// 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 op_sqrtmat
//! @{

//! implementation partly based on:
//! N. J. Higham.
//! A New sqrtm for Matlab.
//! Numerical Analysis Report No. 336, January 1999.
//! Department of Mathematics, University of Manchester.
//! ISSN 1360-1725
//! http://www.maths.manchester.ac.uk/~higham/narep/narep336.ps.gz

template <typename T1>
inline void op_sqrtmat::apply(
    Mat<std::complex<typename T1::elem_type> >& out,
    const mtOp<std::complex<typename T1::elem_type>, T1, op_sqrtmat>& in) {
  arma_debug_sigprint();

  const bool status = op_sqrtmat::apply_direct(out, in.m);

  if (status == false) {
    arma_warn(3, "sqrtmat(): given matrix is singular; may not have a square root");
  }
}

template <typename T1>
inline bool op_sqrtmat::apply_direct(Mat<std::complex<typename T1::elem_type> >& out,
                                     const Op<T1, op_diagmat>& expr) {
  arma_debug_sigprint();

  typedef typename T1::elem_type T;

  const diagmat_proxy<T1> P(expr.m);

  arma_conform_check((P.n_rows != P.n_cols),
                     "sqrtmat(): given matrix must be square sized");

  const uword N = P.n_rows;

  out.zeros(N, N);

  bool singular = false;

  for (uword i = 0; i < N; ++i) {
    const T val = P[i];

    if (val >= T(0)) {
      singular = (singular || (val == T(0)));

      out.at(i, i) = std::sqrt(val);
    } else {
      out.at(i, i) = std::sqrt(std::complex<T>(val));
    }
  }

  return (singular) ? false : true;
}

template <typename T1>
inline bool op_sqrtmat::apply_direct(Mat<std::complex<typename T1::elem_type> >& out,
                                     const Base<typename T1::elem_type, T1>& expr) {
  arma_debug_sigprint();

  typedef typename T1::elem_type in_T;
  typedef typename std::complex<in_T> out_T;

  const quasi_unwrap<T1> expr_unwrap(expr.get_ref());
  const Mat<in_T>& A = expr_unwrap.M;

  arma_conform_check((A.is_square() == false),
                     "sqrtmat(): given matrix must be square sized");

  if (A.n_elem == 0) {
    out.reset();
    return true;
  } else if (A.n_elem == 1) {
    out.set_size(1, 1);
    out[0] = std::sqrt(std::complex<in_T>(A[0]));
    return true;
  }

  if (A.is_diagmat()) {
    arma_debug_print("op_sqrtmat: diag optimisation");

    const uword N = A.n_rows;

    out.zeros(N,
              N);  // aliasing can't happen as op_sqrtmat is defined as cx_mat = op(mat)

    for (uword i = 0; i < N; ++i) {
      const in_T val = A.at(i, i);

      if (val >= in_T(0)) {
        out.at(i, i) = std::sqrt(val);
      } else {
        out.at(i, i) = std::sqrt(out_T(val));
      }
    }

    return true;
  }

  const bool try_sympd = arma_config::optimise_sym && sym_helper::guess_sympd(A);

  if (try_sympd) {
    arma_debug_print("op_sqrtmat: attempting sympd optimisation");

    // if matrix A is sympd, all its eigenvalues are positive

    Col<in_T> eigval;
    Mat<in_T> eigvec;

    const bool eig_status = eig_sym_helper(eigval, eigvec, A, 'd', "sqrtmat()");

    if (eig_status) {
      // ensure each eigenvalue is > 0

      const uword N = eigval.n_elem;
      const in_T* eigval_mem = eigval.memptr();

      bool all_pos = true;

      for (uword i = 0; i < N; ++i) {
        all_pos = (eigval_mem[i] <= in_T(0)) ? false : all_pos;
      }

      if (all_pos) {
        eigval = sqrt(eigval);

        out = conv_to<Mat<out_T> >::from(eigvec * diagmat(eigval) * eigvec.t());

        return true;
      }
    }

    arma_debug_print("op_sqrtmat: sympd optimisation failed");

