text: // Copyright (C) 2014-2016 National ICT Australia (NICTA)
  // 
  // This Source Code Form is subject to the terms of the Mozilla Public
  // License, v. 2.0. If a copy of the MPL was not distributed with this
  // file, You can obtain one at http://mozilla.org/MPL/2.0/.
  // -------------------------------------------------------------------
  // 
  // Written by Conrad Sanderson - http://conradsanderson.id.au
  
  
  
  //! \addtogroup op_expmat
  //! @{
  
  
  //! implementation based on:
  //! Cleve Moler, Charles Van Loan.
  //! Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later.
  //! SIAM Review, Vol. 45, No. 1, 2003, pp. 3-49.
  //! http://dx.doi.org/10.1137/S00361445024180
  
  
  template<typename T1>
  inline
  void
  op_expmat::apply(Mat<typename T1::elem_type>& out, const Op<T1, op_expmat>& expr)
    {
    arma_extra_debug_sigprint();
    
    const bool status = op_expmat::apply_direct(out, expr.m);
    
    if(status == false)
      {
      out.reset();
      arma_stop_runtime_error("expmat(): given matrix appears ill-conditioned");
      }
    }
  
  
  
  template<typename T1>
  inline
  bool
  op_expmat::apply_direct(Mat<typename T1::elem_type>& out, const Base<typename T1::elem_type, T1>& X)
    {
    arma_extra_debug_sigprint();
    
    typedef typename T1::elem_type eT;
    typedef typename T1::pod_type   T;
    
    if(is_op_diagmat<T1>::value)
      {
      out = X.get_ref();  // force the evaluation of diagmat()
      
      arma_debug_check( (out.is_square() == false), "expmat(): given matrix must be square sized" );
      
      const uword N = (std::min)(out.n_rows, out.n_cols);
      
      for(uword i=0; i<N; ++i)
        {
        out.at(i,i) = std::exp( out.at(i,i) );
        }
      }
    else
      {
      Mat<eT> A = X.get_ref();
      
      arma_debug_check( (A.is_square() == false), "expmat(): given matrix must be square sized" );
      
      const T norm_val = arma::norm(A, "inf");
      
      const double log2_val = (norm_val > T(0)) ? double(eop_aux::log2(norm_val)) : double(0);
      
      int exponent = int(0);  std::frexp(log2_val, &exponent);
      
      const uword s = uword( (std::max)(int(0), exponent + int(1)) );
      
      A /= eT(eop_aux::pow(double(2), double(s)));
      
      T c = T(0.5);
      
      Mat<eT> E(A.n_rows, A.n_rows, fill::eye);  E += c * A;
      Mat<eT> D(A.n_rows, A.n_rows, fill::eye);  D -= c * A;
      
      Mat<eT> X = A;
      
      bool positive = true;
      
      const uword N = 6;
      
      for(uword i = 2; i <= N; ++i)
        {
        c = c * T(N - i + 1) / T(i * (2*N - i + 1));
        
        X = A * X;
        
        E += c * X;
        
        if(positive)  { D += c * X; }  else  { D -= c * X; }
        
        positive = (positive) ? false : true;
        }
      
      if( (D.is_finite() == false) || (E.is_finite() == false) )  { return false; }
      
      const bool status = solve(out, D, E);
      
      if(status == false)  { return false; }
      
      for(uword i=0; i < s; ++i)  { out = out * out; }
      }
    
    return true;
    }
  
  
  
  //! @}
  