expm_z.cc

// -*- mode: c++; fill-column: 80; c-basic-offset: 2; indent-tabs-mode: nil -*-
/*
    Copyright (c) 2010 Juan Jose Garcia Ripoll

    Tensor is free software; you can redistribute it and/or modify it
    under the terms of the GNU Library General Public License as published
    by the Free Software Foundation; either version 2 of the License, or
    (at your option) any later version.

    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    GNU Library General Public License for more details.

    You should have received a copy of the GNU General Public License along
    with this program; if not, write to the Free Software Foundation, Inc.,
    51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
*/

#include <algorithm>
#include <tensor/tensor.h>
#include <tensor/linalg.h>

namespace linalg {
  const CTensor
  expm(const CTensor &Aunorm, unsigned int order)
  {
    assert(Aunorm.rank() == 2);
    assert(Aunorm.columns() == Aunorm.rows());

    // Scale A until the norm is < 1/2
    double val = log2(matrix_norminf(Aunorm));
    int e = (int)floor(val);
    size_t j = std::max(0, e+1);

    // Pade approximation for exp(A)
    double c = 1.0/2;
    CTensor A = Aunorm / exp2(j);
    CTensor N = CTensor::eye(A.rows()) + c * A;
    CTensor D = CTensor::eye(A.rows()) - c * A;
    CTensor X = A;

    for (size_t k = 2; k <= order; k++) {
      c = c * (order-k+1) / (k*(2*order-k+1));
      X = fold(A, -1, X, 0);
      CTensor cX = c * X;
      N = N + cX;
      if ((k & 1) == 0) {
        D = D + cX;
      } else {
        D = D - cX;
      }
    }
    X = solve(D, N);
    for (size_t k = 1; k <= j; k++) {
      X = fold(X, -1, X, 0);
    }
    return X;
  }
}