/* Copyright (C) 2009 Carlo de Falco

   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 3 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/>.
*/

#include <iostream>
#include <octave/oct.h>
#include <octave/oct-map.h>
#include "low_level_functions.h"

static double gammaln(double xx)
// Compute logarithm of the gamma function
// Algorithm from 'Numerical Recipes in C, 2nd Edition' pg214.
{
  double x,y,tmp,ser;
  static double cof[6] = {76.18009172947146,-86.50532032291677,
			  24.01409824083091,-1.231739572450155,
			  0.12086650973866179e-2, -0.5395239384953e-5};
  int j;
  y = x = xx;
  tmp = x + 5.5;
  tmp -= (x+0.5) * log(tmp);
  ser = 1.000000000190015;
  for (j=0; j<=5; j++) ser += cof[j]/++y;
  return -tmp+log(2.5066282746310005*ser/x);
}


static double factln(int n)
// computes ln(n!)
// Numerical Recipes in C
// Algorithm from 'Numerical Recipes in C, 2nd Edition' pg215.
{
  static int ntop = 0;
  static double a[101];
  
  if (n <= 1) return 0.0;
  while (n > ntop)
    {
      ++ntop;
      a[ntop] = gammaln(ntop+1.0);
    }
  return a[n];
}

static double bincoeff(int n, int k)
// Computes the binomial coefficient.
//
//     ( n )      n!
//     (   ) = --------
//     ( k )   k!(n-k)!
//
// Algorithm from 'Numerical Recipes in C, 2nd Edition' pg215.
{
  return floor(0.5+exp(factln(n)-factln(k)-factln(n-k)));
}  
   

DEFUN_DLD(nrbsurfderiveval, args, nargout,"\
\nNRBSURFDERIVEVAL: Evaluate n-th order derivatives of a NURBS surface.\n\
\n\
\n usage: skl = nrbsurfderiveval (srf, [u; v], d) \
\n\
\n   INPUT  :\
\n\
\n    srf   : NURBS surface structure, see nrbmak\
\n\
\n    u, v  : parametric coordinates of the point where we compute the\
\n      derivatives\
\n\
\n    d     : number of partial derivatives to compute\
\n\
\n   OUTPUT :\
\n\
\n    skl (i, j, k, l) = i-th component derived j-1,k-1 times at the\
\n      l-th point.\
\n\
\n Adaptation of algorithm A4.4 from the NURBS book\n")
{
  //function skl = nrbsurfderiveval (srf, uv, d) 
  octave_value_list retval;

  octave_scalar_map srf = args(0).scalar_map_value();
  Matrix            uv = args(1).matrix_value ();
  octave_idx_type   d = args(2).idx_type_value ();

  if (! error_state)
    {
      Array<octave_idx_type> idxta (dim_vector (4, 1), 0);
      dim_vector idxa; idxa.resize (4);
      idxa(0) = 3; idxa(1) = d+1; 
      idxa(2) = d+1; idxa(3) = uv.columns (); 
      NDArray skl (idxa, 0.0);
      
      octave_idx_type n = octave_idx_type 
	((srf.contents("number").row_vector_value())(0) - 1);
      octave_idx_type m = octave_idx_type 
	((srf.contents("number").row_vector_value())(1) - 1);
      octave_idx_type p = octave_idx_type 
	((srf.contents("order").row_vector_value())(0) - 1);
      octave_idx_type q = octave_idx_type 
	((srf.contents("order").row_vector_value())(1) - 1);
      
      Cell knots = srf.contents("knots").cell_value();
      RowVector knotsu = knots.elem (0).row_vector_value ();
      RowVector knotsv = knots.elem (1).row_vector_value ();
      
      NDArray coefs  = srf.contents("coefs").array_value();
      
