interpolation.cc

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/* Copyright (C) 2002-2012 Stefan Buehler <sbuehler@ltu.se>
 
   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, 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, write to the Free Software
   Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307,
   USA. */
 
#include <iostream>
#include <cmath>
#include "array.h"
#include "check_input.h"
#include "interpolation.h"
#include "logic.h"
 
 
 
// File-global constants:
 
 
const Numeric sum_check_epsilon = 1e-6;
 
 
 
const Numeric FD_TOL = 1.5e-3;
 
 
 
 
 
#define LOOPIT(x) for ( const Numeric* x=&t##x.fd[1]; x>=&t##x.fd[0]; --x )
 
 
 
 
 
ostream& operator<<(ostream& os, const GridPos& gp)
{
  os << gp.idx << " " << gp.fd[0] << " " << gp.fd[1] << "\n";
  return os;
}
 
 
 
void gridpos( ArrayOfGridPos& gp,
              ConstVectorView old_grid,
              ConstVectorView new_grid,
              const Numeric&  extpolfac )
{
  const Index n_old = old_grid.nelem();
  const Index n_new = new_grid.nelem();
 
  // Assert that gp has the right size:
  assert( is_size(gp,n_new) );
 
  // Assert, that the old grid has more than one element
  assert( 1 < n_old );
 
  // This function hast two parts, depending on whether old_grid is
  // sorted in ascending or descending order. Maybe that's not too
  // elegant, but it's the most efficient, because in this way there
  // is no additional runtime overhead to handle both cases.
 
  // We use only the first two elements to decide how the grid is
  // sorted. (The rest of the grid is checked later by an assert.)
  // If both values are the same, we still assume the grid is
  // ascending. However, this will lead to an assertion fail later on,
  // because the grid has to be strictly sorted.
  bool ascending = ( old_grid[0] <= old_grid[1] );
 
  if (ascending)  
    {
      // So, old_grid should always be sorted in strictly ascending order.
      // (No duplicate values.)
      // Assert that this is so. This may depend on user input, however,
      // inside this elementary function is not the place to check for
      // that. There must be runtime checks on higher levels to insure
      // that all grids are sorted. The assertion here is just as a last
      // safety check.
      assert( is_increasing(old_grid) );
 
      // Limits of extrapolation. 
      const Numeric og_min = old_grid[0] - 
                                     extpolfac * ( old_grid[1] - old_grid[0] );
      const Numeric og_max = old_grid[n_old-1] + 
                         extpolfac * ( old_grid[n_old-1] - old_grid[n_old-2] );
 
      // We will make no firm assumptions about the new grid. But the case
      // that we have in mind is that it is either also sorted, or at
      // least partially sorted, for example like this:
      // 5 4 3 2 2.5 3 4
      // This kind of sequence should be typical if we interpolate
      // atmospheric fields along a limb propagation path.
 
      // Let's get some idea where the first point in the new grid is,
      // relative to the old grid. We use linear interpolation between the
      // maximum and the minimum of the old grid for this. (Taking
      // into account the small allowed extrapolation.)
      Numeric frac = (new_grid[0]-og_min)/(og_max-og_min);
      
      // We are not checking if the value of frac is reasonable,
      // because there is another assertion below to catch the
      // consequences. 
      
      // Initialize current_position: 
      Index   current_position = (Index) rint(frac*(Numeric)(n_old-2));
 
      // The above statement should satisfy
      // 0 <= current_position <= n_old-2
      // Assert that this is indeed the case. 
//       cout << "frac             = " << frac             << "\n";
//       cout << "current_position = " << current_position << "\n";
      assert( 0<= current_position );
      assert( current_position <= n_old-2 );
 
      // The variables lower and upper are used to remember the value of
      // the old grid at current_position and one above current_position:
      Numeric lower = old_grid[current_position];
      Numeric upper = old_grid[current_position+1];
 
      // Loop over all points in the new grid:
      for ( Index i_new=0; i_new<n_new; ++i_new )
        {
          // Get a reference to the current element of gp:
          GridPos& tgp = gp[i_new];
          // And on the current value of the new grid:
          const Numeric tng = new_grid[i_new];
 
          // Verify that the new grid is within the limits of
          // extrapolation that we have defined by og_min and og_max:
          assert( og_min <= tng    );
          assert( tng    <= og_max );
 
//           cout << "lower / tng / upper = "
//                << lower << " / "
//                << tng   << " / "
//                << upper << "\n";
 
          // Is current_position too high?
          // (The current_position>0 condition is there so that the position
          // stays 0 for extrapolation.)
          if ( tng < lower && current_position > 0 )
            {
              do
                {
                  --current_position;
                  lower = old_grid[current_position];
                }
              while ( tng < lower && current_position > 0 );
 
              upper = old_grid[current_position+1];
 
              tgp.idx = current_position;
              tgp.fd[0] = (tng-lower)/(upper-lower);
              tgp.fd[1] = 1.0 - tgp.fd[0];
            }
          else
            {
              // Is it too low? 
              // (The current_position<n_old condition is there so
              // that uppers stays n_old-1 for extrapolation.)
              if ( tng >= upper && current_position < n_old-2 )
                {
                  do
                    {
                      ++current_position;
                      upper = old_grid[current_position+1];
                    }
                  while ( tng >= upper && current_position < n_old-2 );
 
                  lower = old_grid[current_position];
 
                  tgp.idx = current_position;
                  tgp.fd[0] = (tng-lower)/(upper-lower);
                  tgp.fd[1] = 1.0 - tgp.fd[0];
                }
              else
                {
                  // None of the other two conditions were true. That means:
                  // lower <= tng < upper. The current_position is still
                  // valid.
                  
                  // SAB 2010-04-28: Note that if a new grid point is exactly on 
                  // top of an old grid point, then you are now guaranteed to get
                  // fd[0] = 0 and fd[1] = 1. (Except at the upper grid end.)
 
                  tgp.idx = current_position;
                  tgp.fd[0] = (tng-lower)/(upper-lower);
                  tgp.fd[1] = 1.0 - tgp.fd[0];
                }
            }      
        
//          cout << tgp.idx << " " << tgp.fd[0] << " " << tgp.fd[1] << endl;
          
          // Safety check to ensure the above promise:
          assert(old_grid[tgp.idx]<=tng || tgp.idx==0);
        }
    }
  else                          //   if (ascending)  
    {
      // Now we are in the "descending old grid" part. We do exactly
      // the same as in the other part, just accounting for the
      // different order of things. Comments here refer only to
      // interesting differences from the ascending case. See that
      // case for more general comments.
 
      // This time ensure strictly descending order:
      assert( is_decreasing(old_grid) );
 
      // The max is now the first point, the min the last point!
      // I think the sign is right here...
      const Numeric og_max = old_grid[0] - 
                                     extpolfac * ( old_grid[1] - old_grid[0] );
      const Numeric og_min = old_grid[n_old-1] + 
                         extpolfac * ( old_grid[n_old-1] - old_grid[n_old-2] );
 
      // We have to take 1- here, because we are starting from the
      // high end.
      Numeric frac = 1 - (new_grid[0]-og_min)/(og_max-og_min);
 
      // We are not checking if the value of frac is reasonable,
      // because there is another assertion below to catch the
      // consequences. 
 
