM_b(params, true, "`N_LM_b`");  // N_LM_b(k)
        if (los_choice == 1) {
          N_LM_b.compute_ylm_wgtd_quad_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            0, 0
          );
        } else {
          N_LM_b.compute_ylm_wgtd_quad_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            params.ELL, M_
          );
        }
        N_LM_b.fourier_transform();
        update_progbar();
 
        MeshField N_LM_c(params, true, "`N_LM_c`");  // N_LM_c(k)
        if (los_choice == 2) {
          N_LM_c.compute_ylm_wgtd_quad_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            0, 0
          );
        } else {
          N_LM_c.compute_ylm_wgtd_quad_field(
            catalogue_data, catalogue_rand, los_data, los_rand, alpha,
            params.ELL, M_
          );
        }
        N_LM_c.fourier_transform();
        update_progbar();
 
        std::complex<double> Sbar_LM = calc_ylm_wgtd_shotnoise_amp_for_bispec(
          catalogue_data, catalogue_rand, los_data, los_rand, alpha,
          params.ELL, M_
        );  // \bar{S}_LM
 
        if (params.ell1 == 0 && params.ell2 == 0) {
          // When l₁ = l₂ = 0, the Wigner 3-j symbol enforces L = 0
          // and the pre-factors involving degrees and orders become 1.
          std::complex<double> S_ijk = coupling * Sbar_LM;  // S|{i = j = k}
          for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
            sn_dv[idx_dv] += S_ijk;
          }
        }
 
        if (params.ell2 == 0) {  // S|{i ≠ j = k}
          // When l₂ = 0, the Wigner 3-j symbol enforces L = l₁.
          stats_sn.compute_ylm_wgtd_2pt_stats_in_fourier(
            dn_LM_a_for_sn, N_LM_a, Sbar_LM, params.ell1, m1_, kbinning
          );
 
          if (params.shape == "diag") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin = idx_dv;
              std::complex<double> S_i_jk = coupling * (
                stats_sn.pk[ibin] - stats_sn.sn[ibin]
              );
              sn_dv[idx_dv] += S_i_jk + factor_mirror * std::conj(S_i_jk);
            }
          }
 
          if (params.shape == "off-diag") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin_row;
              if (params.idx_bin >= 0) {
                ibin_row = idx_dv;
              } else {
                ibin_row = idx_dv + std::abs(params.idx_bin);
              }
              std::complex<double> S_i_jk = coupling * (
                stats_sn.pk[ibin_row] - stats_sn.sn[ibin_row]
              );
              sn_dv[idx_dv] += S_i_jk + factor_mirror * std::conj(S_i_jk);
            }
          }
 
          if (params.shape == "row") {
            std::complex<double> S_i_jk_row_ = coupling * (
              stats_sn.pk[params.idx_bin] - stats_sn.sn[params.idx_bin]
            );
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              sn_dv[idx_dv] +=
                S_i_jk_row_ + factor_mirror * std::conj(S_i_jk_row_);
            }
          }
 
          if (params.shape == "full") {
            for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
              std::complex<double> S_i_jk_row_ = coupling * (
                stats_sn.pk[idx_row] - stats_sn.sn[idx_row]
              );
              for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
                int idx_dv = idx_row * params.num_bins + idx_col;
                sn_dv[idx_dv] +=
                  S_i_jk_row_ + factor_mirror * std::conj(S_i_jk_row_);
              }
            }
          }
 
          if (params.shape == "triu") {
            for (int idx_row = 0; idx_row < params.num_bins; idx_row++) {
              std::complex<double> S_i_jk_row_ = coupling * (
                stats_sn.pk[idx_row] - stats_sn.sn[idx_row]
              );
              for (int idx_col = idx_row; idx_col < params.num_bins; idx_col++)
              {
                int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2
                  + (idx_col - idx_row);
                sn_dv[idx_dv] +=
                  S_i_jk_row_ + factor_mirror * std::conj(S_i_jk_row_);
              }
            }
          }
        }
 
        if (params.ell1 == 0) {  // S|{j ≠ i = k}
          // When l₁ = 0, the Wigner 3-j symbol enforces L = l₂.
          stats_sn.compute_ylm_wgtd_2pt_stats_in_fourier(
            dn_LM_b_for_sn, N_LM_b, Sbar_LM,
            params.ell2, m2_, kbinning
          );
 
          if (params.shape == "diag") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin = idx_dv;
              std::complex<double> S_ik_j_ki = coupling * (
                stats_sn.pk[ibin] - stats_sn.sn[ibin]
              );
              sn_dv[idx_dv] +=
                S_ik_j_ki + factor_mirror * std::conj(S_ik_j_ki);
            }
          }
 