      Array<idx_vector> idx(dim_vector (3, 1), idx_vector(':'));	 
      idx (0) = idx_vector (3);
      Matrix weights (NDArray (coefs.index (idx).squeeze ()));

      for (octave_idx_type iu(0); iu<uv.cols (); iu++)
	{
	  
	  Matrix wders;
	  surfderiveval (n, p, knotsu, m, q, knotsv, weights, uv(0,iu), uv(1,iu), d, wders);      
	  
	  for (octave_idx_type idim (0); idim<=2; idim++)
	    {

	      Matrix Aders; idx(0) = idx_vector (idim);
	      Matrix P (NDArray (coefs.index (idx).squeeze ()));
	      surfderiveval (n, p, knotsu, m, q, knotsv, P, uv(0,iu), uv(1,iu), d, Aders);;      
	      
	      for (octave_idx_type k(0); k<=d; k++)
		{
		  for (octave_idx_type l(0); l<=d-k; l++)
		    {
		      assert (k < Aders.rows () && l < Aders.cols ());
		      double v = Aders(k, l);
		      for (octave_idx_type j(1); j<=l; j++)
			{
			  assert (idim<idxa(0) && k<idxa(1) && l<idxa(2) && iu<idxa(3));
			  idxta(0) = idim; idxta(1) = k; idxta(2) = l-j; idxta(3) = iu;
			  assert (j < wders.cols ());
			  v -= bincoeff(l,j) * wders(0,j) * skl(idxta);
			}
		      for (octave_idx_type i(1); i<=k; i++)
			{
			  assert (idim<idxa(0) && k-i<idxa(1) && l<idxa(2) && iu<idxa(3));
			  idxta(0) = idim; idxta(1) = k-i; idxta(2) = l; idxta(3) = iu;
			  assert (i < wders.cols ());
			  v -= bincoeff(k,i) * wders(i,0) * skl(idxta);
			  double v2 = 0.0;
			  for (octave_idx_type j(1);j<=l;j++)
			    {
			      idxta(0) = idim; idxta(1) = k-i; idxta(2) = l-j; idxta(3) = iu;
			      v2 += bincoeff(l,j) * wders(i,j) * skl(idxta);
			    }
			  v -= bincoeff(k,i) * v2;
			}
		      assert (idim<idxa(0) && k<idxa(1) && l<idxa(2) && iu<idxa(3));
		      idxta(0) = idim; idxta(1) = k; idxta(2) = l; idxta(3) = iu;
		      skl(idxta) = v/wders(0,0);
		    }
		}
	    }
	  
	} 
      retval(0) = octave_value (skl);
    }
  return retval;
}
/*
%!test
%! k = [0 0  1 1];
%! c = [0 1];
%! [coef(2,:,:), coef(1,:,:)] = meshgrid (c, c);
%! coef(3,:,:) = coef(1,:,:);
%! srf = nrbmak (coef, {k, k});
%! [u, v] = meshgrid (linspace(0,1,11));
%! uv = [u(:)';v(:)'];
%! skl = nrbsurfderiveval (srf, uv, 0);
%! assert (squeeze (skl (1:2,1,1,:)), nrbeval (srf, uv)(1:2,:), 1e3*eps)


%!test
%! k = [0 0  1 1];
%! c = [0 1];
%! [coef(2,:,:), coef(1,:,:)] = meshgrid (c, c);
%! coef(3,:,:) = coef(1,:,:);
%! srf = nrbmak (coef, {k, k});
%! srf = nrbkntins (srf, {[], rand(2,1)});
%! [u, v] = meshgrid (linspace(0,1,11));
%! uv = [u(:)';v(:)'];
%! skl = nrbsurfderiveval (srf, uv, 0);
%! assert (squeeze (skl (1:2,1,1,:)), nrbeval (srf, uv)(1:2,:), 1e3*eps)