      Index   current_position = (Index) rint(frac*(Numeric)(n_old-2));
 
      // The above statement should satisfy
      // 0 <= current_position <= n_old-2
      // Assert that this is indeed the case. 
//       cout << "frac             = " << frac             << "\n";
//       cout << "current_position = " << current_position << "\n";
      assert( 0<= current_position );
      assert( current_position <= n_old-2 );
 
      // Note, that old_grid[lower] has a higher numerical value than
      // old_grid[upper]! 
      Numeric lower = old_grid[current_position];
      Numeric upper = old_grid[current_position+1];
 
      for ( Index i_new=0; i_new<n_new; ++i_new )
        {
          GridPos& tgp = gp[i_new];
          const Numeric tng = new_grid[i_new];
 
          // Verify that the new grid is within the limits of
          // extrapolation that we have defined by og_min and og_max:
          assert( og_min <= tng    );
          assert( tng    <= og_max );
 
//           cout << "lower / tng / upper = "
//                << lower << " / "
//                << tng   << " / "
//                << upper << "\n";
 
          // Is current_position too high? (Sign of comparison changed
          // compared to ascending case!)
          if ( tng > lower && current_position > 0 )
            {
              do
                {
                  --current_position;
                  lower = old_grid[current_position];
                }
              while ( tng > lower && current_position > 0 );
 
              upper = old_grid[current_position+1];
 
              tgp.idx = current_position;
              tgp.fd[0] = (tng-lower)/(upper-lower);
              tgp.fd[1] = 1.0 - tgp.fd[0];
            }
          else
            {
              // Is it too low? (Sign of comparison changed
              // compared to ascending case!)
              if ( tng <= upper && current_position < n_old-2 )
                {
                  do
                    {
                      ++current_position;
                      upper = old_grid[current_position+1];
                    }
                  while ( tng <= upper && current_position < n_old-2 );
 
                  lower = old_grid[current_position];
 
                  tgp.idx = current_position;
                  tgp.fd[0] = (tng-lower)/(upper-lower);
                  tgp.fd[1] = 1.0 - tgp.fd[0];
                }
              else
                {
                  // None of the other two conditions were true. That means:
                  // lower >= tng > upper. The current_position is still
                  // valid. (Note that upper and lower have switched
                  // place compared to the ascending case.)
 
                  // SAB 2010-04-28: Note that if a new grid point is exactly on 
                  // top of an old grid point, then you are now guaranteed to get
                  // fd[0] = 0 and fd[1] = 1. (Except at the upper grid end.)     
                  
                  tgp.idx = current_position;
                  tgp.fd[0] = (tng-lower)/(upper-lower);
                  tgp.fd[1] = 1.0 - tgp.fd[0];
                }
            }      
          
          // Safety check to ensure the above promise:
          assert(old_grid[tgp.idx]>=tng || tgp.idx==0);
        }      
    }
}
 
 
 
 
void gridpos( GridPos& gp,
              ConstVectorView old_grid,
              const Numeric&  new_grid,
              const Numeric&  extpolfac )
{
  ArrayOfGridPos  agp(1);
  Vector          v( 1, new_grid );
  gridpos( agp, old_grid, v, extpolfac );
  gridpos_copy( gp,  agp[0] );  
}
 
 
 
 
void gridpos_1to1( 
   ArrayOfGridPos& gp,
   ConstVectorView grid )
{
  const Index n = grid.nelem();
  gp.resize( n );
  
  for( Index i=0; i<n-1; i++ )
    {
      gp[i].idx   = i;
      gp[i].fd[0] = 0;
      gp[i].fd[1] = 1;
    }
  gp[n-1].idx   = n-2;
  gp[n-1].fd[0] = 1;
  gp[n-1].fd[1] = 0;
}
 
 
 
void gridpos_copy( GridPos&  gp_new,  const GridPos&  gp_old )
{
  gp_new.idx   = gp_old.idx;
  gp_new.fd[0] = gp_old.fd[0];
  gp_new.fd[1] = gp_old.fd[1];
}
 
 
 
 
Numeric fractional_gp( const GridPos&   gp )
{
  return ( Numeric(gp.idx) + gp.fd[0] );
}
 
 
 
 
void gridpos_check_fd( GridPos&   gp )
{
  // Catch values that "must" be wrong
  assert( gp.fd[0] > -FD_TOL );
  assert( gp.fd[0] < 1.0 + FD_TOL );
  assert( gp.fd[1] > -FD_TOL );
  assert( gp.fd[1] < 1.0 + FD_TOL );
 
  if( gp.fd[0] < 0.0 )
    { gp.fd[0] = 0.0; gp.fd[1] = 1.0; }
  else if( gp.fd[0] > 1.0 )
    { gp.fd[0] = 1.0; gp.fd[1] = 0.0; }
 
  if( gp.fd[1] < 0.0 )
    { gp.fd[1] = 0.0; gp.fd[0] = 1.0; }
  else if( gp.fd[1] > 1.0 )
    { gp.fd[1] = 1.0; gp.fd[0] = 0.0; }
}
 
 
 
 
void gridpos_force_end_fd( 
        GridPos&   gp,
  const Index&     n )
{
  assert( gp.idx >= 0 );
 
  // If fd=1, shift to grid index above
  if( gp.fd[0] > 0.5 )
    {
      gp.idx += 1;
    }
  gp.fd[0] = 0;
  gp.fd[1] = 1;
 
  assert( gp.idx < n );
 
  // End of complete grid range must be handled separately
  if( gp.idx == n-1 )
    {
      gp.idx  -= 1;
      gp.fd[0] = 1;
      gp.fd[1] = 0;
    }
}
 
 
 
 
void gridpos_upperend_check( 
        GridPos&   gp,
  const Index&     ie )  
{
  if( gp.idx == ie )
    {
      assert( gp.fd[0] < 0.005 );  // To capture obviously bad cases
      gp.idx   -= 1; 
      gp.fd[0] = 1.0; 
      gp.fd[1] = 0.0; 
    }
}
 
 
 
void gridpos_upperend_check( 
        ArrayOfGridPos&   gp,
  const Index&            ie )   
{
  for (Index i = 0; i < gp.nelem(); i++ ) 
    {
      if( gp[i].idx == ie )
        {
          assert( gp[i].fd[0] < 0.005 );  // To capture obviously bad cases
          gp[i].idx   -= 1; 
          gp[i].fd[0] = 1.0; 
          gp[i].fd[1] = 0.0; 
        }
    }
}
 
 
 
 
bool is_gridpos_at_index_i(  
       const GridPos&   gp,
       const Index&     i,
       const bool&      strict )
{
  if( strict )
    {
      // Assume that gridpos_force_end_fd has been used. The expression 0==0
      // should be safer than 1==1. 
      if( gp.idx == i  &&  gp.fd[0] == 0 )
        { return true; }
      else if( gp.idx == i-1  &&  gp.fd[1] == 0 )
        { return true; }
    }
  else
    {
      if( gp.idx == i  &&  gp.fd[0] < FD_TOL )
        { return true; }
      else if( gp.idx == i-1  &&  gp.fd[1] < FD_TOL )
        { return true; }
    }  
  return false; 
}
 
 
 
 
Index gridpos2gridrange(
       const GridPos&   gp,
       const bool&      upwards )
{
  assert( gp.fd[0] >= 0 );
  assert( gp.fd[1] >= 0 );
 
  // Not at a grid point
  if( gp.fd[0] > 0   &&  gp.fd[1] > 0 )
    {
      return gp.idx;
    }
 
  // Fractional distance 0
  else if( gp.fd[0] == 0 )
    {
      if( upwards )
        return gp.idx;
      else
        return gp.idx - 1;
    }
 
  // Fractional distance 1
  else
    {
      if( upwards )
        return gp.idx + 1;
      else
        return gp.idx;
    }
}
 
 
 
 
 
 
//                      Red Interpolation
 
 
void interpweights( VectorView itw,
                    const GridPos& tc )
{
  assert(is_size(itw,2));       // We must store 2 interpolation
                                // weights.
 
  // Interpolation weights are stored in this order (l=lower
  // u=upper, c=column):
  // 1. l-c
  // 2. u-c
 
  Index iti = 0;
 
  // We could use a straight-forward for loop here:
  //
  //       for ( Index c=1; c>=0; --c )
  //    {
  //      ti[iti] = tc.fd[c];
  //      ++iti;
  //    }
  //
  // However, it is slightly faster to use pointers (I tried it,
  // the speed gain is about 20%). So we should write something
  // like:
  //
  //       for ( const Numeric* c=&tc.fd[1]; c>=&tc.fd[0]; --c )
  //    {
  //      ti[iti] = *c;
  //      ++iti;
  //    }
  //
  // For higher dimensions we have to nest these loops. To avoid
  // typos and safe typing, I use the LOOPIT macro, which expands
  // to the for loop above. Note: NO SEMICOLON AFTER THE LOOPIT
  // COMMAND! 
 