          if (params.shape == "off-diag") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin_col;
              if (params.idx_bin >= 0) {
                ibin_col = idx_dv + params.idx_bin;
              } else {
                ibin_col = idx_dv;
              }
              std::complex<double> S_ik_j_ki = coupling * (
                stats_sn.pk[ibin_col] - stats_sn.sn[ibin_col]
              );
              sn_dv[idx_dv] +=
                S_ik_j_ki + factor_mirror * std::conj(S_ik_j_ki);
            }
          }
 
          if (params.shape == "row") {
            for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
              int ibin_col = idx_dv;
              std::complex<double> S_ik_j_ki = coupling * (
                stats_sn.pk[ibin_col] - stats_sn.sn[ibin_col]
              );
              sn_dv[idx_dv] +=
                S_ik_j_ki + factor_mirror * std::conj(S_ik_j_ki);
            }
          }
 
          if (params.shape == "full") {
            for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
              std::complex<double> S_ik_j_ki_col_ = coupling * (
                stats_sn.pk[idx_col] - stats_sn.sn[idx_col]
              );
              for (int idx_row = 0; idx_row <= params.num_bins; idx_row++) {
                int idx_dv = idx_row * params.num_bins + idx_col;
                sn_dv[idx_dv] +=
                  S_ik_j_ki_col_ + factor_mirror * std::conj(S_ik_j_ki_col_);
              }
            }
          }
 
          if (params.shape == "triu") {
            for (int idx_col = 0; idx_col < params.num_bins; idx_col++) {
              std::complex<double> S_ik_j_ki_col_ = coupling * (
                stats_sn.pk[idx_col] - stats_sn.sn[idx_col]
              );
              for (int idx_row = 0; idx_row <= idx_col; idx_row++) {
                int idx_dv = (2*params.num_bins - idx_row + 1) * idx_row / 2
                  + (idx_col - idx_row);
                sn_dv[idx_dv] +=
                  S_ik_j_ki_col_ + factor_mirror * std::conj(S_ik_j_ki_col_);
              }
            }
          }
        }
 
        for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
          double k_a = k1eff_dv[idx_dv];
          double k_b = k2eff_dv[idx_dv];
 
          std::complex<double> S_ij_k = coupling
            * stats_sn.compute_uncoupled_shotnoise_for_bispec_per_bin(
              dn_LM_c_for_sn, N_LM_c, ylm_r_a, ylm_r_b, sj_a, sj_b,
              Sbar_LM, k_a, k_b
            );  // S|{i = j ≠ k}
          update_progbar();
 
          sn_dv[idx_dv] += factor_phase * (
            S_ij_k + factor_mirror_w3j * std::conj(S_ij_k)
          );
        }
 
        sph_orders_computed.push_back(sph_order_current);
        count_terms++;
        if (trvs::currTask == 0) {
          trvs::logger.stat(
            "Bispectrum term computed at orders (m₁, m₂, M) = ±(%d, %d, %d).",
            m1_, m2_, M_
          );
        }
      }
    }
  }
 
  // ---------------------------------------------------------------------
  // Results
  // ---------------------------------------------------------------------
 
  trv::BispecMeasurements bispec_out;
  for (int idx_dv = 0; idx_dv < dv_dim; idx_dv++) {
    bispec_out.k1_bin.push_back(k1bin_dv[idx_dv]);
    bispec_out.k1_eff.push_back(k1eff_dv[idx_dv]);
    bispec_out.nmodes_1.push_back(nmodes1_dv[idx_dv]);
    bispec_out.k2_bin.push_back(k2bin_dv[idx_dv]);
    bispec_out.k2_eff.push_back(k2eff_dv[idx_dv]);
    bispec_out.nmodes_2.push_back(nmodes2_dv[idx_dv]);
    bispec_out.bk_raw.push_back(norm_factor * bk_dv[idx_dv]);
    bispec_out.bk_shot.push_back(norm_factor * sn_dv[idx_dv]);
  }
  bispec_out.dim = dv_dim;
 
  delete[] nmodes1_dv; delete[] nmodes2_dv;
  delete[] k1bin_dv; delete[] k2bin_dv;
  delete[] k1eff_dv; delete[] k2eff_dv;
  delete[] bk_dv; delete[] sn_dv;
 
  trvs::count_cgrid -= 4;
  trvs::count_grid -= 4;
  trvs::gbytesMem -=
    trvs::size_in_gb< std::complex<double> >(4*params.nmesh);
 
  if (trvs::currTask == 0) {
    trvs::logger.stat(
      "... computed bispectrum from paired survey-type catalogues "
      "for line-of-sight choice %d.",
      los_choice
    );
  }
 
  return bispec_out;
}
#endif  // TRV_USE_LEGACY_CODE
 
}  // namespace trv