// 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_extra_debug_sigprint();
  
  const bool status = op_sqrtmat::apply_direct(out, in.m);
  
  if(status == false)
    {
    arma_debug_warn_level(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_extra_debug_sigprint();
  
  typedef typename T1::elem_type T;
  
  const diagmat_proxy<T1> P(expr.m);
  
  arma_debug_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_extra_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_debug_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_extra_debug_print("op_sqrtmat: detected diagonal matrix");
    
    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_extra_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_extra_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_extra_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_extra_debug_sigprint();
  
  const bool status = op_sqrtmat_cx::apply_direct(out, in.m);
  
  if(status == false)
    {
    arma_debug_warn_level(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_extra_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_extra_debug_sigprint();
  
  typedef typename T1::elem_type eT;
  
  arma_debug_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_extra_debug_sigprint();
  
  typedef typename T1::pod_type   T;
  typedef typename T1::elem_type eT;
  
  Mat<eT> U;
  Mat<eT> S = expr.get_ref();
  
  arma_debug_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_extra_debug_print("op_sqrtmat_cx: detected diagonal matrix");
    
    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_extra_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_extra_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_extra_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_extra_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_extra_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_debug_check( (X.is_square() == false), "sqrtmat_sympd(): given matrix must be square sized" );
    
    if((arma_config::debug) && (is_cx<eT>::yes) && (sym_helper::check_diag_imag(X) == false))
      {
      arma_debug_warn_level(1, "sqrtmat_sympd(): imaginary components on the diagonal are non-zero");
      }
    
    if(is_op_diagmat<T1>::value || X.is_diagmat())
      {
      arma_extra_debug_print("op_sqrtmat_sympd: detected diagonal matrix");
      
      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
  }



//! @}