%!shared srf, uv
%!test 
%! k = [0 0 0 1 1 1];
%! c = [0 1/2 1];
%! [coef(1,:,:), coef(2,:,:)] = meshgrid (c, c);
%! coef(3,:,:) = coef(1,:,:);
%! srf = nrbmak (coef, {k, k});
%! ders= nrbderiv (srf);
%! [u, v] = meshgrid (linspace(0,1,11));
%! uv = [u(:)';v(:)'];
%! skl = nrbsurfderiveval (srf, uv, 1);
%! [fun, der] = nrbdeval (srf, ders, uv);
%! assert (squeeze (skl (1:2,1,1,:)), fun(1:2,:), 1e3*eps)
%! assert (squeeze (skl (1:2,2,1,:)), der{1}(1:2,:), 1e3*eps)
%! assert (squeeze (skl (1:2,1,2,:)), der{2}(1:2,:), 1e3*eps)
%!
%!test 
%! srf = nrbdegelev (srf, [3, 1]);
%! ders= nrbderiv (srf);
%! [fun, der] = nrbdeval (srf, ders, uv);
%! skl = nrbsurfderiveval (srf, uv, 1);
%! assert (squeeze (skl (1:2,1,1,:)), fun(1:2,:), 1e3*eps)
%! assert (squeeze (skl (1:2,2,1,:)), der{1}(1:2,:), 1e3*eps)
%! assert (squeeze (skl (1:2,1,2,:)), der{2}(1:2,:), 1e3*eps)

%!shared uv
%!test 
%! k = [0 0 0 1 1 1];
%! c = [0 1/2 1];
%! [coef(2,:,:), coef(1,:,:)] = meshgrid (c, c);
%! coef(3,:,:) = coef(1,:,:);
%! srf = nrbmak (coef, {k, k});
%! ders= nrbderiv (srf);
%! [u, v] = meshgrid (linspace(0,1,11));
%! uv = [u(:)';v(:)'];
%! skl = nrbsurfderiveval (srf, uv, 1);
%! [fun, der] = nrbdeval (srf, ders, uv);
%! assert (squeeze (skl (1:2,1,1,:)), fun(1:2,:), 1e3*eps)
%! assert (squeeze (skl (1:2,2,1,:)), der{1}(1:2,:), 1e3*eps)
%! assert (squeeze (skl (1:2,1,2,:)), der{2}(1:2,:), 1e3*eps)
%!
%!test 
%! p = q = 3;
%! mcp = 5; ncp = 5;
%! Lx  = Ly  = 10*rand(1);
%! srf = nrbdegelev (nrb4surf ([0 0], [Lx, 0], [0 Ly], [Lx Ly]), [p-1, q-1]);
%! %%srf = nrbkntins (srf, {linspace(0,1,mcp-p+2)(2:end-1), linspace(0,1,ncp-q+2)(2:end-1)});
%! %%srf.coefs = permute (srf.coefs, [1 3 2]);
%! ders= nrbderiv (srf);
%! [fun, der] = nrbdeval (srf, ders, uv);
%! skl = nrbsurfderiveval (srf, uv, 1);
%! assert (squeeze (skl (1:2,1,1,:)), fun(1:2,:), 1e3*eps)
%! assert (squeeze (skl (1:2,2,1,:)), der{1}(1:2,:), 1e3*eps)
%! assert (squeeze (skl (1:2,1,2,:)), der{2}(1:2,:), 1e3*eps)

