EigenQP.cpp

/* 
 
 Author: Luca Di Gaspero
 DIEGM - University of Udine, Italy
 l.digaspero@uniud.it
 http://www.diegm.uniud.it/digaspero/
 
 LICENSE 
 
 This file is part of QuadProg++: a C++ library implementing
 the algorithm of Goldfarb and Idnani for the solution of a (convex) 
 Quadratic Programming problem by means of an active-set dual method.
 Copyright (C) 2007-2009 Luca Di Gaspero.
 Copyright (C) 2009 Eric Moyer.  
 
 QuadProg++ is free software: you can redistribute it and/or modify
 it under the terms of the GNU Lesser General Public License as published by
 the Free Software Foundation, either version 3 of the License, or
 (at your option) any later version.
 
 QuadProg++ 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 Lesser General Public License for more details.
 
 You should have received a copy of the GNU Lesser General Public License
 along with QuadProg++. If not, see <http://www.gnu.org/licenses/>.
 
 */

#include <iostream>
#include <algorithm>
#include <cmath>
#include <limits>
#include <sstream>
#include <stdexcept>
#include "EigenQP.h"
#include <vector>
//#define TRACE_SOLVER
//#include <boost/numeric/ublas/vector.hpp>
//#include <boost/numeric/ublas/matrix.hpp>

using namespace Eigen;
using std::vector;
//namespace ublas = boost::numeric::ublas;

namespace QP {
  
// Utility functions for updating some data needed by the solution method 
void compute_d(VectorXd& d, const MatrixXd& J, const VectorXd& np);
void update_z(VectorXd& z, const MatrixXd& J, const VectorXd& d, int iq);
void update_r(const MatrixXd& R, VectorXd& r, const VectorXd& d, int iq);
bool add_constraint(MatrixXd& R, MatrixXd& J, VectorXd& d, int& iq, double& rnorm);
void delete_constraint(MatrixXd& R, MatrixXd& J, VectorXi& A, VectorXd& u, int n, int p, int& iq, int l);

// Utility functions for computing the Cholesky decomposition and solving
// linear systems
void cholesky_decomposition(MatrixXd& A);
void cholesky_solve(const MatrixXd& L, VectorXd& x, const VectorXd& b);
void forward_elimination(const MatrixXd& L, VectorXd& y, const VectorXd& b);
void backward_elimination(const MatrixXd& U, VectorXd& x, const VectorXd& y);

// Utility functions for computing the scalar product and the euclidean 
// distance between two numbers
double scalar_product(const VectorXd& x, const VectorXd& y);
double distance(double a, double b);

// Utility functions for printing ublas::vectors and matrices
void print_matrix(const char* name, const MatrixXd& A, int n = -1, int m = -1);

  //template<typename T>
//void print_vector(const char* name, const ublas::vector<T>& v, int n = -1);

