// SPDX-License-Identifier: Apache-2.0
//
// Copyright 2008-2016 Conrad Sanderson (http://conradsanderson.id.au)
// Copyright 2008-2016 National ICT Australia (NICTA)
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
// ------------------------------------------------------------------------

//! \addtogroup sym_helper
//! @{

namespace sym_helper {

// computationally inexpensive algorithm to guess whether a matrix is positive definite:
// (1) ensure the matrix is symmetric/hermitian (within a tolerance)
// (2) ensure the diagonal entries are real and greater than zero
// (3) ensure that the value with largest modulus is on the main diagonal
// (4) ensure rudimentary diagonal dominance: (real(A_ii) + real(A_jj)) >
// 2*abs(real(A_ij)) the above conditions are necessary, but not sufficient; doing it
// properly would be too computationally expensive for our purposes more info:
// http://mathworld.wolfram.com/PositiveDefiniteMatrix.html
// http://mathworld.wolfram.com/DiagonallyDominantMatrix.html

template <typename eT>
inline typename enable_if2<is_cx<eT>::no, bool>::result guess_sympd_worker(
    const Mat<eT>& A) {
  arma_debug_sigprint();

  // NOTE: assuming A is square-sized

  const eT tol = eT(100) * std::numeric_limits<eT>::epsilon();  // allow some leeway

  const uword N = A.n_rows;

  const eT* A_mem = A.memptr();
  const eT* A_col = A_mem;

  bool diag_below_tol = true;

  eT max_diag = eT(0);

  for (uword j = 0; j < N; ++j) {
    const eT A_jj = A_col[j];

    if (A_jj <= eT(0)) {
      return false;
    }
    if (arma_isfinite(A_jj) == false) {
      return false;
    }

    if (A_jj >= tol) {
      diag_below_tol = false;
    }

    max_diag = (A_jj > max_diag) ? A_jj : max_diag;

    A_col += N;
  }

  if (diag_below_tol) {
    return false;
  }  // assume matrix is suspect if all diagonal elements are close to zero

  A_col = A_mem;

  const uword Nm1 = N - 1;
  const uword Np1 = N + 1;

  for (uword j = 0; j < Nm1; ++j) {
    const eT A_jj = A_col[j];

    const uword jp1 = j + 1;
    const eT* A_ji_ptr = &(A_mem[j + jp1 * N]);  // &(A.at(j,jp1));
    const eT* A_ii_ptr = &(A_mem[jp1 + jp1 * N]);

    for (uword i = jp1; i < N; ++i) {
      const eT A_ij = A_col[i];
      const eT A_ji = (*A_ji_ptr);

      const eT A_ij_abs = (std::abs)(A_ij);
      const eT A_ji_abs = (std::abs)(A_ji);

      // if( (A_ij_abs >= max_diag) || (A_ji_abs >= max_diag) )  { return false; }
      if (A_ij_abs >= max_diag) {
        return false;
      }

      const eT A_delta = (std::abs)(A_ij - A_ji);
      const eT A_abs_max = (std::max)(A_ij_abs, A_ji_abs);

      if ((A_delta > tol) && (A_delta > (A_abs_max * tol))) {
        return false;
      }

      const eT A_ii = (*A_ii_ptr);

      if ((A_ij_abs + A_ij_abs) >= (A_ii + A_jj)) {
        return false;
      }

      A_ji_ptr += N;
      A_ii_ptr += Np1;
    }

    A_col += N;
  }

  return true;
}

template <typename eT>
inline typename enable_if2<is_cx<eT>::yes, bool>::result guess_sympd_worker(
    const Mat<eT>& A) {
  arma_debug_sigprint();

  // NOTE: assuming A is square-sized

  // NOTE: the function name is required for overloading, but is a misnomer: it processes
  // complex hermitian matrices

  typedef typename get_pod_type<eT>::result T;

  const T tol = T(100) * std::numeric_limits<T>::epsilon();  // allow some leeway

  const uword N = A.n_rows;

  const eT* A_mem = A.memptr();
  const eT* A_col = A_mem;

  bool diag_below_tol = true;

  T max_diag = T(0);

  for (uword j = 0; j < N; ++j) {
    const eT& A_jj = A_col[j];
    const T A_jj_r = std::real(A_jj);
    const T A_jj_i = std::imag(A_jj);
    const T A_jj_rabs = std::abs(A_jj_r);
    const T A_jj_iabs = std::abs(A_jj_i);

    if (A_jj_r <= T(0)) {
      return false;
    }  // real should be positive
    if (arma_isfinite(A_jj_r) == false) {
      return false;
    }

    if (A_jj_iabs > tol) {
      return false;
    }  // imag should be approx zero
    if (A_jj_iabs > A_jj_rabs) {
      return false;
    }  // corner case: real and imag are close to zero, and imag is dominant

    if (A_jj_r >= tol) {
      diag_below_tol = false;
    }

    max_diag = (A_jj_r > max_diag) ? A_jj_r : max_diag;

