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//
//  Copyright (c) 2007
//  Tsai, Dung-Bang     
//  National Taiwan University, Department of Physics
// 
//  E-Mail : dbtsai (at) gmail.com
//  Begine : 2007/11/20
//  Last modify : 2007/11/22
//  Version : v0.1
//
//  EXPGM_PAD computes the matrix exponential exp(H) for general matrixs,
//  including complex and real matrixs using the irreducible (p,p) degree
//  rational Pade approximation to the exponential 
//  exp(z) = r(z)=(+/-)( I+2*(Q(z)/P(z))).
//
//  Usage : 
//
//    U = expm_pad(H)
//    U = expm_pad(H, p)
//    
//    where p is internally set to 6 (recommended and gererally satisfactory).
//
//  See also MATLAB supplied functions, EXPM and EXPM1.
//
//  Reference :
//  EXPOKIT, Software Package for Computing Matrix Exponentials.
//  ACM - Transactions On Mathematical Software, 24(1):130-156, 1998
//
//  Permission to use, copy, modify, distribute and sell this software
//  and its documentation for any purpose is hereby granted without fee,
//  provided that the above copyright notice appear in all copies and
//  that both that copyright notice and this permission notice appear
//  in supporting documentation.  The authors make no representations
//  about the suitability of this software for any purpose.
//  It is provided "as is" without express or implied warranty.
//

#ifndef _BOOST_UBLAS_EXPM_
#define _BOOST_UBLAS_EXPM_
#include <complex>
#include <boost/numeric/ublas/vector.hpp>
#include <boost/numeric/ublas/matrix.hpp>
#include <boost/numeric/ublas/lu.hpp>

namespace boost { namespace numeric { namespace ublas {

template<typename MATRIX> MATRIX expm_pad(const MATRIX &H, const int p = 6)
{
        typedef typename MATRIX::value_type value_type;
        typedef typename MATRIX::size_type size_type;
        typedef double real_value_type; // Correct me. Need to modify.
        assert(H.size1() == H.size2()); 
        const size_type n = H.size1();
        const identity_matrix<value_type> I(n);
        matrix<value_type> U(n,n),H2(n,n),P(n,n),Q(n,n);
        real_value_type norm = 0.0;

// Calcuate Pade coefficients  (1-based instead of 0-based as in the c vector)
        vector<real_value_type> c(p+2);
        c(1)=1;  
        for(size_type i = 1; i <= p; ++i) 
                c(i+1) = c(i) * ((p + 1.0 - i)/(i * (2.0 * p + 1 - i)));
// Calcuate the infinty norm of H, which is defined as the largest row sum of a matrix
        for(size_type i=0; i<n; ++i)
        {
                real_value_type temp = 0.0;
                for(size_type j=0;j<n;j++)
                        temp += std::abs<real_value_type>(H(i,j)); // Correct me, if H is complex, can I use that abs?
                norm = std::max<real_value_type>(norm, temp);
        }
        if (norm == 0.0) 
        {
                std::cerr<<"Error! Null input in the routine EXPM_PAD.\n";
                exit(0);
        }
// Scaling, seek s such that || H*2^(-s) || < 1/2, and set scale = 2^(-s)
        int s = 0;
        real_value_type scale = 1.0;
        if(norm > 0.5)
        {
                s = std::max<int>(0, static_cast<int>((log(norm) / log(2.0) + 2.0)));
                scale /= static_cast<real_value_type>(std::pow(2.0, s));
                U.assign(scale * H); // Here U is used as temp value due to that H is const
        }
// Horner evaluation of the irreducible fraction, see the following ref above.
// Initialise P (numerator) and Q (denominator) 
        H2.assign( prod(U, U) );
        Q.assign( c(p+1)*I );
        P.assign( c(p)*I );
        size_type odd = 1;
        for( size_type k = p - 1; k > 0; --k)
        {
                if( odd == 1)
                {
                        Q = ( prod(Q, H2) + c(k) * I ); 
                }
                else
                {
                        P = ( prod(P, H2) + c(k) * I );
                }
                odd = 1 - odd;
        }
        if( odd == 1)
        {
                Q = ( prod(Q, U) );     
                Q -= P ;
                //U.assign( -(I + 2*(Q\P)));
        }
        else
        {
                P = (prod(P, U));
                Q -= P;
                //U.assign( I + 2*(Q\P));
        }
// In origine expokit package, they use lapack ZGESV to obtain inverse matrix,
// and in that ZGESV routine, it uses LU decomposition for obtaing inverse matrix.
// Since in ublas, there is no matrix inversion template, I simply use the build-in
// LU decompostion package in ublas, and back substitute by myself.
//
        permutation_matrix<size_type> pm(n); 
        int res = lu_factorize(Q, pm);
        if( res != 0)
        {
                std::cerr << "Error in the matrix inversion in template expm_pad.\n";
                exit(0);
        }
        H2 = I;  // H2 is not needed anymore, so it is temporary used as identity matrix for substituting.
        lu_substitute(Q, pm, H2); 
        if( odd == 1)
                U.assign( -(I + 2.0 * prod(H2, P)));
        else
                U.assign( I + 2.0 * prod(H2, P));
// Squaring 
        for(size_t i = 0; i < s; ++i)
        {
                U = (prod(U,U));
        }
        return U;
}

}}}


#endif

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