// The Solving function, implementing the Goldfarb-Idnani method

double solve_quadprog(MatrixXd& G, VectorXd& g0, 
                      const MatrixXd& CE, const VectorXd& ce0,  
                      const MatrixXd& CI, const VectorXd& ci0, 
                      VectorXd& x)
{
  std::ostringstream msg;
  {
    //Ensure that the dimensions of the matrices and ublas::vectors can be
    //safely converted from unsigned int into to int without overflow.
    unsigned mx = std::numeric_limits<int>::max();
    if(G.cols() >= mx || G.rows() >= mx || 
       CE.rows() >= mx || CE.cols() >= mx ||
       CI.rows() >= mx || CI.cols() >= mx || 
       ci0.size() >= mx || ce0.size() >= mx || g0.size() >= mx){
      msg << "The dimensions of one of the input matrices or ublas::vectors were "
	  << "too large." << std::endl
	  << "The maximum allowable size for inputs to solve_quadprog is:"
	  << mx << std::endl;
      throw std::logic_error(msg.str());
    }
  }
  int n = G.cols(), p = CE.cols(), m = CI.cols();
  if ((int)G.rows() != n)
  {
    msg << "The ublas::matrix G is not a square ublas::matrix (" << G.rows() << " x " 
	<< G.cols() << ")";
    throw std::logic_error(msg.str());
  }
  if ((int)CE.rows() != n)
  {
    msg << "The ublas::matrix CE is incompatible (incorrect number of rows " 
	<< CE.rows() << " , expecting " << n << ")";
    throw std::logic_error(msg.str());
  }
  if ((int)ce0.size() != p)
  {
    msg << "The ublas::vector ce0 is incompatible (incorrect dimension " 
	<< ce0.size() << ", expecting " << p << ")";
    throw std::logic_error(msg.str());
  }
  if ((int)CI.rows() != n)
  {
    msg << "The ublas::matrix CI is incompatible (incorrect number of rows " 
	<< CI.rows() << " , expecting " << n << ")";
    throw std::logic_error(msg.str());
  }
  if ((int)ci0.size() != m)
  {
    msg << "The ublas::vector ci0 is incompatible (incorrect dimension " 
	<< ci0.size() << ", expecting " << m << ")";
    throw std::logic_error(msg.str());
  }
  x.resize(n);
  register int i, j, k, l; /* indices */
  int ip; // this is the index of the constraint to be added to the active set
  MatrixXd R(n, n), J(n, n);
  VectorXd s(m + p), z(n), r(m + p), d(n), np(n), u(m + p), x_old(n), u_old(m + p);
  double f_value, psi, c1, c2, sum, ss, R_norm;
  double inf;
  if (std::numeric_limits<double>::has_infinity)
    inf = std::numeric_limits<double>::infinity();
  else
    inf = 1.0E300;
  double t, t1, t2; /* t is the step lenght, which is the minimum of the partial step length t1 
    * and the full step length t2 */
  VectorXi A(m + p), A_old(m + p), iai(m + p);
  int q, iq, iter = 0;
  vector<bool> iaexcl(m + p);
	
  /* p is the number of equality constraints */
  /* m is the number of inequality constraints */
  q = 0;  /* size of the active set A (containing the indices of the active constraints) */
#ifdef TRACE_SOLVER
  std::cout << std::endl << "Starting solve_quadprog" << std::endl;
  print_ublas::matrix("G", G);
  print_ublas::vector("g0", g0);
  print_ublas::matrix("CE", CE);
  print_ublas::vector("ce0", ce0);
  print_ublas::matrix("CI", CI);
  print_ublas::vector("ci0", ci0);
#endif  
  
  /*
   * Preprocessing phase
   */
	
  /* compute the trace of the original ublas::matrix G */
  c1 = 0.0;
  for (i = 0; i < n; i++)
  {
    c1 += G(i, i);
  }
  /* decompose the ublas::matrix G in the form L^T L */
  cholesky_decomposition(G);
#ifdef TRACE_SOLVER
  print_matrix("G", G);
#endif
  /* initialize the ublas::matrix R */
  for (i = 0; i < n; i++)
  {
    d(i) = 0.0;
    for (j = 0; j < n; j++)
      R(i, j) = 0.0;
  }
  R_norm = 1.0; /* this variable will hold the norm of the ublas::matrix R */
  
  /* compute the inverse of the factorized ublas::matrix G^-1, this is the initial value for H */
  c2 = 0.0;
  for (i = 0; i < n; i++) 
  {
    d(i) = 1.0;
    forward_elimination(G, z, d);
    for (j = 0; j < n; j++)
      J(i, j) = z(j);
    c2 += z(i);
    d(i) = 0.0;
  }
#ifdef TRACE_SOLVER
  print_ublas::matrix("J", J);
#endif
  