    A_col += N;
  }

  if (diag_below_tol) {
    return false;
  }  // assume matrix is suspect if all diagonal elements are close to zero

  const T square_max_diag = max_diag * max_diag;

  if (arma_isfinite(square_max_diag) == false) {
    return false;
  }

  A_col = A_mem;

  const uword Nm1 = N - 1;
  const uword Np1 = N + 1;

  for (uword j = 0; j < Nm1; ++j) {
    const uword jp1 = j + 1;
    const eT* A_ji_ptr = &(A_mem[j + jp1 * N]);  // &(A.at(j,jp1));
    const eT* A_ii_ptr = &(A_mem[jp1 + jp1 * N]);

    const T A_jj_real = std::real(A_col[j]);

    for (uword i = jp1; i < N; ++i) {
      const eT& A_ij = A_col[i];
      const T A_ij_real = std::real(A_ij);
      const T A_ij_imag = std::imag(A_ij);

      // avoid using std::abs(), as that is time consuming due to division and std::sqrt()
      const T square_A_ij_abs = (A_ij_real * A_ij_real) + (A_ij_imag * A_ij_imag);

      if (arma_isfinite(square_A_ij_abs) == false) {
        return false;
      }

      if (square_A_ij_abs >= square_max_diag) {
        return false;
      }

      const T A_ij_real_abs = (std::abs)(A_ij_real);
      const T A_ij_imag_abs = (std::abs)(A_ij_imag);

      const eT& A_ji = (*A_ji_ptr);
      const T A_ji_real = std::real(A_ji);
      const T A_ji_imag = std::imag(A_ji);

      const T A_ji_real_abs = (std::abs)(A_ji_real);
      const T A_ji_imag_abs = (std::abs)(A_ji_imag);

      const T A_real_delta = (std::abs)(A_ij_real - A_ji_real);
      const T A_real_abs_max = (std::max)(A_ij_real_abs, A_ji_real_abs);

      if ((A_real_delta > tol) && (A_real_delta > (A_real_abs_max * tol))) {
        return false;
      }

      const T A_imag_delta =
          (std::abs)(A_ij_imag + A_ji_imag);  // take into account complex conjugate
      const T A_imag_abs_max = (std::max)(A_ij_imag_abs, A_ji_imag_abs);

      if ((A_imag_delta > tol) && (A_imag_delta > (A_imag_abs_max * tol))) {
        return false;
      }

      const T A_ii_real = std::real(*A_ii_ptr);

      if ((A_ij_real_abs + A_ij_real_abs) >= (A_ii_real + A_jj_real)) {
        return false;
      }

      A_ji_ptr += N;
      A_ii_ptr += Np1;
    }

    A_col += N;
  }

  return true;
}

template <typename eT>
inline bool guess_sympd(const Mat<eT>& A) {
  arma_debug_sigprint();

  // analyse matrices with size >= 4x4

  if ((A.n_rows != A.n_cols) || (A.n_rows < uword(4))) {
    return false;
  }

  return guess_sympd_worker(A);
}

template <typename eT>
inline bool guess_sympd(const Mat<eT>& A, const uword min_n_rows) {
  arma_debug_sigprint();

  if ((A.n_rows != A.n_cols) || (A.n_rows < min_n_rows)) {
    return false;
  }

  return guess_sympd_worker(A);
}

//

template <typename eT>
inline typename enable_if2<is_cx<eT>::no, bool>::result is_approx_sym_worker(
    const Mat<eT>& A) {
  arma_debug_sigprint();

  const eT tol = eT(100) * std::numeric_limits<eT>::epsilon();  // allow some leeway

  const uword N = A.n_rows;

  const eT* A_mem = A.memptr();
  const eT* A_col = A_mem;

  bool diag_below_tol = true;

  for (uword j = 0; j < N; ++j) {
    const eT A_jj = A_col[j];

    if (arma_isfinite(A_jj) == false) {
      return false;
    }

    if (std::abs(A_jj) >= tol) {
      diag_below_tol = false;
    }

    A_col += N;
  }

  if (diag_below_tol) {
    return false;
  }  // assume matrix is suspect if all diagonal elements are close to zero

  A_col = A_mem;

  const uword Nm1 = N - 1;

  for (uword j = 0; j < Nm1; ++j) {
    const uword jp1 = j + 1;
    const eT* A_ji_ptr = &(A_mem[j + jp1 * N]);  // &(A.at(j,jp1));

    for (uword i = jp1; i < N; ++i) {
      const eT A_ij = A_col[i];
      const eT A_ji = (*A_ji_ptr);

      const eT A_ij_abs = (std::abs)(A_ij);
      const eT A_ji_abs = (std::abs)(A_ji);

      const eT A_delta = (std::abs)(A_ij - A_ji);
      const eT A_abs_max = (std::max)(A_ij_abs, A_ji_abs);

      if ((A_delta > tol) && (A_delta > (A_abs_max * tol))) {
        return false;
      }