    // fallthrough if eigen decomposition failed or an eigenvalue is <= 0
  }

  Mat<out_T> U;
  Mat<out_T> S(A.n_rows, A.n_cols, arma_nozeros_indicator());

  const in_T* Amem = A.memptr();
  out_T* Smem = S.memptr();

  const uword n_elem = A.n_elem;

  for (uword i = 0; i < n_elem; ++i) {
    Smem[i] = std::complex<in_T>(Amem[i]);
  }

  const bool schur_ok = auxlib::schur(U, S);

  if (schur_ok == false) {
    arma_debug_print("sqrtmat(): schur decomposition failed");
    out.soft_reset();
    return false;
  }

  const bool status = op_sqrtmat_cx::helper(S);

  const Mat<out_T> X = U * S;

  S.reset();

  out = X * U.t();

  return status;
}

template <typename T1>
inline void op_sqrtmat_cx::apply(Mat<typename T1::elem_type>& out,
                                 const Op<T1, op_sqrtmat_cx>& in) {
  arma_debug_sigprint();

  const bool status = op_sqrtmat_cx::apply_direct(out, in.m);

  if (status == false) {
    arma_warn(3, "sqrtmat(): given matrix is singular; may not have a square root");
  }
}

template <typename T1>
inline bool op_sqrtmat_cx::apply_direct(Mat<typename T1::elem_type>& out,
                                        const Op<T1, op_diagmat>& expr) {
  arma_debug_sigprint();

  typedef typename T1::elem_type eT;

  const diagmat_proxy<T1> P(expr.m);

  bool status = false;

  if (P.is_alias(out)) {
    Mat<eT> tmp;

    status = op_sqrtmat_cx::apply_direct_noalias(tmp, P);

    out.steal_mem(tmp);
  } else {
    status = op_sqrtmat_cx::apply_direct_noalias(out, P);
  }

  return status;
}

template <typename T1>
inline bool op_sqrtmat_cx::apply_direct_noalias(Mat<typename T1::elem_type>& out,
                                                const diagmat_proxy<T1>& P) {
  arma_debug_sigprint();

  typedef typename T1::elem_type eT;

  arma_conform_check((P.n_rows != P.n_cols),
                     "sqrtmat(): given matrix must be square sized");

  const uword N = P.n_rows;

  out.zeros(N, N);

  const eT zero = eT(0);

  bool singular = false;

  for (uword i = 0; i < N; ++i) {
    const eT val = P[i];

    singular = (singular || (val == zero));

    out.at(i, i) = std::sqrt(val);
  }

  return (singular) ? false : true;
}

template <typename T1>
inline bool op_sqrtmat_cx::apply_direct(Mat<typename T1::elem_type>& out,
                                        const Base<typename T1::elem_type, T1>& expr) {
  arma_debug_sigprint();

  typedef typename T1::pod_type T;
  typedef typename T1::elem_type eT;

  Mat<eT> U;
  Mat<eT> S = expr.get_ref();

  arma_conform_check((S.n_rows != S.n_cols),
                     "sqrtmat(): given matrix must be square sized");

  if (S.n_elem == 0) {
    out.reset();
    return true;
  } else if (S.n_elem == 1) {
    out.set_size(1, 1);
    out[0] = std::sqrt(S[0]);
    return true;
  }

  if (S.is_diagmat()) {
    arma_debug_print("op_sqrtmat_cx: diag optimisation");

    const uword N = S.n_rows;

    out.zeros(N, N);  // aliasing can't happen as S is generated

    for (uword i = 0; i < N; ++i) {
      out.at(i, i) = std::sqrt(S.at(i, i));
    }

    return true;
  }

  const bool try_sympd = arma_config::optimise_sym && sym_helper::guess_sympd(S);

  if (try_sympd) {
    arma_debug_print("op_sqrtmat_cx: attempting sympd optimisation");

    // if matrix S is sympd, all its eigenvalues are positive

    Col<T> eigval;
    Mat<eT> eigvec;

    const bool eig_status = eig_sym_helper(eigval, eigvec, S, 'd', "sqrtmat()");

    if (eig_status) {
      // ensure each eigenvalue is > 0

      const uword N = eigval.n_elem;
      const T* eigval_mem = eigval.memptr();

      bool all_pos = true;

      for (uword i = 0; i < N; ++i) {
        all_pos = (eigval_mem[i] <= T(0)) ? false : all_pos;
      }

      if (all_pos) {
        eigval = sqrt(eigval);

        out = eigvec * diagmat(eigval) * eigvec.t();

        return true;
      }
    }

    arma_debug_print("op_sqrtmat_cx: sympd optimisation failed");