%!shared srf, uv, P, dPdx, d2Pdx2, c1, c2
%!test
%! [u, v] = meshgrid (linspace(0,1,10));
%! uv = [u(:)';v(:)'];
%! c1 = nrbmak([0 1/2 1; 0 1 0],[0 0 0 1 1 1]);
%! c1 = nrbtform (c1, vecrotx (pi/2));
%! c2  = nrbtform(c1, vectrans([0 1 0]));
%! srf = nrbdegelev (nrbruled (c1, c2), [3, 1]);
%! skl = nrbsurfderiveval (srf, uv, 2);
%! P = squeeze(skl(:,1,1,:));
%! dPdx = squeeze(skl(:,2,1,:));
%! d2Pdx2 = squeeze(skl(:,3,1,:));
%!assert(P(3,:), 2*(P(1,:)-P(1,:).^2),100*eps)
%!assert(dPdx(3,:), 2-4*P(1,:), 100*eps)
%!assert(d2Pdx2(3,:), -4+0*P(1,:), 100*eps)
%! srf = nrbdegelev (nrbruled (c1, c2), [5, 6]);
%! skl = nrbsurfderiveval (srf, uv, 2);
%! P = squeeze(skl(:,1,1,:));
%! dPdx = squeeze(skl(:,2,1,:));
%! d2Pdx2 = squeeze(skl(:,3,1,:));
%! assert (squeeze (skl (1:2,1,1,:)), nrbeval (srf, uv)(1:2,:), 1e3*eps)
%!assert(P(3,:), 2*(P(1,:)-P(1,:).^2),100*eps)
%!assert(dPdx(3,:), 2-4*P(1,:), 100*eps)
%!assert(d2Pdx2(3,:), -4+0*P(1,:), 100*eps)
%!
%!test
%! skl = nrbsurfderiveval (srf, uv, 0);
%! assert (squeeze (skl (1:2,1,1,:)), nrbeval (srf, uv)(1:2,:), 1e3*eps)

%!shared dPdu, d2Pdu2, P, srf, uv
%!test
%! [u, v] = meshgrid (linspace(0,1,10));
%! uv = [u(:)';v(:)'];
%! c1 = nrbmak([0 1/2 1; 0.1 1.6 1.1; 0 0 0],[0 0 0 1 1 1]);
%! c2 = nrbmak([0 1/2 1; 0.1 1.6 1.1; 1 1 1],[0 0 0 1 1 1]);
%! srf = nrbdegelev (nrbruled (c1, c2), [0, 1]);
%! skl = nrbsurfderiveval (srf, uv, 2);
%! P = squeeze(skl(:,1,1,:));
%! dPdu = squeeze(skl(:,2,1,:));
%! dPdv = squeeze(skl(:,1,2,:));
%! d2Pdu2 = squeeze(skl(:,3,1,:));
%! assert (squeeze (skl (1:2,1,1,:)), nrbeval (srf, uv)(1:2,:), 1e3*eps)
%!assert(dPdu(2,:), 3-4*P(1,:),100*eps)
%!assert(d2Pdu2(2,:), -4+0*P(1,:),100*eps)
%!
%!test
%! skl = nrbsurfderiveval (srf, uv, 0);
%! assert (squeeze (skl (1:2,1,1,:)), nrbeval (srf, uv)(1:2,:), 1e3*eps)

%!test
%! srf = nrb4surf([0 0], [1 0], [0 1], [1 1]);
%! geo = nrbdegelev (srf, [3 3]);
%! geo.coefs (4, 2:end-1, 2:end-1) += .1 * rand (1, geo.number(1)-2, geo.number(2)-2);
%! geo = nrbkntins (geo, {[.1:.1:.9], [.2:.2:.8]});
%! [u, v] = meshgrid (linspace(0,1,10));
%! uv = [u(:)';v(:)'];
%! skl = nrbsurfderiveval (geo, uv, 2);
%! dgeo = nrbderiv (geo);
%! [pnts, ders] = nrbdeval (geo, dgeo, uv);
%! assert (ders{1}, squeeze(skl(:,2,1,:)), 1e-9)
%! assert (ders{2}, squeeze(skl(:,1,2,:)), 1e-9)