  LOOPIT(c)
    {
      itw.get(iti) = *c;
      ++iti;
    }
}
 
 
void interpweights( VectorView itw,
                    const GridPos& tr,
                    const GridPos& tc )
{
  assert(is_size(itw,4));       // We must store 4 interpolation
                                // weights.
  Index iti = 0;
 
  LOOPIT(r)
  LOOPIT(c)
    {
      itw.get(iti) = (*r) * (*c);
      ++iti;
    }
}
 
 
void interpweights( VectorView itw,
                    const GridPos& tp,
                    const GridPos& tr,
                    const GridPos& tc )
{
  assert(is_size(itw,8));       // We must store 8 interpolation
                                // weights.
  Index iti = 0;
 
  LOOPIT(p)
  LOOPIT(r)
  LOOPIT(c)
    {
      itw.get(iti) = (*p) * (*r) * (*c);
      ++iti;
    }
}
 
 
void interpweights( VectorView itw,
                    const GridPos& tb,
                    const GridPos& tp,
                    const GridPos& tr,
                    const GridPos& tc )
{
  assert(is_size(itw,16));      // We must store 16 interpolation
                                // weights.
  Index iti = 0;
 
  LOOPIT(b)
  LOOPIT(p)
  LOOPIT(r)
  LOOPIT(c)
    {
      itw.get(iti) = (*b) * (*p) * (*r) * (*c);
      ++iti;
    }
}
 
 
void interpweights( VectorView itw,
                    const GridPos& ts,
                    const GridPos& tb,
                    const GridPos& tp,
                    const GridPos& tr,
                    const GridPos& tc )
{
  assert(is_size(itw,32));      // We must store 32 interpolation
                                // weights.
  Index iti = 0;
 
  LOOPIT(s)
  LOOPIT(b)
  LOOPIT(p)
  LOOPIT(r)
  LOOPIT(c)
    {
      itw.get(iti) = (*s) * (*b) * (*p) * (*r) * (*c);
      ++iti;
    }
}
 
 
void interpweights( VectorView itw,
                    const GridPos& tv,
                    const GridPos& ts,
                    const GridPos& tb,
                    const GridPos& tp,
                    const GridPos& tr,
                    const GridPos& tc )
{
  assert(is_size(itw,64));      // We must store 64 interpolation
                                // weights.
  Index iti = 0;
 
  LOOPIT(v)
  LOOPIT(s)
  LOOPIT(b)
  LOOPIT(p)
  LOOPIT(r)
  LOOPIT(c)
    {
      itw.get(iti) = (*v) * (*s) * (*b) * (*p) * (*r) * (*c);
      ++iti;
    }
}
 
 
Numeric interp( ConstVectorView itw,
                ConstVectorView a,    
                const GridPos&  tc )
{
  assert(is_size(itw,2));       // We need 2 interpolation
                                // weights.
 
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one.
  assert( is_same_within_epsilon( itw.sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // To store interpolated value:
  Numeric tia = 0;
 
  Index iti = 0;
  for ( Index c=0; c<2; ++c )
    {
      tia += a.get(tc.idx+c) * itw.get(iti);
      ++iti;
    }
 
  return tia;
}
 
 
Numeric interp( ConstVectorView  itw,
                ConstMatrixView  a,    
                const GridPos&   tr,
                const GridPos&   tc )
{
  assert(is_size(itw,4));       // We need 4 interpolation
                                // weights.
 
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one.
  assert( is_same_within_epsilon( itw.sum(),
                                  1,
                                  sum_check_epsilon ) );
 
  // To store interpolated value:
  Numeric tia = 0;
 
  Index iti = 0;
  for ( Index r=0; r<2; ++r )
    for ( Index c=0; c<2; ++c )
      {
        tia += a.get(tr.idx+r,
                     tc.idx+c) * itw.get(iti);
        ++iti;
      }
 
  return tia;
}
 
 
Numeric interp( ConstVectorView  itw,
                ConstTensor3View a,    
                const GridPos&   tp,
                const GridPos&   tr,
                const GridPos&   tc )
{
  assert(is_size(itw,8));       // We need 8 interpolation
                                // weights.
 
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one.
  assert( is_same_within_epsilon( itw.sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // To store interpolated value:
  Numeric tia = 0;
 
  Index iti = 0;
  for ( Index p=0; p<2; ++p )
    for ( Index r=0; r<2; ++r )
      for ( Index c=0; c<2; ++c )
        {
          tia += a.get(tp.idx+p,
                       tr.idx+r,
                       tc.idx+c) * itw.get(iti);
          ++iti;
        }
 
  return tia;
}
 
 
Numeric interp( ConstVectorView  itw,
                ConstTensor4View a,    
                const GridPos&   tb,
                const GridPos&   tp,
                const GridPos&   tr,
                const GridPos&   tc )
{
  assert(is_size(itw,16));      // We need 16 interpolation
                                // weights.
 
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one.
  assert( is_same_within_epsilon( itw.sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // To store interpolated value:
  Numeric tia = 0;
 
  Index iti = 0;
  for ( Index b=0; b<2; ++b )
    for ( Index p=0; p<2; ++p )
      for ( Index r=0; r<2; ++r )
        for ( Index c=0; c<2; ++c )
          {
            tia += a.get(tb.idx+b,
                         tp.idx+p,
                         tr.idx+r,
                         tc.idx+c) * itw.get(iti);
            ++iti;
          }
 
  return tia;
}
 
 
Numeric interp( ConstVectorView  itw,
                ConstTensor5View a,    
                const GridPos&   ts,
                const GridPos&   tb,
                const GridPos&   tp,
                const GridPos&   tr,
                const GridPos&   tc )
{
  assert(is_size(itw,32));      // We need 32 interpolation
                                // weights.
 
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one.
  assert( is_same_within_epsilon( itw.sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // To store interpolated value:
  Numeric tia = 0;
 
  Index iti = 0;
  for ( Index s=0; s<2; ++s )
    for ( Index b=0; b<2; ++b )
      for ( Index p=0; p<2; ++p )
        for ( Index r=0; r<2; ++r )
          for ( Index c=0; c<2; ++c )
            {
              tia += a.get(ts.idx+s,
                           tb.idx+b,
                           tp.idx+p,
                           tr.idx+r,
                           tc.idx+c) * itw.get(iti);
              ++iti;
            }
 
  return tia;
}
 
 
Numeric interp( ConstVectorView  itw,
                ConstTensor6View a,    
                const GridPos&   tv,
                const GridPos&   ts,
                const GridPos&   tb,
                const GridPos&   tp,
                const GridPos&   tr,
                const GridPos&   tc )
{
  assert(is_size(itw,64));      // We need 64 interpolation
                                // weights.
 
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one.
  assert( is_same_within_epsilon( itw.sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // To store interpolated value:
  Numeric tia = 0;
 
  Index iti = 0;
  for ( Index v=0; v<2; ++v )
    for ( Index s=0; s<2; ++s )
      for ( Index b=0; b<2; ++b )
        for ( Index p=0; p<2; ++p )
          for ( Index r=0; r<2; ++r )
            for ( Index c=0; c<2; ++c )
              {
                tia += a.get(tv.idx+v,
                             ts.idx+s,
                             tb.idx+b,
                             tp.idx+p,
                             tr.idx+r,
                             tc.idx+c) * itw.get(iti);
                ++iti;
              }
 
  return tia;
}
 
 
//                      Blue interpolation
 
 
void interpweights( MatrixView itw,
                    const ArrayOfGridPos& cgp )
{
  Index n = cgp.nelem();
  assert(is_size(itw,n,2));     // We must store 2 interpolation
                                // weights for each position.
 
