expm-1.h

//
//  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(j,i)); // Correct me, if H is complex, can I use that abs?
      norm = std::max<real_value_type>(norm, temp);
    }
  if (norm == 0.0) 
    {
      boost::throw_exception(boost::numeric::ublas::bad_argument());
      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
  }
  else
    U.assign(H);
  // 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