%!test
%! ku = kv = [0 0 0 1 1 1];
%! c(1,:,:) = [1 1 1]'*[0 0 1] - 1;
%! c(2,:,:) = (1+[1 1 1]'*[0 1/2 1]) .* ([0 1/2 1]'*[1 1 1]);
%! c(3,:,:) = ([1 1 1]'*[0 1/2 1]) .* ([0 1/2 1]'*[1 1 1]) ;
%! c(4,:,:) = (1+[1 1 1]'*[0 1/2 1]);
%! c = permute (c, [1 3 2]);
%! geo = nrbmak (c, {ku, kv});
%!
%! [u, v] = meshgrid (linspace(0,1,50));
%! uv = [u(:), v(:)]';
%! dF = nrbsurfderiveval (geo, uv, 2);
%!
%! assert (dF(1,1,1,:)(:), u(:)-1, 10*eps)
%! assert (dF(2,1,1,:)(:), v(:), 10*eps)
%! assert (dF(3,1,1,:)(:), u(:).*v(:)./(u(:)+1), 10*eps)
%! assert (dF(1,2,1,:)(:), ones (size (u(:))), 10*eps)
%! assert (dF(1,1,2,:)(:), zeros (size (u(:))), 10*eps)
%! assert (dF(2,2,1,:)(:), zeros (size (u(:))), 10*eps)
%! assert (dF(2,1,2,:)(:), ones (size (u(:))), 10*eps)
%! assert (dF(3,1,2,:)(:), u(:)./(u(:)+1), 10*eps)
%! assert (dF(3,2,1,:)(:), v(:)./(u(:)+1) - u(:).*v(:)./(u(:)+1).^2, 10*eps)
%! assert (dF(1:2,3,:,:)(:), zeros (size (dF(1:2,3,:,:)(:))), 10*eps)
%! assert (dF(1:2,:,3,:)(:), zeros (size (dF(1:2,:,3,:)(:))), 10*eps)
%! assert (dF(3,3,1,:)(:),  -2*v(:)./(u(:)+1).^3, 10*eps)
%! assert (dF(3,1,3,:)(:), zeros (size (dF(3,1,3,:)(:))), 10*eps)

%!test
%! ku = kv = [0 0 0 1 1 1];
%! c(1,:,:) = [1 1 1]'*[0 0 1] - 1;
%! c(2,:,:) = ([1 1 1]'*[0 1/2 1]) .* ([0 1/2 1]'*[1 1 1]) ;
%! c(4,:,:) = (1+[1 1 1]'*[0 1/2 1]);
%! c = permute (c, [1 3 2]);
%! geo = nrbmak (c, {ku, kv});
%!
%! [u, v] = meshgrid (linspace(0,1,50));
%! uv = [u(:), v(:)]';
%! dF = nrbsurfderiveval (geo, uv, 2);
%!
%! assert (dF(1,1,1,:)(:), u(:)-1, 10*eps)
%! assert (dF(3,1,1,:)(:), zeros (size (u(:))), 10*eps)
%! assert (dF(2,1,1,:)(:), u(:).*v(:)./(u(:)+1), 10*eps)
%! assert (dF(1,2,1,:)(:), ones (size (u(:))), 10*eps)
%! assert (dF(1,1,2,:)(:), zeros (size (u(:))), 10*eps)
%! assert (dF(3,2,1,:)(:), zeros (size (u(:))), 10*eps)
%! assert (dF(3,1,2,:)(:), zeros (size (u(:))), 10*eps)
%! assert (dF(2,1,2,:)(:), u(:)./(u(:)+1), 10*eps)
%! assert (dF(2,2,1,:)(:), v(:)./(u(:)+1) - u(:).*v(:)./(u(:)+1).^2, 10*eps)
%! assert (dF([1 3],3,:,:)(:), zeros (size (dF([1 3],3,:,:)(:))), 10*eps)
%! assert (dF([1 3],:,3,:)(:), zeros (size (dF([1 3],:,3,:)(:))), 10*eps)
%! assert (dF(2,3,1,:)(:),  -2*v(:)./(u(:)+1).^3, 10*eps)
%! assert (dF(2,1,3,:)(:), zeros (size (dF(3,1,3,:)(:))), 10*eps)