  // We have to loop all the points in the sequence:
  for ( Index i=0; i<n; ++i )
    {
      // Current grid positions:
      const GridPos& tc = cgp[i];
 
      // Interpolation weights are stored in this order (l=lower
      // u=upper, c=column):
      // 1. l-c
      // 2. u-c
 
      Index iti = 0;
 
      // We could use a straight-forward for loop here:
      //
      //       for ( Index c=1; c>=0; --c )
      //        {
      //          ti[iti] = tc.fd[c];
      //          ++iti;
      //        }
      //
      // However, it is slightly faster to use pointers (I tried it,
      // the speed gain is about 20%). So we should write something
      // like:
      //
      //       for ( const Numeric* c=&tc.fd[1]; c>=&tc.fd[0]; --c )
      //        {
      //          ti[iti] = *c;
      //          ++iti;
      //        }
      //
      // For higher dimensions we have to nest these loops. To avoid
      // typos and safe typing, I use the LOOPIT macro, which expands
      // to the for loop above. Note: NO SEMICOLON AFTER THE LOOPIT
      // COMMAND! 
 
      LOOPIT(c)
        {
          itw.get(i,iti) = *c;
          ++iti;
        }
    }
}
 
 
void interpweights( MatrixView itw,
                    const ArrayOfGridPos& rgp,
                    const ArrayOfGridPos& cgp )
{
  Index n = cgp.nelem();
  assert(is_size(rgp,n));       // rgp must have same size as cgp.
  assert(is_size(itw,n,4));     // We must store 4 interpolation
                                // weights for each position.
 
  // We have to loop all the points in the sequence:
  for ( Index i=0; i<n; ++i )
    {
      // Current grid positions:
      const GridPos& tr = rgp[i];
      const GridPos& tc = cgp[i];
 
      // Interpolation weights are stored in this order (l=lower
      // u=upper, r=row, c=column):
      // 1. l-r l-c
      // 2. l-r u-c
      // 3. u-r l-c
      // 4. u-r u-c
 
      Index iti = 0;
 
      LOOPIT(r)
      LOOPIT(c)
          {
            itw.get(i,iti) = (*r) * (*c);
            ++iti;
          }
    }
}
 
 
void interpweights( MatrixView itw,
                    const ArrayOfGridPos& pgp,
                    const ArrayOfGridPos& rgp,
                    const ArrayOfGridPos& cgp )
{
  Index n = cgp.nelem();
  assert(is_size(pgp,n));       // pgp must have same size as cgp.
  assert(is_size(rgp,n));       // rgp must have same size as cgp.
  assert(is_size(itw,n,8));     // We must store 8 interpolation
                                // weights for each position.
 
  // We have to loop all the points in the sequence:
  for ( Index i=0; i<n; ++i )
    {
      // Current grid positions:
      const GridPos& tp = pgp[i];
      const GridPos& tr = rgp[i];
      const GridPos& tc = cgp[i];
 
      Index iti = 0;
      LOOPIT(p)
      LOOPIT(r)
      LOOPIT(c)
        {
          itw.get(i,iti) = (*p) * (*r) * (*c);
          ++iti;
        }
    }
}
 
 
void interpweights( MatrixView itw,
                    const ArrayOfGridPos& bgp,
                    const ArrayOfGridPos& pgp,
                    const ArrayOfGridPos& rgp,
                    const ArrayOfGridPos& cgp )
{
  Index n = cgp.nelem();
  assert(is_size(bgp,n));       // bgp must have same size as cgp.
  assert(is_size(pgp,n));       // pgp must have same size as cgp.
  assert(is_size(rgp,n));       // rgp must have same size as cgp.
  assert(is_size(itw,n,16));    // We must store 16 interpolation
                                // weights for each position.
 
  // We have to loop all the points in the sequence:
  for ( Index i=0; i<n; ++i )
    {
      // Current grid positions:
      const GridPos& tb = bgp[i];
      const GridPos& tp = pgp[i];
      const GridPos& tr = rgp[i];
      const GridPos& tc = cgp[i];
 
      Index iti = 0;
      LOOPIT(b)
      LOOPIT(p)
      LOOPIT(r)
      LOOPIT(c)
        {
          itw.get(i,iti) = (*b) * (*p) * (*r) * (*c);
          ++iti;
        }
    }
}
 
 
void interpweights( MatrixView itw,
                    const ArrayOfGridPos& sgp,
                    const ArrayOfGridPos& bgp,
                    const ArrayOfGridPos& pgp,
                    const ArrayOfGridPos& rgp,
                    const ArrayOfGridPos& cgp )
{
  Index n = cgp.nelem();
  assert(is_size(sgp,n));       // sgp must have same size as cgp.
  assert(is_size(bgp,n));       // bgp must have same size as cgp.
  assert(is_size(pgp,n));       // pgp must have same size as cgp.
  assert(is_size(rgp,n));       // rgp must have same size as cgp.
  assert(is_size(itw,n,32));    // We must store 32 interpolation
                                // weights for each position.
 
  // We have to loop all the points in the sequence:
  for ( Index i=0; i<n; ++i )
    {
      // Current grid positions:
      const GridPos& ts = sgp[i];
      const GridPos& tb = bgp[i];
      const GridPos& tp = pgp[i];
      const GridPos& tr = rgp[i];
      const GridPos& tc = cgp[i];
 
      Index iti = 0;
      LOOPIT(s)
      LOOPIT(b)
      LOOPIT(p)
      LOOPIT(r)
      LOOPIT(c)
        {
          itw.get(i,iti) = (*s) * (*b) * (*p) * (*r) * (*c);
          ++iti;
        }
    }
}
 
 
void interpweights( MatrixView itw,
                    const ArrayOfGridPos& vgp,
                    const ArrayOfGridPos& sgp,
                    const ArrayOfGridPos& bgp,
                    const ArrayOfGridPos& pgp,
                    const ArrayOfGridPos& rgp,
                    const ArrayOfGridPos& cgp )
{
  Index n = cgp.nelem();
  assert(is_size(vgp,n));       // vgp must have same size as cgp.
  assert(is_size(sgp,n));       // sgp must have same size as cgp.
  assert(is_size(bgp,n));       // bgp must have same size as cgp.
  assert(is_size(pgp,n));       // pgp must have same size as cgp.
  assert(is_size(rgp,n));       // rgp must have same size as cgp.
  assert(is_size(itw,n,64));    // We must store 64 interpolation
                                // weights for each position.
 
  // We have to loop all the points in the sequence:
  for ( Index i=0; i<n; ++i )
    {
      // Current grid positions:
      const GridPos& tv = vgp[i];
      const GridPos& ts = sgp[i];
      const GridPos& tb = bgp[i];
      const GridPos& tp = pgp[i];
      const GridPos& tr = rgp[i];
      const GridPos& tc = cgp[i];
 
      Index iti = 0;
      LOOPIT(v)
      LOOPIT(s)
      LOOPIT(b)
      LOOPIT(p)
      LOOPIT(r)
      LOOPIT(c)
        {
          itw.get(i,iti) = (*v) * (*s) * (*b) * (*p) * (*r) * (*c);
          ++iti;
        }
    }
}
 
 
void interp( VectorView            ia,
             ConstMatrixView       itw,
             ConstVectorView       a,    
             const ArrayOfGridPos& cgp)
{
  Index n = cgp.nelem();
  assert(is_size(ia,n));        //  ia must have same size as cgp.
  assert(is_size(itw,n,2));     // We need 2 interpolation
                                // weights for each position.
 