  /* c1 * c2 is an estimate for cond(G) */
  
  /* 
    * Find the unconstrained minimizer of the quadratic form 0.5 * x G x + g0 x 
   * this is a feasible point in the dual space
   * x = G^-1 * g0
   */
  cholesky_solve(G, x, g0);
  for (i = 0; i < n; i++)
    x(i) = -x(i);
  /* and compute the current solution value */ 
  f_value = 0.5 * scalar_product(g0, x);
#ifdef TRACE_SOLVER
  std::cout << "Unconstrained solution: " << f_value << std::endl;
  print_ublas::vector("x", x);
#endif
  
  /* Add equality constraints to the working set A */
  iq = 0;
  for (i = 0; i < p; i++)
  {
    for (j = 0; j < n; j++)
      np(j) = CE(j, i);
    compute_d(d, J, np);
    update_z(z, J, d, iq);
    update_r(R, r, d, iq);
#ifdef TRACE_SOLVER
    print_ublas::matrix("R", R, n, iq);
    print_ublas::vector("z", z);
    print_ublas::vector("r", r, iq);
    print_ublas::vector("d", d);
#endif
    
    /* compute full step length t2: i.e., the minimum step in primal space s.t. the contraint 
      becomes feasible */
    t2 = 0.0;
    if (fabs(scalar_product(z, z)) > std::numeric_limits<double>::epsilon()) // i.e. z != 0
      t2 = (-scalar_product(np, x) - ce0(i)) / scalar_product(z, np);
    
    /* set x = x + t2 * z */
    for (k = 0; k < n; k++)
      x(k) += t2 * z(k);
    
    /* set u = u+ */
    u(iq) = t2;
    for (k = 0; k < iq; k++)
      u(k) -= t2 * r(k);
    
    /* compute the new solution value */
    f_value += 0.5 * (t2 * t2) * scalar_product(z, np);
    A(i) = -i - 1;
    
    if (!add_constraint(R, J, d, iq, R_norm))
    {	  
      // Equality constraints are linearly dependent
      throw std::runtime_error("Constraints are linearly dependent");
      return f_value;
    }
  }
  
  /* set iai = K \ A */
  for (i = 0; i < m; i++)
    iai(i) = i;
  
l1:	iter++;
#ifdef TRACE_SOLVER
  print_ublas::vector("x", x);
#endif
  /* step 1: choose a violated constraint */
  for (i = p; i < iq; i++)
  {
    ip = A(i);
    iai(ip) = -1;
  }
	
  /* compute s(x) = ci^T * x + ci0 for all elements of K \ A */
  ss = 0.0;
  psi = 0.0; /* this value will contain the sum of all infeasibilities */
  ip = 0; /* ip will be the index of the chosen violated constraint */
  for (i = 0; i < m; i++)
  {
    iaexcl[i] = true;
    sum = 0.0;
    for (j = 0; j < n; j++)
      sum += CI(j, i) * x(j);
    sum += ci0(i);
    s(i) = sum;
    psi += std::min(0.0, sum);
  }
#ifdef TRACE_SOLVER
  print_ublas::vector("s", s, m);
#endif
  
  
  if (fabs(psi) <= m * std::numeric_limits<double>::epsilon() * c1 * c2* 100.0)
  {
    /* numerically there are not infeasibilities anymore */
    q = iq;
    
    return f_value;
  }
  
  /* save old values for u and A */
  for (i = 0; i < iq; i++)
  {
    u_old(i) = u(i);
    A_old(i) = A(i);
  }
  /* and for x */
  for (i = 0; i < n; i++)
    x_old(i) = x(i);
  
l2: /* Step 2: check for feasibility and determine a new S-pair */
    for (i = 0; i < m; i++)
    {
      if (s(i) < ss && iai(i) != -1 && iaexcl[i])
      {
        ss = s(i);
        ip = i;
      }
    }
  if (ss >= 0.0)
  {
    q = iq;
    
    return f_value;
  }
  
  /* set np = n(ip) */
  for (i = 0; i < n; i++)
    np(i) = CI(i, ip);
  /* set u = (u 0)^T */
  u(iq) = 0.0;
  /* add ip to the active set A */
  A(iq) = ip;
  