%!test
%! crv = nrbline ([1 0], [2 0]);
%! srf = nrbrevolve (crv, [0 0 0], [0 0 1], pi/2);
%! srf = nrbtransp (srf);
%! [v, u] = meshgrid (linspace (0, 1, 11));
%! uv = [u(:)'; v(:)'];
%! skl = nrbsurfderiveval (srf, uv, 2);
%! c = sqrt(2);
%! w      = @(x, y) (2 - c)*y.^2 + (c-2)*y + 1;
%! dwdy   = @(x, y) 2*(2-c)*y + c - 2;
%! d2wdy2 = @(x, y) 2*(2-c);
%! F1 = @(x, y) (x+1) .* ((1-y).^2 + c*y.*(1-y)) ./ w(x,y);
%! F2 = @(x, y) (x+1) .* (y.^2 + c*y.*(1-y)) ./ w(x,y);
%! dF1dx = @(x, y) ((1-y).^2 + c*y.*(1-y)) ./ w(x,y);
%! dF2dx = @(x, y) (y.^2 + c*y.*(1-y)) ./ w(x,y);
%! dF1dy = @(x, y) (x+1) .* ((2 - 2*c)*y + c - 2) ./ w(x,y) - (x+1) .* ((1-y).^2 + c*y.*(1-y)) .* dwdy(x,y) ./ w(x,y).^2;
%! dF2dy = @(x, y) (x+1) .* ((2 - 2*c)*y + c) ./ w(x,y) - (x+1) .* (y.^2 + c*y.*(1-y)) .* dwdy(x,y) ./ w(x,y).^2;
%! d2F1dx2 = @(x, y) zeros (size (x));
%! d2F2dx2 = @(x, y) zeros (size (x));
%! d2F1dxdy = @(x, y) ((2 - 2*c)*y + c - 2) ./ w(x,y) - ((1-y).^2 + c*y.*(1-y)) .* dwdy(x,y) ./ w(x,y).^2;
%! d2F2dxdy = @(x, y) ((2 - 2*c)*y + c) ./ w(x,y) - (y.^2 + c*y.*(1-y)) .* dwdy(x,y) ./ w(x,y).^2;
%! d2F1dy2  = @(x, y) (x+1)*(2 - 2*c) ./ w(x,y) - 2*(x+1) .* ((2 - 2*c)*y + c - 2) .* dwdy(x,y) ./ w(x,y).^2 - ...
%!                    (x+1) .* ((1-y).^2 + c*y.*(1-y)) * d2wdy2(x,y) ./ w(x,y).^2 + ...
%!                    2 * (x+1) .* ((1-y).^2 + c*y.*(1-y)) .* w(x,y) .*dwdy(x,y).^2 ./ w(x,y).^4;
%! d2F2dy2  = @(x, y) (x+1)*(2 - 2*c) ./ w(x,y) - 2*(x+1) .* ((2 - 2*c)*y + c) .* dwdy(x,y) ./ w(x,y).^2 - ...
%!                    (x+1) .* (y.^2 + c*y.*(1-y)) * d2wdy2(x,y) ./ w(x,y).^2 + ...
%!                    2 * (x+1) .* (y.^2 + c*y.*(1-y)) .* w(x,y) .*dwdy(x,y).^2 ./ w(x,y).^4;
%! assert ([F1(u(:),v(:)), F2(u(:),v(:))], squeeze(skl(1:2,1,1,:))', 1e2*eps);
%! assert ([dF1dx(u(:),v(:)), dF2dx(u(:),v(:))], squeeze(skl(1:2,2,1,:))', 1e2*eps);
%! assert ([dF1dy(u(:),v(:)), dF2dy(u(:),v(:))], squeeze(skl(1:2,1,2,:))', 1e2*eps);
%! assert ([d2F1dx2(u(:),v(:)), d2F2dx2(u(:),v(:))], squeeze(skl(1:2,3,1,:))', 1e2*eps);
%! assert ([d2F1dxdy(u(:),v(:)), d2F2dxdy(u(:),v(:))], squeeze(skl(1:2,2,2,:))', 1e2*eps);
%! assert ([d2F1dy2(u(:),v(:)), d2F2dy2(u(:),v(:))], squeeze(skl(1:2,1,3,:))', 1e2*eps);