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one. We
  // only check the first element.
  assert( is_same_within_epsilon( itw(0,Range(joker)).sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // We have to loop all the points in the sequence:
  for ( Index i=0; i<n; ++i )
    {
      // Current grid positions:
      const GridPos& tc = cgp[i];
 
      // Get handle to current element of output vector and initialize
      // it to zero:
      Numeric& tia = ia[i];
      tia = 0;
 
      Index iti = 0;
      for ( Index c=0; c<2; ++c )
        {
          tia += a.get(tc.idx+c) * itw.get(i,iti);
          ++iti;
        }
    }
}
 
 
void interp( VectorView            ia,
             ConstMatrixView       itw,
             ConstMatrixView       a,    
             const ArrayOfGridPos& rgp,
             const ArrayOfGridPos& cgp)
{
  Index n = cgp.nelem();
  assert(is_size(ia,n));        //  ia must have same size as cgp.
  assert(is_size(rgp,n));       // rgp must have same size as cgp.
  assert(is_size(itw,n,4));     // We need 4 interpolation
                                // weights for each position.
  
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one. We
  // only check the first element.
  assert( is_same_within_epsilon( itw(0,Range(joker)).sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // We have to loop all the points in the sequence:
  for ( Index i=0; i<n; ++i )
    {
      // Current grid positions:
      const GridPos& tr = rgp[i];
      const GridPos& tc = cgp[i];
 
      // Get handle to current element of output vector and initialize
      // it to zero:
      Numeric& tia = ia[i];
      tia = 0;
 
      Index iti = 0;
      for ( Index r=0; r<2; ++r )
        for ( Index c=0; c<2; ++c )
          {
            tia += a.get(tr.idx+r,
                         tc.idx+c) * itw.get(i,iti);
            ++iti;
          }
    }
}
 
 
void interp( VectorView            ia,
             ConstMatrixView       itw,
             ConstTensor3View      a,    
             const ArrayOfGridPos& pgp,
             const ArrayOfGridPos& rgp,
             const ArrayOfGridPos& cgp)
{
  Index n = cgp.nelem();
  assert(is_size(ia,n));        //  ia must have same size as cgp.
  assert(is_size(pgp,n));       // pgp must have same size as cgp.
  assert(is_size(rgp,n));       // rgp must have same size as cgp.
  assert(is_size(itw,n,8));     // We need 8 interpolation
                                // weights for each position.
  
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one. We
  // only check the first element.
  assert( is_same_within_epsilon( itw(0,Range(joker)).sum(),
                                  1,
                                  sum_check_epsilon ) );
 
  // We have to loop all the points in the sequence:
  for ( Index i=0; i<n; ++i )
    {
      // Current grid positions:
      const GridPos& tp = pgp[i];
      const GridPos& tr = rgp[i];
      const GridPos& tc = cgp[i];
 
      // Get handle to current element of output vector and initialize
      // it to zero:
      Numeric& tia = ia[i];
      tia = 0;
 
      Index iti = 0;
      for ( Index p=0; p<2; ++p )
        for ( Index r=0; r<2; ++r )
          for ( Index c=0; c<2; ++c )
            {
              tia += a.get(tp.idx+p,
                           tr.idx+r,
                           tc.idx+c) * itw.get(i,iti);
              ++iti;
            }
    }
}
 
 
void interp( VectorView            ia,
             ConstMatrixView       itw,
             ConstTensor4View      a,    
             const ArrayOfGridPos& bgp,
             const ArrayOfGridPos& pgp,
             const ArrayOfGridPos& rgp,
             const ArrayOfGridPos& cgp)
{
  Index n = cgp.nelem();
  assert(is_size(ia,n));        //  ia must have same size as cgp.
  assert(is_size(bgp,n));       // bgp must have same size as cgp.
  assert(is_size(pgp,n));       // pgp must have same size as cgp.
  assert(is_size(rgp,n));       // rgp must have same size as cgp.
  assert(is_size(itw,n,16));    // We need 16 interpolation
                                // weights for each position.
  
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one. We
  // only check the first element.
  assert( is_same_within_epsilon( itw(0,Range(joker)).sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // We have to loop all the points in the sequence:
  for ( Index i=0; i<n; ++i )
    {
      // Current grid positions:
      const GridPos& tb = bgp[i];
      const GridPos& tp = pgp[i];
      const GridPos& tr = rgp[i];
      const GridPos& tc = cgp[i];
 
      // Get handle to current element of output vector and initialize
      // it to zero:
      Numeric& tia = ia[i];
      tia = 0;
 
      Index iti = 0;
      for ( Index b=0; b<2; ++b )
        for ( Index p=0; p<2; ++p )
          for ( Index r=0; r<2; ++r )
            for ( Index c=0; c<2; ++c )
              {
                tia += a.get(tb.idx+b,
                             tp.idx+p,
                             tr.idx+r,
                             tc.idx+c) * itw.get(i,iti);
                ++iti;
              }
    }
}
 
 
void interp( VectorView            ia,
             ConstMatrixView       itw,
             ConstTensor5View      a,    
             const ArrayOfGridPos& sgp,
             const ArrayOfGridPos& bgp,
             const ArrayOfGridPos& pgp,
             const ArrayOfGridPos& rgp,
             const ArrayOfGridPos& cgp)
{
  Index n = cgp.nelem();
  assert(is_size(ia,n));        //  ia must have same size as cgp.
  assert(is_size(sgp,n));       // sgp must have same size as cgp.
  assert(is_size(bgp,n));       // bgp must have same size as cgp.
  assert(is_size(pgp,n));       // pgp must have same size as cgp.
  assert(is_size(rgp,n));       // rgp must have same size as cgp.
  assert(is_size(itw,n,32));    // We need 32 interpolation
                                // weights for each position.
  
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one. We
  // only check the first element.
  assert( is_same_within_epsilon( itw(0,Range(joker)).sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // We have to loop all the points in the sequence:
  for ( Index i=0; i<n; ++i )
    {
      // Current grid positions:
      const GridPos& ts = sgp[i];
      const GridPos& tb = bgp[i];
      const GridPos& tp = pgp[i];
      const GridPos& tr = rgp[i];
      const GridPos& tc = cgp[i];
 
      // Get handle to current element of output vector and initialize
      // it to zero:
      Numeric& tia = ia[i];
      tia = 0;
 
      Index iti = 0;
      for ( Index s=0; s<2; ++s )
        for ( Index b=0; b<2; ++b )
          for ( Index p=0; p<2; ++p )
            for ( Index r=0; r<2; ++r )
              for ( Index c=0; c<2; ++c )
                {
                  tia += a.get(ts.idx+s,
                               tb.idx+b,
                               tp.idx+p,
                               tr.idx+r,
                               tc.idx+c) * itw.get(i,iti);
                  ++iti;
                }
    }
}
 
 
void interp( VectorView            ia,
             ConstMatrixView       itw,
             ConstTensor6View      a,    
             const ArrayOfGridPos& vgp,
             const ArrayOfGridPos& sgp,
             const ArrayOfGridPos& bgp,
             const ArrayOfGridPos& pgp,
             const ArrayOfGridPos& rgp,
             const ArrayOfGridPos& cgp)
{
  Index n = cgp.nelem();
  assert(is_size(ia,n));        //  ia must have same size as cgp.
  assert(is_size(vgp,n));       // vgp must have same size as cgp.
  assert(is_size(sgp,n));       // sgp must have same size as cgp.
  assert(is_size(bgp,n));       // bgp must have same size as cgp.
  assert(is_size(pgp,n));       // pgp must have same size as cgp.
  assert(is_size(rgp,n));       // rgp must have same size as cgp.
  assert(is_size(itw,n,64));    // We need 64 interpolation
                                // weights for each position.
  
