NICE_VisLearning/optimization/quadprog/QuadProg++.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 "vislearning/optimization/quadprog/QuadProg++.h"
#include "core/algebra/CholeskyRobustAuto.h"
#include "core/algebra/CholeskyRobust.h"

#include "core/vector/VectorT.h"
#include "core/vector/MatrixT.h"
#include "core/image/ImageT.h"
#include "core/imagedisplay/ImageDisplay.h"

//#define TRACE_SOLVER
namespace QuadProgPP{
// Utility functions for updating some data needed by the solution method 
void compute_d(Vector<double>& d, const Matrix<double>& J, const Vector<double>& np);
void update_z(Vector<double>& z, const Matrix<double>& J, const Vector<double>& d, int iq);
void update_r(const Matrix<double>& R, Vector<double>& r, const Vector<double>& d, int iq);
bool add_constraint(Matrix<double>& R, Matrix<double>& J, Vector<double>& d, int& iq, double& rnorm);
void delete_constraint(Matrix<double>& R, Matrix<double>& J, Vector<int>& A, Vector<double>& u, int n, int p, int& iq, int l);

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

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

// Utility functions for printing vectors and matrices
void print_matrix(const char* name, const Matrix<double>& A, int n = -1, int m = -1);

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

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

double solve_quadprog(Matrix<double>& G, Vector<double>& g0, 
                      const Matrix<double>& CE, const Vector<double>& ce0,  
                      const Matrix<double>& CI, const Vector<double>& ci0, 
                      Vector<double>& x)
{
  std::ostringstream msg;
  {
    //Ensure that the dimensions of the matrices and vectors can be
    //safely converted from unsigned int into to int without overflow.
    unsigned mx = std::numeric_limits<int>::max();
    if(G.ncols() >= mx || G.nrows() >= mx || 
       CE.nrows() >= mx || CE.ncols() >= mx ||
       CI.nrows() >= mx || CI.ncols() >= mx || 
       ci0.size() >= mx || ce0.size() >= mx || g0.size() >= mx){
      msg << "The dimensions of one of the input matrices or 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.ncols(), p = CE.ncols(), m = CI.ncols();
  if ((int)G.nrows() != n)
  {
    msg << "The matrix G is not a square matrix (" << G.nrows() << " x " 
	<< G.ncols() << ")";
    throw std::logic_error(msg.str());
  }
  if ((int)CE.nrows() != n)
  {
    msg << "The matrix CE is incompatible (incorrect number of rows " 
	<< CE.nrows() << " , expecting " << n << ")";
    throw std::logic_error(msg.str());
  }
  if ((int)ce0.size() != p)
  {
    msg << "The vector ce0 is incompatible (incorrect dimension " 
	<< ce0.size() << ", expecting " << p << ")";
    throw std::logic_error(msg.str());
  }
  if ((int)CI.nrows() != n)
  {
    msg << "The matrix CI is incompatible (incorrect number of rows " 
	<< CI.nrows() << " , expecting " << n << ")";
    throw std::logic_error(msg.str());
  }
  if ((int)ci0.size() != m)
  {
    msg << "The 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
  Matrix<double> R(n, n), J(n, n);
  Vector<double> 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 */
  Vector<int> 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_matrix("G", G);
  print_vector("g0", g0);
  print_matrix("CE", CE);
  print_vector("ce0", ce0);
  print_matrix("CI", CI);
  print_vector("ci0", ci0);
#endif  
  
  /*
   * Preprocessing phase
   */
	
  /* compute the trace of the original matrix G */
  c1 = 0.0;
  for (i = 0; i < n; i++)
  {
    c1 += G[i][i];
  }
  /* decompose the matrix G in the form L^T L */
  cholesky_decomposition(G);
#ifdef TRACE_SOLVER
  print_matrix("G", G);
#endif
  /* initialize the 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 matrix R */
  
  /* compute the inverse of the factorized 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_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_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_matrix("R", R, n, iq);
    print_vector("z", z);
    print_vector("r", r, iq);
    print_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_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_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_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_vector("z", z);
  print_vector("r", r, iq + 1);
  print_vector("u", u, iq + 1);
  print_vector("d", d);
  print_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_vector("x", x);
    print_vector("z", z);
    print_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_vector("x", x);
  print_vector("u", u, iq + 1);
  print_vector("r", r, iq + 1);
  print_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_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_matrix("R", R);
      print_vector("A", A, iq);
			print_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_matrix("R", R);
    print_vector("A", A, iq);
		print_vector("iai", iai);
#endif
    goto l1;
  }
  
  /* a patial step has taken */
#ifdef TRACE_SOLVER
  std::cout << "Partial step has taken " << t << std::endl;
  print_vector("x", x);
#endif
  /* drop constraint l */
  iai[l] = l;
  delete_constraint(R, J, A, u, n, p, iq, l);
#ifdef TRACE_SOLVER
  print_matrix("R", R);
  print_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_vector("s", s, m);
#endif
  goto l2a;
}

inline void compute_d(Vector<double>& d, const Matrix<double>& J, const Vector<double>& 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(Vector<double>& z, const Matrix<double>& J, const Vector<double>& 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 Matrix<double>& R, Vector<double>& r, const Vector<double>& 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(Matrix<double>& R, Matrix<double>& J, Vector<double>& 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 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 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_matrix("R", R, iq, iq);
  print_matrix("J", J);
  print_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(Matrix<double>& R, Matrix<double>& J, Vector<int>& A, Vector<double>& 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 Vector<double>& x, const Vector<double>& 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(Matrix<double>& A) 
{

	register int n = A.nrows(),i,j;
	NICE::Matrix M;
	M.resize(n,n);
	//copy array to nice matrix
	for (i = 0; i < n; i++)
	{
		for (j = 0; j < n; j++)
		{
			M(n,n)= A[i][j];
		}
	}
	NICE::Matrix L;
	L.resize(n,n);
	NICE::CholeskyRobust *cra = new NICE::CholeskyRobustAuto(false);
	cra->robustChol ( M, L );
	//copy back
	for (i = 0; i < n; i++)
	{
		for (j = 0; j <= i; j++)
		{
			A[i][j]=L(i,j);
			A[j][i]=L(i,j);
		}
	}
	delete cra;


/*
  register int i, j, k, n = A.nrows();
  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 Matrix<double>& L, Vector<double>& x, const Vector<double>& b)
{
  int n = L.nrows();
  Vector<double> 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 Matrix<double>& L, Vector<double>& y, const Vector<double>& b)
{
  register int i, j, n = L.nrows();
	
  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 Matrix<double>& U, Vector<double>& x, const Vector<double>& y)
{
  register int i, j, n = U.nrows();
	
  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 Matrix<double>& A, int n, int m)
{
  std::ostringstream s;
  std::string t;
  if (n == -1)
    n = A.nrows();
  if (m == -1)
    m = A.ncols();
	
  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 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;
}
}