text: /*
    spline.h
   *
   * simple cubic spline interpolation library without external
   * dependencies
   *
   * ---------------------------------------------------------------------
   * Copyright (C) 2011, 2014 Tino Kluge (ttk448 at gmail.com)
   *
   *  This program is free software; you can redistribute it and/or
   *  modify it under the terms of the GNU 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 General Public License for more details.
   *
   *  You should have received a copy of the GNU General Public License
   *  along with this program.  If not, see <http://www.gnu.org/licenses/>.
   * ---------------------------------------------------------------------
   *
   */
  
  
  #ifndef TK_SPLINE_H
  #define TK_SPLINE_H
  
  #include <cstdio>
  #include <cassert>
  #include <vector>
  #include <algorithm>
  
  
  // unnamed namespace only because the implementation is in this
  // header file and we don't want to export symbols to the obj files
  namespace
  {
  
  namespace tk
  {
  
  // band matrix solver
  class band_matrix
  {
  private:
      std::vector< std::vector<double> > m_upper;  // upper band
      std::vector< std::vector<double> > m_lower;  // lower band
  public:
      band_matrix() {};                             // constructor
      band_matrix(int dim, int n_u, int n_l);       // constructor
      ~band_matrix() {};                            // destructor
      void resize(int dim, int n_u, int n_l);      // init with dim,n_u,n_l
      int dim() const;                             // matrix dimension
      int num_upper() const
      {
          return m_upper.size()-1;
      }
      int num_lower() const
      {
          return m_lower.size()-1;
      }
      // access operator
      double & operator () (int i, int j);            // write
      double   operator () (int i, int j) const;      // read
      // we can store an additional diogonal (in m_lower)
      double& saved_diag(int i);
      double  saved_diag(int i) const;
      void lu_decompose();
      std::vector<double> r_solve(const std::vector<double>& b) const;
      std::vector<double> l_solve(const std::vector<double>& b) const;
      std::vector<double> lu_solve(const std::vector<double>& b,
                                   bool is_lu_decomposed=false);
  
  };
  
  
  // spline interpolation
  class spline
  {
  public:
      enum bd_type {
          first_deriv = 1,
          second_deriv = 2
      };
  
  private:
      std::vector<double> m_x,m_y;            // x,y coordinates of points
      // interpolation parameters
      // f(x) = a*(x-x_i)^3 + b*(x-x_i)^2 + c*(x-x_i) + y_i
      std::vector<double> m_a,m_b,m_c;        // spline coefficients
      double  m_b0, m_c0;                     // for left extrapol
      bd_type m_left, m_right;
      double  m_left_value, m_right_value;
      bool    m_force_linear_extrapolation;
  
  public:
      // set default boundary condition to be zero curvature at both ends
      spline(): m_left(second_deriv), m_right(second_deriv),
          m_left_value(0.0), m_right_value(0.0),
          m_force_linear_extrapolation(false)
      {
          ;
      }
  
      // optional, but if called it has to come be before set_points()
      void set_boundary(bd_type left, double left_value,
                        bd_type right, double right_value,
                        bool force_linear_extrapolation=false);
      void set_points(const std::vector<double>& x,
                      const std::vector<double>& y, bool cubic_spline=true);
      double operator() (double x) const;
  };
  
  
  
  // ---------------------------------------------------------------------
  // implementation part, which could be separated into a cpp file
  // ---------------------------------------------------------------------
  
  
  // band_matrix implementation
  // -------------------------
  
  band_matrix::band_matrix(int dim, int n_u, int n_l)
  {
      resize(dim, n_u, n_l);
  }
  void band_matrix::resize(int dim, int n_u, int n_l)
  {
      assert(dim>0);
      assert(n_u>=0);
      assert(n_l>=0);
      m_upper.resize(n_u+1);
      m_lower.resize(n_l+1);
      for(size_t i=0; i<m_upper.size(); i++) {
          m_upper[i].resize(dim);
      }
      for(size_t i=0; i<m_lower.size(); i++) {
          m_lower[i].resize(dim);
      }
  }
  int band_matrix::dim() const
  {
      if(m_upper.size()>0) {
          return m_upper[0].size();
      } else {
          return 0;
      }
  }
  