#ifdef TRACE_SOLVER
  std::cout << "Trying with constraint " << ip << std::endl;
  print_ublas::vector("np", np);
#endif
  
l2a:/* Step 2a: determine step direction */
    /* compute z = H np: the step direction in the primal space (through J, see the paper) */
    compute_d(d, J, np);
  update_z(z, J, d, iq);
  /* compute N* np (if q > 0): the negative of the step direction in the dual space */
  update_r(R, r, d, iq);
#ifdef TRACE_SOLVER
  std::cout << "Step direction z" << std::endl;
  print_ublas::vector("z", z);
  print_ublas::vector("r", r, iq + 1);
  print_ublas::vector("u", u, iq + 1);
  print_ublas::vector("d", d);
  print_ublas::vector("A", A, iq + 1);
#endif
  
  /* Step 2b: compute step length */
  l = 0;
  /* Compute t1: partial step length (maximum step in dual space without violating dual feasibility */
  t1 = inf; /* +inf */
  /* find the index l s.t. it reaches the minimum of u+(x) / r */
  for (k = p; k < iq; k++)
  {
    if (r(k) > 0.0)
    {
      if (u(k) / r(k) < t1)
	    {
	      t1 = u(k) / r(k);
	      l = A(k);
	    }
    }
  }
  /* Compute t2: full step length (minimum step in primal space such that the constraint ip becomes feasible */
  if (fabs(scalar_product(z, z))  > std::numeric_limits<double>::epsilon()) // i.e. z != 0
    t2 = -s(ip) / scalar_product(z, np);
  else
    t2 = inf; /* +inf */
  
  /* the step is chosen as the minimum of t1 and t2 */
  t = std::min(t1, t2);
#ifdef TRACE_SOLVER
  std::cout << "Step sizes: " << t << " (t1 = " << t1 << ", t2 = " << t2 << ") ";
#endif
  
  /* Step 2c: determine new S-pair and take step: */
  
  /* case (i): no step in primal or dual space */
  if (t >= inf)
  {
    /* QPP is infeasible */
    // FIXME: unbounded to raise
    q = iq;
    return inf;
  }
  /* case (ii): step in dual space */
  if (t2 >= inf)
  {
    /* set u = u +  t * (-r 1) and drop constraint l from the active set A */
    for (k = 0; k < iq; k++)
      u(k) -= t * r(k);
    u(iq) += t;
    iai(l) = l;
    delete_constraint(R, J, A, u, n, p, iq, l);
#ifdef TRACE_SOLVER
    std::cout << " in dual space: " 
      << f_value << std::endl;
    print_ublas::vector("x", x);
    print_ublas::vector("z", z);
    print_ublas::vector("A", A, iq + 1);
#endif
    goto l2a;
  }
  
  /* case (iii): step in primal and dual space */
  
  /* set x = x + t * z */
  for (k = 0; k < n; k++)
    x(k) += t * z(k);
  /* update the solution value */
  f_value += t * scalar_product(z, np) * (0.5 * t + u(iq));
  /* u = u + t * (-r 1) */
  for (k = 0; k < iq; k++)
    u(k) -= t * r(k);
  u(iq) += t;
#ifdef TRACE_SOLVER
  std::cout << " in both spaces: " 
    << f_value << std::endl;
  print_ublas::vector("x", x);
  print_ublas::vector("u", u, iq + 1);
  print_ublas::vector("r", r, iq + 1);
  print_ublas::vector("A", A, iq + 1);
#endif
  