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one. We
  // only check the first element.
  assert( is_same_within_epsilon( itw(0,Range(joker)).sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // We have to loop all the points in the sequence:
  for ( Index i=0; i<n; ++i )
    {
      // Current grid positions:
      const GridPos& tv = vgp[i];
      const GridPos& ts = sgp[i];
      const GridPos& tb = bgp[i];
      const GridPos& tp = pgp[i];
      const GridPos& tr = rgp[i];
      const GridPos& tc = cgp[i];
 
      // Get handle to current element of output vector and initialize
      // it to zero:
      Numeric& tia = ia[i];
      tia = 0;
 
      Index iti = 0;
      for ( Index v=0; v<2; ++v )
        for ( Index s=0; s<2; ++s )
          for ( Index b=0; b<2; ++b )
            for ( Index p=0; p<2; ++p )
              for ( Index r=0; r<2; ++r )
                for ( Index c=0; c<2; ++c )
                  {
                    tia += a.get(tv.idx+v,
                                 ts.idx+s,
                                 tb.idx+b,
                                 tp.idx+p,
                                 tr.idx+r,
                                 tc.idx+c) * itw.get(i,iti);
                    ++iti;
                  }
    }
}
 
//                      Green interpolation
 
 
void interpweights( Tensor3View itw,
                    const ArrayOfGridPos& rgp,
                    const ArrayOfGridPos& cgp )
{
  Index nr = rgp.nelem();
  Index nc = cgp.nelem();
  assert(is_size(itw,nr,nc,4)); // We must store 4 interpolation
                                // weights for each position.
 
  // We have to loop all the points in the new grid:
  for ( Index ir=0; ir<nr; ++ir )
    {
      // Current grid position:
      const GridPos& tr = rgp[ir];
 
      for ( Index ic=0; ic<nc; ++ic )
        {
          // Current grid position:
          const GridPos& tc = cgp[ic];
 
          // Interpolation weights are stored in this order (l=lower
          // u=upper, r=row, c=column):
          // 1. l-r l-c
          // 2. l-r u-c
          // 3. u-r l-c
          // 4. u-r u-c
 
          Index iti = 0;
 
          LOOPIT(r)
            LOOPIT(c)
            {
              itw.get(ir,ic,iti) = (*r) * (*c);
              ++iti;
            }
        }
    }
}
 
 
void interpweights( Tensor4View itw,
                    const ArrayOfGridPos& pgp,
                    const ArrayOfGridPos& rgp,
                    const ArrayOfGridPos& cgp )
{
  Index np = pgp.nelem();
  Index nr = rgp.nelem();
  Index nc = cgp.nelem();
  // We must store 8 interpolation weights for each position:
  assert(is_size(itw,np,nr,nc,8));      
 
  // We have to loop all the points in the new grid:
  for ( Index ip=0; ip<np; ++ip )
    {
      const GridPos& tp = pgp[ip];
      for ( Index ir=0; ir<nr; ++ir )
        {
          const GridPos& tr = rgp[ir];
          for ( Index ic=0; ic<nc; ++ic )
            {
              const GridPos& tc = cgp[ic];
 
              Index iti = 0;
 
              LOOPIT(p)
                LOOPIT(r)
                LOOPIT(c)
                {
                  itw.get(ip,ir,ic,iti) =
                    (*p) * (*r) * (*c);
                  ++iti;
                }
            }
        }
    }
}
 
 
void interpweights( Tensor5View itw,
                    const ArrayOfGridPos& bgp,
                    const ArrayOfGridPos& pgp,
                    const ArrayOfGridPos& rgp,
                    const ArrayOfGridPos& cgp )
{
  Index nb = bgp.nelem();
  Index np = pgp.nelem();
  Index nr = rgp.nelem();
  Index nc = cgp.nelem();
  // We must store 16 interpolation weights for each position:
  assert(is_size(itw,nb,np,nr,nc,16));  
 
  // We have to loop all the points in the new grid:
  for ( Index ib=0; ib<nb; ++ib )
    {
      const GridPos& tb = bgp[ib];
      for ( Index ip=0; ip<np; ++ip )
        {
          const GridPos& tp = pgp[ip];
          for ( Index ir=0; ir<nr; ++ir )
            {
              const GridPos& tr = rgp[ir];
              for ( Index ic=0; ic<nc; ++ic )
                {
                  const GridPos& tc = cgp[ic];
 
                  Index iti = 0;
 
                  LOOPIT(b)
                    LOOPIT(p)
                    LOOPIT(r)
                    LOOPIT(c)
                    {
                      itw.get(ib,ip,ir,ic,iti) =
                        (*b) * (*p) * (*r) * (*c);
                      ++iti;
                    }
                }
            }
        }
    }
}
 
 
void interpweights( Tensor6View itw,
                    const ArrayOfGridPos& sgp,
                    const ArrayOfGridPos& bgp,
                    const ArrayOfGridPos& pgp,
                    const ArrayOfGridPos& rgp,
                    const ArrayOfGridPos& cgp )
{
  Index ns = sgp.nelem();
  Index nb = bgp.nelem();
  Index np = pgp.nelem();
  Index nr = rgp.nelem();
  Index nc = cgp.nelem();
  // We must store 32 interpolation weights for each position:
  assert(is_size(itw,ns,nb,np,nr,nc,32));       
 
  // We have to loop all the points in the new grid:
  for ( Index is=0; is<ns; ++is )
    {
      const GridPos& ts = sgp[is];
      for ( Index ib=0; ib<nb; ++ib )
        {
          const GridPos& tb = bgp[ib];
          for ( Index ip=0; ip<np; ++ip )
            {
              const GridPos& tp = pgp[ip];
              for ( Index ir=0; ir<nr; ++ir )
                {
                  const GridPos& tr = rgp[ir];
                  for ( Index ic=0; ic<nc; ++ic )
                    {
                      const GridPos& tc = cgp[ic];
 
                      Index iti = 0;
 
                      LOOPIT(s)
                        LOOPIT(b)
                        LOOPIT(p)
                        LOOPIT(r)
                        LOOPIT(c)
                        {
                          itw.get(is,ib,ip,ir,ic,iti) =
                            (*s) * (*b) * (*p) * (*r) * (*c);
                          ++iti;
                        }
                    }
                }
            }
        }
    }
}
 
 
void interpweights( Tensor7View itw,
                    const ArrayOfGridPos& vgp,
                    const ArrayOfGridPos& sgp,
                    const ArrayOfGridPos& bgp,
                    const ArrayOfGridPos& pgp,
                    const ArrayOfGridPos& rgp,
                    const ArrayOfGridPos& cgp )
{
  Index nv = vgp.nelem();
  Index ns = sgp.nelem();
  Index nb = bgp.nelem();
  Index np = pgp.nelem();
  Index nr = rgp.nelem();
  Index nc = cgp.nelem();
  // We must store 64 interpolation weights for each position:
  assert(is_size(itw,nv,ns,nb,np,nr,nc,64));    
 
  // We have to loop all the points in the new grid:
  for ( Index iv=0; iv<nv; ++iv )
    {
      const GridPos& tv = vgp[iv];
      for ( Index is=0; is<ns; ++is )
        {
          const GridPos& ts = sgp[is];
          for ( Index ib=0; ib<nb; ++ib )
            {
              const GridPos& tb = bgp[ib];
              for ( Index ip=0; ip<np; ++ip )
                {
                  const GridPos& tp = pgp[ip];
                  for ( Index ir=0; ir<nr; ++ir )
                    {
                      const GridPos& tr = rgp[ir];
                      for ( Index ic=0; ic<nc; ++ic )
                        {
                          const GridPos& tc = cgp[ic];
 
                          Index iti = 0;
 
                          LOOPIT(v)
                            LOOPIT(s)
                            LOOPIT(b)
                            LOOPIT(p)
                            LOOPIT(r)
                            LOOPIT(c)
                            {
                              itw.get(iv,is,ib,ip,ir,ic,iti) =
                                (*v) * (*s) * (*b) * (*p) * (*r) * (*c);
                              ++iti;
                            }
                        }
                    }
                }
            }
        }
    }
}
 
 
void interp( MatrixView            ia,
             ConstTensor3View      itw,
             ConstMatrixView       a,   
             const ArrayOfGridPos& rgp,
             const ArrayOfGridPos& cgp)
{
  Index nr = rgp.nelem();
  Index nc = cgp.nelem();
  assert(is_size(ia,nr,nc));    
  assert(is_size(itw,nr,nc,4)); // We need 4 interpolation
                                // weights for each position.
 