    // fallthrough if eigen decomposition failed or an eigenvalue is <= 0
  }

  const bool schur_ok = auxlib::schur(U, S);

  if (schur_ok == false) {
    arma_debug_print("sqrtmat(): schur decomposition failed");
    out.soft_reset();
    return false;
  }

  const bool status = op_sqrtmat_cx::helper(S);

  const Mat<eT> X = U * S;

  S.reset();

  out = X * U.t();

  return status;
}

template <typename T>
inline bool op_sqrtmat_cx::helper(Mat<std::complex<T> >& S) {
  typedef typename std::complex<T> eT;

  if (S.is_empty()) {
    return true;
  }

  const uword N = S.n_rows;

  const eT zero = eT(0);

  eT& S_00 = S[0];

  bool singular = (S_00 == zero);

  S_00 = std::sqrt(S_00);

  for (uword j = 1; j < N; ++j) {
    eT* S_j = S.colptr(j);

    eT& S_jj = S_j[j];

    singular = (singular || (S_jj == zero));

    S_jj = std::sqrt(S_jj);

    for (uword ii = 0; ii <= (j - 1); ++ii) {
      const uword i = (j - 1) - ii;

      const eT* S_i = S.colptr(i);

      // S_j[i] /= (S_i[i] + S_j[j]);
      S_j[i] /= (S_i[i] + S_jj);

      for (uword k = 0; k < i; ++k) {
        S_j[k] -= S_i[k] * S_j[i];
      }
    }
  }

  return (singular) ? false : true;
}

template <typename T1>
inline void op_sqrtmat_sympd::apply(Mat<typename T1::elem_type>& out,
                                    const Op<T1, op_sqrtmat_sympd>& in) {
  arma_debug_sigprint();

  const bool status = op_sqrtmat_sympd::apply_direct(out, in.m);

  if (status == false) {
    out.soft_reset();
    arma_stop_runtime_error("sqrtmat_sympd(): transformation failed");
  }
}

template <typename T1>
inline bool op_sqrtmat_sympd::apply_direct(Mat<typename T1::elem_type>& out,
                                           const Base<typename T1::elem_type, T1>& expr) {
  arma_debug_sigprint();

#if defined(ARMA_USE_LAPACK)
  {
    typedef typename T1::elem_type eT;
    typedef typename T1::pod_type T;

    const unwrap<T1> U(expr.get_ref());
    const Mat<eT>& X = U.M;

    arma_conform_check((X.is_square() == false),
                       "sqrtmat_sympd(): given matrix must be square sized");

    if ((arma_config::check_conform) && (is_cx<eT>::yes) &&
        (sym_helper::check_diag_imag(X) == false)) {
      arma_warn(1, "sqrtmat_sympd(): imaginary components on the diagonal are non-zero");
    }

    if (is_op_diagmat<T1>::value || X.is_diagmat()) {
      arma_debug_print("op_sqrtmat_sympd: diag optimisation");

      out = X;

      eT* colmem = out.memptr();

      const uword N = X.n_rows;

      for (uword i = 0; i < N; ++i) {
        eT& out_ii = colmem[i];
        T out_ii_real = access::tmp_real(out_ii);

        if (out_ii_real < T(0)) {
          return false;
        }

        out_ii = std::sqrt(out_ii);

        colmem += N;
      }

      return true;
    }

    Col<T> eigval;
    Mat<eT> eigvec;

    const bool status = eig_sym_helper(eigval, eigvec, X, 'd', "sqrtmat_sympd()");

    if (status == false) {
      return false;
    }

    const uword N = eigval.n_elem;
    const T* eigval_mem = eigval.memptr();

    bool all_pos = true;

    for (uword i = 0; i < N; ++i) {
      all_pos = (eigval_mem[i] < T(0)) ? false : all_pos;
    }

    if (all_pos == false) {
      return false;
    }

    eigval = sqrt(eigval);

    out = eigvec * diagmat(eigval) * eigvec.t();

    return true;
  }
#else
  {
    arma_ignore(out);
    arma_ignore(expr);
    arma_stop_logic_error("sqrtmat_sympd(): use of LAPACK must be enabled");
    return false;
  }
#endif
}

//! @}