  if (fabs(t - t2) < std::numeric_limits<double>::epsilon())
  {
#ifdef TRACE_SOLVER
    std::cout << "Full step has taken " << t << std::endl;
    print_ublas::vector("x", x);
#endif
    /* full step has taken */
    /* add constraint ip to the active set*/
    if (!add_constraint(R, J, d, iq, R_norm))
    {
      iaexcl[ip] = false;
      delete_constraint(R, J, A, u, n, p, iq, ip);
#ifdef TRACE_SOLVER
      print_ublas::matrix("R", R);
      print_ublas::vector("A", A, iq);
			print_ublas::vector("iai", iai);
#endif
      for (i = 0; i < m; i++)
        iai(i) = i;
      for (i = p; i < iq; i++)
	    {
	      A(i) = A_old(i);
	      u(i) = u_old(i);
				iai(A(i)) = -1;
	    }
      for (i = 0; i < n; i++)
        x(i) = x_old(i);
      goto l2; /* go to step 2 */
    }    
    else
      iai(ip) = -1;
#ifdef TRACE_SOLVER
    print_ublas::matrix("R", R);
    print_ublas::vector("A", A, iq);
		print_ublas::vector("iai", iai);
#endif
    goto l1;
  }
  
  /* a patial step has taken */
#ifdef TRACE_SOLVER
  std::cout << "Partial step has taken " << t << std::endl;
  print_ublas::vector("x", x);
#endif
  /* drop constraint l */
  iai(l) = l;
  delete_constraint(R, J, A, u, n, p, iq, l);
#ifdef TRACE_SOLVER
  print_ublas::matrix("R", R);
  print_ublas::vector("A", A, iq);
#endif
  
  /* update s(ip) = CI * x + ci0 */
  sum = 0.0;
  for (k = 0; k < n; k++)
    sum += CI(k, ip) * x(k);
  s(ip) = sum + ci0(ip);
  
#ifdef TRACE_SOLVER
  print_ublas::vector("s", s, m);
#endif
  goto l2a;
}

inline void compute_d(VectorXd& d, const MatrixXd& J, const VectorXd& np)
{
  register int i, j, n = d.size();
  register double sum;
  
  /* compute d = H^T * np */
  for (i = 0; i < n; i++)
  {
    sum = 0.0;
    for (j = 0; j < n; j++)
      sum += J(j, i) * np(j);
    d(i) = sum;
  }
}

inline void update_z(VectorXd& z, const MatrixXd& J, const VectorXd& d, int iq)
{
  register int i, j, n = z.size();
	
  /* setting of z = H * d */
  for (i = 0; i < n; i++)
  {
    z(i) = 0.0;
    for (j = iq; j < n; j++)
      z(i) += J(i, j) * d(j);
  }
}

inline void update_r(const MatrixXd& R, VectorXd& r, const VectorXd& d, int iq)
{
  register int i, j;/*, n = d.size();*/
  register double sum;
  
  /* setting of r = R^-1 d */
  for (i = iq - 1; i >= 0; i--)
  {
    sum = 0.0;
    for (j = i + 1; j < iq; j++)
      sum += R(i, j) * r(j);
    r(i) = (d(i) - sum) / R(i, i);
  }
}

bool add_constraint(MatrixXd& R, MatrixXd& J, VectorXd& d, int& iq, double& R_norm)
{
  int n = d.size();
#ifdef TRACE_SOLVER
  std::cout << "Add constraint " << iq << '/';
#endif
  register int i, j, k;
  double cc, ss, h, t1, t2, xny;
	