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one. We
  // only check the first element.
  assert( is_same_within_epsilon( itw(0,0,Range(joker)).sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // We have to loop all the points in the new grid:
  for ( Index ir=0; ir<nr; ++ir )
    {
      // Current grid position:
      const GridPos& tr = rgp[ir];
 
      for ( Index ic=0; ic<nc; ++ic )
        {
          // Current grid position:
          const GridPos& tc = cgp[ic];
 
          // Get handle to current element of output tensor and initialize
          // it to zero:
          Numeric& tia = ia(ir,ic);
          tia = 0;
 
          Index iti = 0;
          for ( Index r=0; r<2; ++r )
            for ( Index c=0; c<2; ++c )
            {
              tia += a.get(tr.idx+r,
                           tc.idx+c) * itw.get(ir,ic,iti);
              ++iti;
            }
        }
    }
}
 
 
void interp( Tensor3View           ia,
             ConstTensor4View      itw,
             ConstTensor3View      a,   
             const ArrayOfGridPos& pgp,
             const ArrayOfGridPos& rgp,
             const ArrayOfGridPos& cgp)
{
  Index np = pgp.nelem();
  Index nr = rgp.nelem();
  Index nc = cgp.nelem();
  assert(is_size(ia,
                 np,nr,nc));    
  assert(is_size(itw,
                 np,nr,nc,
                 8));
 
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one. We
  // only check the first element.
  assert( is_same_within_epsilon( itw(0,0,0,Range(joker)).sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // We have to loop all the points in the new grid:
  for ( Index ip=0; ip<np; ++ip )
    {
      const GridPos& tp = pgp[ip];
      for ( Index ir=0; ir<nr; ++ir )
        {
          const GridPos& tr = rgp[ir];
          for ( Index ic=0; ic<nc; ++ic )
            {
              // Current grid position:
              const GridPos& tc = cgp[ic];
 
              // Get handle to current element of output tensor and
              // initialize it to zero:
              Numeric& tia = ia(ip,ir,ic);
              tia = 0;
 
              Index iti = 0;
              for ( Index p=0; p<2; ++p )
                for ( Index r=0; r<2; ++r )
                  for ( Index c=0; c<2; ++c )
                    {
                      tia += a.get(tp.idx+p,
                                   tr.idx+r,
                                   tc.idx+c) * itw.get(ip,ir,ic,
                                                       iti);
                      ++iti;
                    }
            }
        }
    }
}
 
 
void interp( Tensor4View           ia,
             ConstTensor5View      itw,
             ConstTensor4View      a,   
             const ArrayOfGridPos& bgp,
             const ArrayOfGridPos& pgp,
             const ArrayOfGridPos& rgp,
             const ArrayOfGridPos& cgp)
{
  Index nb = bgp.nelem();
  Index np = pgp.nelem();
  Index nr = rgp.nelem();
  Index nc = cgp.nelem();
  assert(is_size(ia,
                 nb,np,nr,nc));    
  assert(is_size(itw,
                 nb,np,nr,nc,
                 16));
 
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one. We
  // only check the first element.
  assert( is_same_within_epsilon( itw(0,0,0,0,Range(joker)).sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // We have to loop all the points in the new grid:
  for ( Index ib=0; ib<nb; ++ib )
    {
      const GridPos& tb = bgp[ib];
      for ( Index ip=0; ip<np; ++ip )
        {
          const GridPos& tp = pgp[ip];
          for ( Index ir=0; ir<nr; ++ir )
            {
              const GridPos& tr = rgp[ir];
              for ( Index ic=0; ic<nc; ++ic )
                {
                  // Current grid position:
                  const GridPos& tc = cgp[ic];
 
                  // Get handle to current element of output tensor and
                  // initialize it to zero:
                  Numeric& tia = ia(ib,ip,ir,ic);
                  tia = 0;
 
                  Index iti = 0;
                  for ( Index b=0; b<2; ++b )
                    for ( Index p=0; p<2; ++p )
                      for ( Index r=0; r<2; ++r )
                        for ( Index c=0; c<2; ++c )
                          {
                            tia += a.get(tb.idx+b,
                                         tp.idx+p,
                                         tr.idx+r,
                                         tc.idx+c) * itw.get(ib,ip,ir,ic,
                                                             iti);
                            ++iti;
                          }
                }
            }
        }
    }
}
 
 
void interp( Tensor5View           ia,
             ConstTensor6View      itw,
             ConstTensor5View      a,   
             const ArrayOfGridPos& sgp,
             const ArrayOfGridPos& bgp,
             const ArrayOfGridPos& pgp,
             const ArrayOfGridPos& rgp,
             const ArrayOfGridPos& cgp)
{
  Index ns = sgp.nelem();
  Index nb = bgp.nelem();
  Index np = pgp.nelem();
  Index nr = rgp.nelem();
  Index nc = cgp.nelem();
  assert(is_size(ia,
                 ns,nb,np,nr,nc));    
  assert(is_size(itw,
                 ns,nb,np,nr,nc,
                 32));
 
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one. We
  // only check the first element.
  assert( is_same_within_epsilon( itw(0,0,0,0,0,Range(joker)).sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // We have to loop all the points in the new grid:
  for ( Index is=0; is<ns; ++is )
    {
      const GridPos& ts = sgp[is];
      for ( Index ib=0; ib<nb; ++ib )
        {
          const GridPos& tb = bgp[ib];
          for ( Index ip=0; ip<np; ++ip )
            {
              const GridPos& tp = pgp[ip];
              for ( Index ir=0; ir<nr; ++ir )
                {
                  const GridPos& tr = rgp[ir];
                  for ( Index ic=0; ic<nc; ++ic )
                    {
                      // Current grid position:
                      const GridPos& tc = cgp[ic];
 
                      // Get handle to current element of output tensor and
                      // initialize it to zero:
                      Numeric& tia = ia(is,ib,ip,ir,ic);
                      tia = 0;
 
                      Index iti = 0;
                      for ( Index s=0; s<2; ++s )
                        for ( Index b=0; b<2; ++b )
                          for ( Index p=0; p<2; ++p )
                            for ( Index r=0; r<2; ++r )
                              for ( Index c=0; c<2; ++c )
                                {
                                  tia += a.get(ts.idx+s,
                                               tb.idx+b,
                                               tp.idx+p,
                                               tr.idx+r,
                                               tc.idx+c) * itw.get(is,ib,ip,ir,ic,
                                                                   iti);
                                  ++iti;
                                }
                    }
                }
            }
        }
    }
}
 
 
void interp( Tensor6View           ia,
             ConstTensor7View      itw,
             ConstTensor6View      a,   
             const ArrayOfGridPos& vgp,
             const ArrayOfGridPos& sgp,
             const ArrayOfGridPos& bgp,
             const ArrayOfGridPos& pgp,
             const ArrayOfGridPos& rgp,
             const ArrayOfGridPos& cgp)
{
  Index nv = vgp.nelem();
  Index ns = sgp.nelem();
  Index nb = bgp.nelem();
  Index np = pgp.nelem();
  Index nr = rgp.nelem();
  Index nc = cgp.nelem();
  assert(is_size(ia,
                 nv,ns,nb,np,nr,nc));    
  assert(is_size(itw,
                 nv,ns,nb,np,nr,nc,
                 64));
 
  // Check that interpolation weights are valid. The sum of all
  // weights (last dimension) must always be approximately one. We
  // only check the first element.
  assert( is_same_within_epsilon( itw(0,0,0,0,0,0,Range(joker)).sum(),
                                  1,
                                  sum_check_epsilon ) );
  
