// 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_logmat
//! @{


// Partly based on algorithm 11.9 (inverse scaling and squaring algorithm with Schur decomposition) in:
// Nicholas J. Higham.
// Functions of Matrices: Theory and Computation.
// SIAM, 2008.
// ISBN 978-0-89871-646-7


template<typename T1>
inline
void
op_logmat::apply(Mat< std::complex<typename T1::elem_type> >& out, const mtOp<std::complex<typename T1::elem_type>,T1,op_logmat>& in)
  {
  arma_debug_sigprint();
  
  const bool status = op_logmat::apply_direct(out, in.m, in.aux_uword_a);
  
  if(status == false)
    {
    out.soft_reset();
    arma_stop_runtime_error("logmat(): transformation failed");
    }
  }



template<typename T1>
inline
bool
op_logmat::apply_direct(Mat< std::complex<typename T1::elem_type> >& out, const Op<T1,op_diagmat>& expr, const uword)
  {
  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), "logmat(): given matrix must be square sized" );
  
  const uword N = P.n_rows;
  
  out.zeros(N,N);  // aliasing can't happen as op_logmat is defined as cx_mat = op(mat)
  
  for(uword i=0; i<N; ++i)
    {
    const T val = P[i];
    
    if(val >= T(0))
      {
      out.at(i,i) = std::log(val);
      }
    else
      {
      out.at(i,i) = std::log( std::complex<T>(val) );
      }
    }
  
  return true;
  }



template<typename T1>
inline
bool
op_logmat::apply_direct(Mat< std::complex<typename T1::elem_type> >& out, const Base<typename T1::elem_type,T1>& expr, const uword n_iters)
  {
  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), "logmat(): 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::log( std::complex<in_T>( A[0] ) );
    return true;
    }
  
  if(A.is_diagmat())
    {
    arma_debug_print("op_logmat: diag optimisation");
    
    const uword N = A.n_rows;
    
    out.zeros(N,N);  // aliasing can't happen as op_logmat 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::log(val);
        }
      else
        {
        out.at(i,i) = std::log( 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_logmat: 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', "logmat()");
    
    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 = log(eigval);
        
        out = conv_to< Mat<out_T> >::from( eigvec * diagmat(eigval) * eigvec.t() );
        
        return true;
        }
      }
    
    arma_debug_print("op_logmat: sympd optimisation failed");
    
    // fallthrough if eigen decomposition failed or an eigenvalue is <= 0
    }
  
  
  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] );
    }
  
  return op_logmat_cx::apply_common(out, S, n_iters);
  }



template<typename T1>
inline
void
op_logmat_cx::apply(Mat<typename T1::elem_type>& out, const Op<T1,op_logmat_cx>& in)
  {
  arma_debug_sigprint();
  
  const bool status = op_logmat_cx::apply_direct(out, in.m, in.aux_uword_a);
  
  if(status == false)
    {
    out.soft_reset();
    arma_stop_runtime_error("logmat(): transformation failed");
    }
  }



template<typename T1>
inline
bool
op_logmat_cx::apply_direct(Mat<typename T1::elem_type>& out, const Op<T1,op_diagmat>& expr, const uword)
  {
  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_logmat_cx::apply_direct_noalias(tmp, P);
    
    out.steal_mem(tmp);
    }
  else
    {
    status = op_logmat_cx::apply_direct_noalias(out, P);
    }
  
  return status;
  }



template<typename T1>
inline
bool
op_logmat_cx::apply_direct_noalias(Mat<typename T1::elem_type>& out, const diagmat_proxy<T1>& P)
  {
  arma_debug_sigprint();
  
  arma_conform_check( (P.n_rows != P.n_cols), "logmat(): given matrix must be square sized" );
  
  const uword N = P.n_rows;
  
  out.zeros(N,N);
  
  for(uword i=0; i<N; ++i)
    {
    out.at(i,i) = std::log(P[i]);
    }
  
  return true;
  }



template<typename T1>
inline
bool
op_logmat_cx::apply_direct(Mat<typename T1::elem_type>& out, const Base<typename T1::elem_type,T1>& expr, const uword n_iters)
  {
  arma_debug_sigprint();
  
  typedef typename T1::pod_type   T;
  typedef typename T1::elem_type eT;
  
  Mat<eT> S = expr.get_ref();
  
  arma_conform_check( (S.n_rows != S.n_cols), "logmat(): 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::log(S[0]);
    return true;
    }
  
  if(S.is_diagmat())
    {
    arma_debug_print("op_logmat_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::log( 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_logmat_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', "logmat()");
    
    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 = log(eigval);
        
        out = eigvec * diagmat(eigval) * eigvec.t();
        
        return true;
        }
      }
    
    arma_debug_print("op_logmat_cx: sympd optimisation failed");
    
    // fallthrough if eigen decomposition failed or an eigenvalue is <= 0
    }
  
  return op_logmat_cx::apply_common(out, S, n_iters);
  }



template<typename T>
inline
bool
op_logmat_cx::apply_common(Mat< std::complex<T> >& out, Mat< std::complex<T> >& S, const uword n_iters)
  {
  arma_debug_sigprint();
  
  typedef typename std::complex<T> eT;
  
  Mat<eT> U;
  
  const bool schur_ok = auxlib::schur(U,S);
  
  if(schur_ok == false)  { arma_debug_print("logmat(): schur decomposition failed"); return false; }
  