  /* we have to find the Givens rotation which will reduce the element
    d(j) to zero.
    if it is already zero we don't have to do anything, except of
    decreasing j */  
  for (j = n - 1; j >= iq + 1; j--)
  {
    /* The Givens rotation is done with the ublas::matrix (cc cs, cs -cc).
    If cc is one, then element (j) of d is zero compared with element
    (j - 1). Hence we don't have to do anything. 
    If cc is zero, then we just have to switch column (j) and column (j - 1) 
    of J. Since we only switch columns in J, we have to be careful how we
    update d depending on the sign of gs.
    Otherwise we have to apply the Givens rotation to these columns.
    The i - 1 element of d has to be updated to h. */
    cc = d(j - 1);
    ss = d(j);
    h = distance(cc, ss);
    if (fabs(h) < std::numeric_limits<double>::epsilon()) // h == 0
      continue;
    d(j) = 0.0;
    ss = ss / h;
    cc = cc / h;
    if (cc < 0.0)
    {
      cc = -cc;
      ss = -ss;
      d(j - 1) = -h;
    }
    else
      d(j - 1) = h;
    xny = ss / (1.0 + cc);
    for (k = 0; k < n; k++)
    {
      t1 = J(k, j - 1);
      t2 = J(k, j);
      J(k, j - 1) = t1 * cc + t2 * ss;
      J(k, j) = xny * (t1 + J(k, j - 1)) - t2;
    }
  }
  /* update the number of constraints added*/
  iq++;
  /* To update R we have to put the iq components of the d ublas::vector
    into column iq - 1 of R
    */
  for (i = 0; i < iq; i++)
    R(i, iq - 1) = d(i);
#ifdef TRACE_SOLVER
  std::cout << iq << std::endl;
  print_ublas::matrix("R", R, iq, iq);
  print_ublas::matrix("J", J);
  print_ublas::vector("d", d, iq);
#endif
  
  if (fabs(d(iq - 1)) <= std::numeric_limits<double>::epsilon() * R_norm) 
  {
    // problem degenerate
    return false;
  }
  R_norm = std::max<double>(R_norm, fabs(d(iq - 1)));
  return true;
}

void delete_constraint(MatrixXd& R, MatrixXd& J, VectorXi& A, VectorXd& u, int n, int p, int& iq, int l)
{
#ifdef TRACE_SOLVER
  std::cout << "Delete constraint " << l << ' ' << iq;
#endif
  register int i, j, k, qq = -1; // just to prevent warnings from smart compilers
  double cc, ss, h, xny, t1, t2;
  
  /* Find the index qq for active constraint l to be removed */
  for (i = p; i < iq; i++)
    if (A(i) == l)
    {
      qq = i;
      break;
    }
      
  /* remove the constraint from the active set and the duals */
  for (i = qq; i < iq - 1; i++)
    {
      A(i) = A(i + 1);
      u(i) = u(i + 1);
      for (j = 0; j < n; j++)
        R(j, i) = R(j, i + 1);
    }
      
  A(iq - 1) = A(iq);
  u(iq - 1) = u(iq);
  A(iq) = 0; 
  u(iq) = 0.0;
  for (j = 0; j < iq; j++)
    R(j, iq - 1) = 0.0;
  /* constraint has been fully removed */
  iq--;
#ifdef TRACE_SOLVER
  std::cout << '/' << iq << std::endl;
#endif 
  
  if (iq == 0)
    return;
  
  for (j = qq; j < iq; j++)
  {
    cc = R(j, j);
    ss = R(j + 1, j);
    h = distance(cc, ss);
    if (fabs(h) < std::numeric_limits<double>::epsilon()) // h == 0
      continue;
    cc = cc / h;
    ss = ss / h;
    R(j + 1, j) = 0.0;
    if (cc < 0.0)
    {
      R(j, j) = -h;
      cc = -cc;
      ss = -ss;
    }
    else
      R(j, j) = h;
    
    xny = ss / (1.0 + cc);
    for (k = j + 1; k < iq; k++)
    {
      t1 = R(j, k);
      t2 = R(j + 1, k);
      R(j, k) = t1 * cc + t2 * ss;
      R(j + 1, k) = xny * (t1 + R(j, k)) - t2;
    }
    for (k = 0; k < n; k++)
    {
      t1 = J(k, j);
      t2 = J(k, j + 1);
      J(k, j) = t1 * cc + t2 * ss;
      J(k, j + 1) = xny * (J(k, j) + t1) - t2;
    }
  }
}