      A_ji_ptr += N;
    }

    A_col += N;
  }

  return true;
}

template <typename eT>
inline typename enable_if2<is_cx<eT>::yes, bool>::result is_approx_sym_worker(
    const Mat<eT>& A) {
  arma_debug_sigprint();

  // NOTE: the function name is required for overloading, but is a misnomer: it processes
  // complex hermitian matrices

  typedef typename get_pod_type<eT>::result T;

  const T tol = T(100) * std::numeric_limits<T>::epsilon();  // allow some leeway

  const uword N = A.n_rows;

  const eT* A_mem = A.memptr();
  const eT* A_col = A_mem;

  bool diag_below_tol = true;

  // ensure diagonal has approx real-only elements
  for (uword j = 0; j < N; ++j) {
    const eT& A_jj = A_col[j];
    const T A_jj_r = std::real(A_jj);
    const T A_jj_i = std::imag(A_jj);
    const T A_jj_rabs = std::abs(A_jj_r);
    const T A_jj_iabs = std::abs(A_jj_i);

    if (A_jj_iabs > tol) {
      return false;
    }  // imag should be approx zero
    if (A_jj_iabs > A_jj_rabs) {
      return false;
    }  // corner case: real and imag are close to zero, and imag is dominant

    if (arma_isfinite(A_jj_r) == false) {
      return false;
    }

    if (A_jj_rabs >= tol) {
      diag_below_tol = false;
    }

    A_col += N;
  }

  if (diag_below_tol) {
    return false;
  }  // assume matrix is suspect if all diagonal elements are close to zero

  A_col = A_mem;

  const uword Nm1 = N - 1;

  for (uword j = 0; j < Nm1; ++j) {
    const uword jp1 = j + 1;
    const eT* A_ji_ptr = &(A_mem[j + jp1 * N]);  // &(A.at(j,jp1));

    for (uword i = jp1; i < N; ++i) {
      const eT& A_ij = A_col[i];
      const T A_ij_real = std::real(A_ij);
      const T A_ij_imag = std::imag(A_ij);

      const T A_ij_real_abs = (std::abs)(A_ij_real);
      const T A_ij_imag_abs = (std::abs)(A_ij_imag);

      const eT& A_ji = (*A_ji_ptr);
      const T A_ji_real = std::real(A_ji);
      const T A_ji_imag = std::imag(A_ji);

      const T A_ji_real_abs = (std::abs)(A_ji_real);
      const T A_ji_imag_abs = (std::abs)(A_ji_imag);

      const T A_real_delta = (std::abs)(A_ij_real - A_ji_real);
      const T A_real_abs_max = (std::max)(A_ij_real_abs, A_ji_real_abs);

      if ((A_real_delta > tol) && (A_real_delta > (A_real_abs_max * tol))) {
        return false;
      }

      const T A_imag_delta =
          (std::abs)(A_ij_imag + A_ji_imag);  // take into account complex conjugate
      const T A_imag_abs_max = (std::max)(A_ij_imag_abs, A_ji_imag_abs);

      if ((A_imag_delta > tol) && (A_imag_delta > (A_imag_abs_max * tol))) {
        return false;
      }

      A_ji_ptr += N;
    }

    A_col += N;
  }

  return true;
}

template <typename eT>
inline bool is_approx_sym(const Mat<eT>& A) {
  arma_debug_sigprint();

  // analyse matrices with size >= 4x4

  if ((A.n_rows != A.n_cols) || (A.n_rows < uword(4))) {
    return false;
  }

  return is_approx_sym_worker(A);
}

template <typename eT>
inline bool is_approx_sym(const Mat<eT>& A, const uword min_n_rows) {
  arma_debug_sigprint();

  if ((A.n_rows != A.n_cols) || (A.n_rows < min_n_rows)) {
    return false;
  }

  return is_approx_sym_worker(A);
}

//

template <typename eT>
inline bool check_diag_imag(const Mat<eT>& A) {
  arma_debug_sigprint();

  // NOTE: assuming matrix A is square-sized

  typedef typename get_pod_type<eT>::result T;

  const T tol = T(10000) * std::numeric_limits<T>::epsilon();  // allow some leeway

  const eT* colmem = A.memptr();

  const uword N = A.n_rows;

  for (uword i = 0; i < N; ++i) {
    const eT& A_ii = colmem[i];
    const T A_ii_imag = access::tmp_imag(A_ii);

    if (std::abs(A_ii_imag) > tol) {
      return false;
    }

    colmem += N;
  }

  return true;
}

}  // end of namespace sym_helper

//! @}