%!test
%! knots = {[0 0 1 1] [0 0 1 1]};
%! coefs(:,1,1) = [0;0;0;1];
%! coefs(:,2,1) = [1;0;0;1];
%! coefs(:,1,2) = [0;1;0;1];
%! coefs(:,2,2) = [1;1;1;2];
%! srf = nrbmak (coefs, knots);
%! [v, u] = meshgrid (linspace (0, 1, 3));
%! uv = [u(:)'; v(:)'];
%! skl = nrbsurfderiveval (srf, uv, 2);
%! w = @(x, y) x.*y + 1;
%! F1 = @(x, y) x ./ w(x,y);
%! F2 = @(x, y) y ./ w(x,y);
%! F3 = @(x, y) x .* y ./ w(x,y);
%! dF1dx = @(x, y) 1./w(x,y) - x.*y./w(x,y).^2;
%! dF1dy = @(x, y)  - x.^2./w(x,y).^2;
%! dF2dx = @(x, y)  - y.^2./w(x,y).^2;
%! dF2dy = @(x, y) 1./w(x,y) - x.*y./w(x,y).^2;
%! dF3dx = @(x, y) y./w(x,y) - x.*(y./w(x,y)).^2;
%! dF3dy = @(x, y) x./w(x,y) - y.*(x./w(x,y)).^2;
%! d2F1dx2  = @(x, y) -2*y./w(x,y).^2 + 2*x.*y.^2./w(x,y).^3;
%! d2F1dy2  = @(x, y) 2*x.^3./w(x,y).^3;
%! d2F1dxdy = @(x, y) -x./w(x,y).^2 - x./w(x,y).^2 + 2*x.^2.*y./w(x,y).^3;
%! d2F2dx2  = @(x, y) 2*y.^3./w(x,y).^3;
%! d2F2dy2  = @(x, y) -2*x./w(x,y).^2 + 2*y.*x.^2./w(x,y).^3;
%! d2F2dxdy = @(x, y) -y./w(x,y).^2 - y./w(x,y).^2 + 2*y.^2.*x./w(x,y).^3;
%! d2F3dx2  = @(x, y) -2*y.^2./w(x,y).^2 + 2*x.*y.^3./w(x,y).^3;
%! d2F3dy2  = @(x, y) -2*x.^2./w(x,y).^2 + 2*y.*x.^3./w(x,y).^3;
%! d2F3dxdy = @(x, y) 1./w(x,y) - 3*x.*y./w(x,y).^2 + 2*(x.*y).^2./w(x,y).^3;
%! assert ([F1(u(:),v(:)), F2(u(:),v(:)), F3(u(:),v(:))], squeeze(skl(1:3,1,1,:))', 1e2*eps);
%! assert ([dF1dx(u(:),v(:)), dF2dx(u(:),v(:)), dF3dx(u(:),v(:))], squeeze(skl(1:3,2,1,:))', 1e2*eps);
%! assert ([dF1dy(u(:),v(:)), dF2dy(u(:),v(:)), dF3dy(u(:),v(:))], squeeze(skl(1:3,1,2,:))', 1e2*eps);
%! assert ([d2F1dx2(u(:),v(:)), d2F2dx2(u(:),v(:)), d2F3dx2(u(:),v(:))], squeeze(skl(1:3,3,1,:))', 1e2*eps);
%! assert ([d2F1dy2(u(:),v(:)), d2F2dy2(u(:),v(:)), d2F3dy2(u(:),v(:))], squeeze(skl(1:3,1,3,:))', 1e2*eps);
%! assert ([d2F1dxdy(u(:),v(:)), d2F2dxdy(u(:),v(:)), d2F3dxdy(u(:),v(:))], squeeze(skl(1:3,2,2,:))', 1e2*eps);

*/