inline double distance(double a, double b)
{
  register double a1, b1, t;
  a1 = fabs(a);
  b1 = fabs(b);
  if (a1 > b1) 
  {
    t = (b1 / a1);
    return a1 * ::std::sqrt(1.0 + t * t);
  }
  else
    if (b1 > a1)
    {
      t = (a1 / b1);
      return b1 * ::std::sqrt(1.0 + t * t);
    }
  return a1 * ::std::sqrt(2.0);
}


inline double scalar_product(const VectorXd& x, const VectorXd& y)
{
  register int i, n = x.size();
  register double sum;
	
  sum = 0.0;
  for (i = 0; i < n; i++)
    sum += x(i) * y(i);
  return sum;			
}

void cholesky_decomposition(MatrixXd& A) 
{
  register int i, j, k, n = A.rows();
  register double sum;
	
  for (i = 0; i < n; i++)
  {
    for (j = i; j < n; j++)
    {
      sum = A(i, j);
      for (k = i - 1; k >= 0; k--)
        sum -= A(i, k)*A(j, k);
      if (i == j) 
	    {
	      if (sum <= 0.0)
        {
          std::ostringstream os;
          // raise error
          print_matrix("A", A);
          os << "Error in cholesky decomposition, sum: " << sum;
          throw std::logic_error(os.str());
          exit(-1);
        }
	      A(i, i) = ::std::sqrt(sum);
	    }
      else
        A(j, i) = sum / A(i, i);
    }
    for (k = i + 1; k < n; k++)
      A(i, k) = A(k, i);
  } 
}

void cholesky_solve(const MatrixXd& L, VectorXd& x, const VectorXd& b)
{
  int n = L.rows();
  VectorXd y(n);
	
  /* Solve L * y = b */
  forward_elimination(L, y, b);
  /* Solve L^T * x = y */
  backward_elimination(L, x, y);
}

inline void forward_elimination(const MatrixXd& L, VectorXd& y, const VectorXd& b)
{
  register int i, j, n = L.rows();
	
  y(0) = b(0) / L(0, 0);
  for (i = 1; i < n; i++)
  {
    y(i) = b(i);
    for (j = 0; j < i; j++)
      y(i) -= L(i, j) * y(j);
    y(i) = y(i) / L(i, i);
  }
}

inline void backward_elimination(const MatrixXd& U, VectorXd& x, const VectorXd& y)
{
  register int i, j, n = U.rows();
	
  x(n - 1) = y(n - 1) / U(n - 1, n - 1);
  for (i = n - 2; i >= 0; i--)
  {
    x(i) = y(i);
    for (j = i + 1; j < n; j++)
      x(i) -= U(i, j) * x(j);
    x(i) = x(i) / U(i, i);
  }
}

void print_matrix(const char* name, const MatrixXd& A, int n, int m)
{
  std::ostringstream s;
  std::string t;
  if (n == -1)
    n = A.rows();
  if (m == -1)
    m = A.cols();
	
  s << name << ": " << std::endl;
  for (int i = 0; i < n; i++)
  {
    s << " ";
    for (int j = 0; j < m; j++)
      s << A(i, j) << ", ";
    s << std::endl;
  }
  t = s.str();
  t = t.substr(0, t.size() - 3); // To remove the trailing space, comma and newline
	
  std::cout << t << std::endl;
}
}  /*
template<typename T>
void print_vector(const char* name, const ublas::vector<T>& v, int n)
{
  std::ostringstream s;
  std::string t;
  if (n == -1)
    n = v.size();
	
  s << name << ": " << std::endl << " ";
  for (int i = 0; i < n; i++)
  {
    s << v(i) << ", ";
  }
  t = s.str();
  t = t.substr(0, t.size() - 2); // To remove the trailing space and comma
	
  std::cout << t << std::endl;
}
}
  */