  // NOTE: theta[0] and theta[1] not really used
  double theta[] = { 1.10e-5, 1.82e-3, 1.6206284795015624e-2, 5.3873532631381171e-2, 1.1352802267628681e-1, 1.8662860613541288e-1, 2.642960831111435e-1 };
  
  const uword N = S.n_rows;
  
  uword p = 0;
  uword m = 6;
  
  uword iter = 0;
  
  while(iter < n_iters)
    {
    const T tau = norm( (S - eye< Mat<eT> >(N,N)), 1 );
    
    if(tau <= theta[6])
      {
      p++;
      
      uword j1 = 0;
      uword j2 = 0;
      
      for(uword i=2; i<=6; ++i)  { if( tau      <= theta[i])  { j1 = i; break; } }
      for(uword i=2; i<=6; ++i)  { if((tau/2.0) <= theta[i])  { j2 = i; break; } }
      
      // sanity check, for development purposes only
      arma_conform_check( (j2 > j1), "internal error: op_logmat::apply_direct(): j2 > j1" );
      
      if( ((j1 - j2) <= 1) || (p == 2) )  { m = j1; break; }
      }
    
    const bool sqrtmat_ok = op_sqrtmat_cx::apply_direct(S,S);
    
    if(sqrtmat_ok == false)  { arma_debug_print("logmat(): sqrtmat() failed"); return false; }
    
    iter++;
    }
  
  if(iter >= n_iters)  { arma_warn(2, "logmat(): reached max iterations without full convergence"); }
  
  S.diag() -= eT(1);
  
  if(m >= 1)
    {
    const bool helper_ok = op_logmat_cx::helper(S,m);
    
    if(helper_ok == false)  { return false; }
    }
  
  out = U * S * U.t();
  
  out *= eT(eop_aux::pow(double(2), double(iter)));
  
  return true;
  }



template<typename eT>
inline
bool
op_logmat_cx::helper(Mat<eT>& A, const uword m)
  {
  arma_debug_sigprint();
  
  if(A.internal_has_nonfinite())  { return false; }
  
  const vec indices = regspace<vec>(1,m-1);
  
  mat tmp(m, m, arma_zeros_indicator());
  
  tmp.diag(-1) = indices / sqrt(square(2.0*indices) - 1.0);
  tmp.diag(+1) = indices / sqrt(square(2.0*indices) - 1.0);
  
  vec eigval;
  mat eigvec;
  
  const bool eig_ok = eig_sym_helper(eigval, eigvec, tmp, 'd', "logmat()");
  
  if(eig_ok == false)  { arma_debug_print("logmat(): eig_sym() failed"); return false; }
  
  const vec nodes   = (eigval + 1.0) / 2.0;
  const vec weights = square(eigvec.row(0).t());
  
  const uword N = A.n_rows;
  
  Mat<eT> B(N, N, arma_zeros_indicator());
  
  Mat<eT> X;
  
  for(uword i=0; i < m; ++i)
    {
    // B += weights(i) * solve( (nodes(i)*A + eye< Mat<eT> >(N,N)), A );
    
    //const bool solve_ok = solve( X, (nodes(i)*A + eye< Mat<eT> >(N,N)), A, solve_opts::fast );
    const bool solve_ok = solve( X, trimatu(nodes(i)*A + eye< Mat<eT> >(N,N)), A, solve_opts::no_approx );
    
    if(solve_ok == false)  { arma_debug_print("logmat(): solve() failed"); return false; }
    
    B += weights(i) * X;
    }
  
  A = B;
  
  return true;
  }



template<typename T1>
inline
void
op_logmat_sympd::apply(Mat<typename T1::elem_type>& out, const Op<T1,op_logmat_sympd>& in)
  {
  arma_debug_sigprint();
  
  const bool status = op_logmat_sympd::apply_direct(out, in.m);
  
  if(status == false)
    {
    out.soft_reset();
    arma_stop_runtime_error("logmat_sympd(): transformation failed");
    }
  }



template<typename T1>
inline
bool
op_logmat_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::pod_type   T;
    typedef typename T1::elem_type eT;
    
    const unwrap<T1>   U(expr.get_ref());
    const Mat<eT>& X = U.M;
    
    arma_conform_check( (X.is_square() == false), "logmat_sympd(): given matrix must be square sized" );
    
    if((arma_config::check_conform) && (arma_config::warn_level > 0) && (is_cx<eT>::yes) && (sym_helper::check_diag_imag(X) == false))
      {
      arma_warn(1, "logmat_sympd(): imaginary components on diagonal are non-zero");
      }
    
    if(is_op_diagmat<T1>::value || X.is_diagmat())
      {
      arma_debug_print("op_logmat_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::log(out_ii);
        
        colmem += N;
        }
      
      return true;
      }
    
    Col< T> eigval;
    Mat<eT> eigvec;
    
    const bool status = eig_sym_helper(eigval, eigvec, X, 'd', "logmat_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 = log(eigval);
    
    out = eigvec * diagmat(eigval) * eigvec.t();
    
    return true;
    }
  #else
    {
    arma_ignore(out);
    arma_ignore(expr);
    arma_stop_logic_error("logmat_sympd(): use of LAPACK must be enabled");
    return false;
    }
  #endif
  }



//! @}