  // We have to loop all the points in the new grid:
  for ( Index iv=0; iv<nv; ++iv )
    {
      const GridPos& tv = vgp[iv];
      for ( Index is=0; is<ns; ++is )
        {
          const GridPos& ts = sgp[is];
          for ( Index ib=0; ib<nb; ++ib )
            {
              const GridPos& tb = bgp[ib];
              for ( Index ip=0; ip<np; ++ip )
                {
                  const GridPos& tp = pgp[ip];
                  for ( Index ir=0; ir<nr; ++ir )
                    {
                      const GridPos& tr = rgp[ir];
                      for ( Index ic=0; ic<nc; ++ic )
                        {
                          // Current grid position:
                          const GridPos& tc = cgp[ic];
 
                          // Get handle to current element of output tensor and
                          // initialize it to zero:
                          Numeric& tia = ia(iv,is,ib,ip,ir,ic);
                          tia = 0;
 
                          Index iti = 0;
                          for ( Index v=0; v<2; ++v )
                            for ( Index s=0; s<2; ++s )
                              for ( Index b=0; b<2; ++b )
                                for ( Index p=0; p<2; ++p )
                                  for ( Index r=0; r<2; ++r )
                                    for ( Index c=0; c<2; ++c )
                                      {
                                        tia += a.get(tv.idx+v,
                                                     ts.idx+s,
                                                     tb.idx+b,
                                                     tp.idx+p,
                                                     tr.idx+r,
                                                     tc.idx+c) * itw.get(iv,is,ib,ip,ir,ic,
                                                                         iti);
                                        ++iti;
                                      }
                        }
                    }
                }
            }
        }
    }
}
 
 
 
Numeric interp_poly(ConstVectorView x,
                    ConstVectorView y,
                    const Numeric& x_i, 
                    const GridPos& gp)
{
  Index N_x = x.nelem();
 
  assert(N_x == y.nelem());
  assert(N_x > 2);
 
  Vector xa(4), ya(4);
  Numeric y_int;
  Numeric dy_int;
  y_int = 0.;
 
  // 1 - polynomial interpolation (3 points) with grid position search
  // 2 - polynomial interpolation (3 points) without grid position search
  // 3 - polynomial interpolation (4 points)
 
  Index interp_method = 1; 
 
  if (interp_method == 1)
    { 
      
      // Pick out three points for interpolation
      if((gp.fd[0] <= 0.5 && gp.idx > 0) || gp.idx == N_x-2 )
        {
          xa[0] = x[gp.idx - 1];
          xa[1] = x[gp.idx];
          xa[2] = x[gp.idx + 1];
      
          ya[0] = y[gp.idx - 1];
          ya[1] = y[gp.idx];
          ya[2] = y[gp.idx + 1];
        }
 
      else if((gp.fd[0] > 0.5 && gp.idx < N_x-2) || gp.idx == 0 )
        {
          xa[0] = x[gp.idx];
          xa[1] = x[gp.idx + 1];
          xa[2] = x[gp.idx + 2];
      
          ya[0] = y[gp.idx];
          ya[1] = y[gp.idx + 1];
          ya[2] = y[gp.idx + 2];
        } 
  
      else if(gp.idx == N_x-1)
        {
          xa[0] = x[N_x - 2];
          xa[1] = x[N_x - 1];
          xa[2] = x[N_x];
      
          ya[0] = y[N_x - 2];
          ya[1] = y[N_x - 1];
          ya[2] = y[N_x];
        }  
      else
        {
          assert(false);
          arts_exit();
        }
      
      polint(y_int, dy_int, xa, ya, 3, x_i); 
  
    }
  
  else if (interp_method == 2) 
    {
      if( gp.idx == 0 )
        {
          xa[0] = x[gp.idx];
          xa[1] = x[gp.idx + 1];
          xa[2] = x[gp.idx + 2];
      
          ya[0] = y[gp.idx];
          ya[1] = y[gp.idx + 1];
          ya[2] = y[gp.idx + 2];
        }
      else if(gp.idx == N_x-1)
        {
          xa[0] = x[gp.idx - 2];
          xa[1] = x[gp.idx - 1];
          xa[2] = x[gp.idx];
      
          ya[0] = y[gp.idx - 2];
          ya[1] = y[gp.idx - 1];
          ya[2] = y[gp.idx];
        }
      else 
        {
          xa[0] = x[gp.idx - 1];
          xa[1] = x[gp.idx];
          xa[2] = x[gp.idx + 1];
      
          ya[0] = y[gp.idx - 1];
          ya[1] = y[gp.idx];
          ya[2] = y[gp.idx + 1]; 
        }
      
      // Polinominal interpolation, n = 3
      polint(y_int, dy_int, xa, ya, 3, x_i); 
    }
  
  else if (interp_method == 3)
    { 
      // Take 4 points
      if( gp.idx == 0 )
        {
          xa[0] = - x[gp.idx + 1];
          xa[1] = x[gp.idx + 0];
          xa[2] = x[gp.idx + 1];
          xa[3] = x[gp.idx + 2];
          
          ya[0] = y[gp.idx + 1];
          ya[1] = y[gp.idx + 0];
          ya[2] = y[gp.idx + 1];
          ya[3] = y[gp.idx + 2]; 
        }
      else if(gp.idx == N_x-1)
        {
          xa[0] = x[gp.idx - 1];
          xa[1] = x[gp.idx - 0];
          xa[2] = 2*x[gp.idx] - x[gp.idx-1];
          xa[3] = 2*x[gp.idx] - x[gp.idx-2];
          
          ya[0] = y[gp.idx - 1];
          ya[1] = y[gp.idx - 0];
          ya[2] = y[gp.idx - 1];
          ya[3] = y[gp.idx - 2];
        }
      else if(gp.idx == N_x-2)
        {
          xa[0] = x[gp.idx - 2];
          xa[1] = x[gp.idx - 1];
          xa[2] = x[gp.idx ];
          xa[3] = x[gp.idx + 1];
          
          ya[0] = y[gp.idx - 2];
          ya[1] = y[gp.idx - 1];
          ya[2] = y[gp.idx];
          ya[3] = y[gp.idx + 1];
        }
      else 
        {
          xa[0] = x[gp.idx - 1];
          xa[1] = x[gp.idx];
          xa[2] = x[gp.idx + 1];
          xa[3] = x[gp.idx + 2];
          
          ya[0] = y[gp.idx - 1];
          ya[1] = y[gp.idx];
          ya[2] = y[gp.idx + 1];
          ya[3] = y[gp.idx + 2];
        }
      // Polinominal interpolation, n = 4
      polint(y_int, dy_int, xa, ya, 4, x_i); 
    }
      
  
  return y_int;
}
 
 
 
 
 
void polint(Numeric& y_int,
            Numeric& dy_int,
            ConstVectorView xa,
            ConstVectorView ya,
            const Index& n,
            const Numeric& x)
{
  Index ns = 1;
  Numeric den, dif, dift, ho, hp, w;
  
  dif = abs(x-xa[0]);
 
  Vector c(n);
  Vector d(n); 
  
  // Find the index of the closest table entry
  for(Index i=0; i<n; i++)
    {
      if( (dift = abs(x-xa[i])) < dif)
        {
          ns = i;
          dif = dift;
        }
      // Initialize c and d
      c[i] = ya[i];
      d[i] = ya[i];
    }
  // Initial approximation to y
  y_int = ya[ns--];
  
  for(Index m=1; m<n; m++)
    {
      for(Index i=0; i < n-m; i++)
        {
          ho = xa[i] - x;
          hp = xa[i+m] - x;
          w = c[i+1] - d[i];
          den = ho-hp;
          // This error occurs when two input xa's are identical. 
          assert(den != 0.);
          den = w/den;
          d[i] = hp * den;
          c[i] = ho * den;
        }
      y_int += (dy_int = (2*(ns+1) < (n-m) ? c[ns+1] : d[ns--] ));
    }
}