  
  // defines the new operator (), so that we can access the elements
  // by A(i,j), index going from i=0,...,dim()-1
  double & band_matrix::operator () (int i, int j)
  {
      int k=j-i;       // what band is the entry
      assert( (i>=0) && (i<dim()) && (j>=0) && (j<dim()) );
      assert( (-num_lower()<=k) && (k<=num_upper()) );
      // k=0 -> diogonal, k<0 lower left part, k>0 upper right part
      if(k>=0)   return m_upper[k][i];
      else	    return m_lower[-k][i];
  }
  double band_matrix::operator () (int i, int j) const
  {
      int k=j-i;       // what band is the entry
      assert( (i>=0) && (i<dim()) && (j>=0) && (j<dim()) );
      assert( (-num_lower()<=k) && (k<=num_upper()) );
      // k=0 -> diogonal, k<0 lower left part, k>0 upper right part
      if(k>=0)   return m_upper[k][i];
      else	    return m_lower[-k][i];
  }
  // second diag (used in LU decomposition), saved in m_lower
  double band_matrix::saved_diag(int i) const
  {
      assert( (i>=0) && (i<dim()) );
      return m_lower[0][i];
  }
  double & band_matrix::saved_diag(int i)
  {
      assert( (i>=0) && (i<dim()) );
      return m_lower[0][i];
  }
  
  // LR-Decomposition of a band matrix
  void band_matrix::lu_decompose()
  {
      int  i_max,j_max;
      int  j_min;
      double x;
  
      // preconditioning
      // normalize column i so that a_ii=1
      for(int i=0; i<this->dim(); i++) {
          assert(this->operator()(i,i)!=0.0);
          this->saved_diag(i)=1.0/this->operator()(i,i);
          j_min=std::max(0,i-this->num_lower());
          j_max=std::min(this->dim()-1,i+this->num_upper());
          for(int j=j_min; j<=j_max; j++) {
              this->operator()(i,j) *= this->saved_diag(i);
          }
          this->operator()(i,i)=1.0;          // prevents rounding errors
      }
  
      // Gauss LR-Decomposition
      for(int k=0; k<this->dim(); k++) {
          i_max=std::min(this->dim()-1,k+this->num_lower());  // num_lower not a mistake!
          for(int i=k+1; i<=i_max; i++) {
              assert(this->operator()(k,k)!=0.0);
              x=-this->operator()(i,k)/this->operator()(k,k);
              this->operator()(i,k)=-x;                         // assembly part of L
              j_max=std::min(this->dim()-1,k+this->num_upper());
              for(int j=k+1; j<=j_max; j++) {
                  // assembly part of R
                  this->operator()(i,j)=this->operator()(i,j)+x*this->operator()(k,j);
              }
          }
      }
  }
  // solves Ly=b
  std::vector<double> band_matrix::l_solve(const std::vector<double>& b) const
  {
      assert( this->dim()==(int)b.size() );
      std::vector<double> x(this->dim());
      int j_start;
      double sum;
      for(int i=0; i<this->dim(); i++) {
          sum=0;
          j_start=std::max(0,i-this->num_lower());
          for(int j=j_start; j<i; j++) sum += this->operator()(i,j)*x[j];
          x[i]=(b[i]*this->saved_diag(i)) - sum;
      }
      return x;
  }
  // solves Rx=y
  std::vector<double> band_matrix::r_solve(const std::vector<double>& b) const
  {
      assert( this->dim()==(int)b.size() );
      std::vector<double> x(this->dim());
      int j_stop;
      double sum;
      for(int i=this->dim()-1; i>=0; i--) {
          sum=0;
          j_stop=std::min(this->dim()-1,i+this->num_upper());
          for(int j=i+1; j<=j_stop; j++) sum += this->operator()(i,j)*x[j];
          x[i]=( b[i] - sum ) / this->operator()(i,i);
      }
      return x;
  }
  
  std::vector<double> band_matrix::lu_solve(const std::vector<double>& b,
          bool is_lu_decomposed)
  {
      assert( this->dim()==(int)b.size() );
      std::vector<double>  x,y;
      if(is_lu_decomposed==false) {
          this->lu_decompose();
      }
      y=this->l_solve(b);
      x=this->r_solve(y);
      return x;
  }
  
  
  
  
  // spline implementation
  // -----------------------
  
  void spline::set_boundary(spline::bd_type left, double left_value,
                            spline::bd_type right, double right_value,
                            bool force_linear_extrapolation)
  {
      assert(m_x.size()==0);          // set_points() must not have happened yet
      m_left=left;
      m_right=right;
      m_left_value=left_value;
      m_right_value=right_value;
      m_force_linear_extrapolation=force_linear_extrapolation;
  }
  
  
  void spline::set_points(const std::vector<double>& x,
                          const std::vector<double>& y, bool cubic_spline)
  {
      assert(x.size()==y.size());
      assert(x.size()>2);
      m_x=x;
      m_y=y;
      int   n=x.size();
      // TODO: maybe sort x and y, rather than returning an error
      for(int i=0; i<n-1; i++) {
          assert(m_x[i]<m_x[i+1]);
      }
  
      if(cubic_spline==true) { // cubic spline interpolation
          // setting up the matrix and right hand side of the equation system
          // for the parameters b[]
          band_matrix A(n,1,1);
          std::vector<double>  rhs(n);
          for(int i=1; i<n-1; i++) {
              A(i,i-1)=1.0/3.0*(x[i]-x[i-1]);
              A(i,i)=2.0/3.0*(x[i+1]-x[i-1]);
              A(i,i+1)=1.0/3.0*(x[i+1]-x[i]);
              rhs[i]=(y[i+1]-y[i])/(x[i+1]-x[i]) - (y[i]-y[i-1])/(x[i]-x[i-1]);
          }
          // boundary conditions
          if(m_left == spline::second_deriv) {
              // 2*b[0] = f''
              A(0,0)=2.0;
              A(0,1)=0.0;
              rhs[0]=m_left_value;
          } else if(m_left == spline::first_deriv) {
              // c[0] = f', needs to be re-expressed in terms of b:
              // (2b[0]+b[1])(x[1]-x[0]) = 3 ((y[1]-y[0])/(x[1]-x[0]) - f')
              A(0,0)=2.0*(x[1]-x[0]);
              A(0,1)=1.0*(x[1]-x[0]);
              rhs[0]=3.0*((y[1]-y[0])/(x[1]-x[0])-m_left_value);
          } else {
              assert(false);
          }
          if(m_right == spline::second_deriv) {
              // 2*b[n-1] = f''
              A(n-1,n-1)=2.0;
              A(n-1,n-2)=0.0;
              rhs[n-1]=m_right_value;
          } else if(m_right == spline::first_deriv) {
              // c[n-1] = f', needs to be re-expressed in terms of b:
              // (b[n-2]+2b[n-1])(x[n-1]-x[n-2])
              // = 3 (f' - (y[n-1]-y[n-2])/(x[n-1]-x[n-2]))
              A(n-1,n-1)=2.0*(x[n-1]-x[n-2]);
              A(n-1,n-2)=1.0*(x[n-1]-x[n-2]);
              rhs[n-1]=3.0*(m_right_value-(y[n-1]-y[n-2])/(x[n-1]-x[n-2]));
          } else {
              assert(false);
          }
  
          // solve the equation system to obtain the parameters b[]
          m_b=A.lu_solve(rhs);
  
          // calculate parameters a[] and c[] based on b[]
          m_a.resize(n);
          m_c.resize(n);
          for(int i=0; i<n-1; i++) {
              m_a[i]=1.0/3.0*(m_b[i+1]-m_b[i])/(x[i+1]-x[i]);
              m_c[i]=(y[i+1]-y[i])/(x[i+1]-x[i])
                     - 1.0/3.0*(2.0*m_b[i]+m_b[i+1])*(x[i+1]-x[i]);
          }
      } else { // linear interpolation
          m_a.resize(n);
          m_b.resize(n);
          m_c.resize(n);
          for(int i=0; i<n-1; i++) {
              m_a[i]=0.0;
              m_b[i]=0.0;
              m_c[i]=(m_y[i+1]-m_y[i])/(m_x[i+1]-m_x[i]);
          }
      }
  
      // for left extrapolation coefficients
      m_b0 = (m_force_linear_extrapolation==false) ? m_b[0] : 0.0;
      m_c0 = m_c[0];
  
      // for the right extrapolation coefficients
      // f_{n-1}(x) = b*(x-x_{n-1})^2 + c*(x-x_{n-1}) + y_{n-1}
      double h=x[n-1]-x[n-2];
      // m_b[n-1] is determined by the boundary condition
      m_a[n-1]=0.0;
      m_c[n-1]=3.0*m_a[n-2]*h*h+2.0*m_b[n-2]*h+m_c[n-2];   // = f'_{n-2}(x_{n-1})
      if(m_force_linear_extrapolation==true)
          m_b[n-1]=0.0;
  }
  
  double spline::operator() (double x) const
  {
      size_t n=m_x.size();
      // find the closest point m_x[idx] < x, idx=0 even if x<m_x[0]
      std::vector<double>::const_iterator it;
      it=std::lower_bound(m_x.begin(),m_x.end(),x);
      int idx=std::max( int(it-m_x.begin())-1, 0);
  
      double h=x-m_x[idx];
      double interpol;
      if(x<m_x[0]) {
          // extrapolation to the left
          interpol=(m_b0*h + m_c0)*h + m_y[0];
      } else if(x>m_x[n-1]) {
          // extrapolation to the right
          interpol=(m_b[n-1]*h + m_c[n-1])*h + m_y[n-1];
      } else {
          // interpolation
          interpol=((m_a[idx]*h + m_b[idx])*h + m_c[idx])*h + m_y[idx];
      }
      return interpol;
  }
  
  
  } // namespace tk
  
  
  } // namespace
  
  #endif /* TK_SPLINE_H */
  