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## -*- texinfo -*-
## @deftypefn  {} {@var{xy} =} stream2 (@var{x}, @var{y}, @var{u}, @var{v}, @var{sx}, @var{sy})
## @deftypefnx {} {@var{xy} =} stream2 (@var{u}, @var{v}, @var{sx}, @var{sy})
## @deftypefnx {} {@var{xy} =} stream2 (@dots{}, @var{options})
## Compute 2-D streamline data.
##
## Calculates streamlines of a vector field given by @code{[@var{u}, @var{v}]}.
## The vector field is defined over a rectangular grid given by
## @code{[@var{x}, @var{y}]}.  The streamlines start at the seed points
## @code{[@var{sx}, @var{sy}]}.  The returned value @var{xy} contains a cell
## array of vertex arrays.  If the starting point is outside the vector field,
## @code{[]} is returned.
##
## The input parameter @var{options} is a 2-D vector of the form
## @code{[@var{stepsize}, @var{max_vertices}]}.  The first parameter
## specifies the step size used for trajectory integration (default 0.1).  A
## negative value is allowed which will reverse the direction of integration.
## The second parameter specifies the maximum number of segments used to
## create a streamline (default 10,000).
##
## The return value @var{xy} is a @nospell{nverts x 2} matrix containing the
## coordinates of the field line segments.
##
## Example:
##
## @example
## @group
## [x, y] = meshgrid (0:3);
## u = 2 * x;
## v = y;
## xy = stream2 (x, y, u, v, 1.0, 0.5);
## @end group
## @end example
##
## @seealso{streamline, stream3}
## @end deftypefn

## References:
##
## @article{
##    title = {Particle Tracing Algorithms for 3D Curvilinear Grids},
##    year = {2000},
##    author = {Nielson, Gregory and Uller, H. and Sadarjoen, I. and Walsum, Theo and Hin, Andrea and Post, Frits}
## }
##
## @article{
##    title = {Sources of error in the graphical analysis of CFD results},
##    publisher = {Journal of Scientific Computing},
##    year = {1988},
##    volume = {3},
##    number = {2},
##    pages = {149--164},
##    author = {Buning, Pieter G.},
## }

function xy = stream2 (varargin)

  options = [];
  switch (numel (varargin))
    case {4,5}
      if (numel (varargin) == 4)
        [u, v, spx, spy] = varargin{:};
      else
        [u, v, spx, spy, options] = varargin{:};
      endif
      [m, n] = size (u);
      [x, y] = meshgrid (1:n, 1:m);
    case 6
      [x, y, u, v, spx, spy] = varargin{:};
    case 7
      [x, y, u, v, spx, spy, options] = varargin{:};
    otherwise
      print_usage ();
  endswitch

  stepsize = 0.1;
  max_vertices = 10_000;
  if (! isempty (options))
    switch (numel (options))
      case 1
        stepsize = options(1);
      case 2
        stepsize = options(1);
        max_vertices = options(2);
      otherwise
        error ("stream2: OPTIONS must be a 1- or 2-element vector");
    endswitch

    if (! isreal (stepsize) || stepsize == 0)
      error ("stream2: STEPSIZE must be a real scalar != 0");
    endif
    if (! isreal (max_vertices) || max_vertices < 1)
      error ("stream2: MAX_VERTICES must be an integer > 0");
    endif
    max_vertices = fix (max_vertices);
  endif

  if (! (size_equal (u, v, x, y) && size_equal (spx, spy)))
    error ("stream2: matrix dimensions must match");
  endif
  if (iscomplex (u) || iscomplex (v) || iscomplex (x) || iscomplex (y)
      || iscomplex (spx) || iscomplex (spy))
    error ("stream2: all inputs must be real-valued");
  endif

  gx = x(1,:);
  gy = y(:,1).';

  ## Jacobian Matrix
  dx = diff (gx);
  dy = diff (gy);
  ## "<" used to check if the mesh is ascending
  if (any (dx <= 0) || any (dy <= 0)
      || any (isnan (dx)) || any (isnan (dy)))
    error ("stream2: non-monotonically increasing or NaN values found in mesh");
  endif
  tx = 1 ./ dx;
  ty = 1 ./ dy;
  ## "Don't cares" used for handling points located on the border
  tx(end + 1) = 0;
  ty(end + 1) = 0;
  dx(end + 1) = 0;
  dy(end + 1) = 0;

  px = spx(:);
  py = spy(:);

  for nseed = 1 : numel (px)

    xp = px(nseed);
    yp = py(nseed);
    idx = find (diff (gx <= xp), 1);
    if (gx(end) == xp)
      idx = numel (gx);
    endif
    idy = find (diff (gy <= yp), 1);
    if (gy(end) == yp)
      idy = numel (gy);
    endif

    if (isempty (idx) || isempty (idy))
      xy{nseed} = [];
    else
      ## Transform seed from P coordinates to C coordinates
      zeta = (idx - 1) + (xp - gx(idx)) * tx(idx);
      xi = (idy - 1) + (yp - gy(idy)) * ty(idy);

      C = __streameuler2d__ (u, v, tx, ty, zeta, xi, stepsize, max_vertices);

      ## Transform from C coordinates to P coordinates
      idu = floor (C(:,1));
      idv = floor (C(:,2));
      xy{nseed} = [gx(idu + 1).' + (C(:,1) - idu).*(dx(idu + 1).'), ...
                   gy(idv + 1).' + (C(:,2) - idv).*(dy(idv + 1).')];
    endif

  endfor

endfunction


%!demo
%! clf;
%! [x, y] = meshgrid (-5:5, -4:4);
%! u = x - 2 * y;
%! v = 2 * x - 3 * y;
%! sx = [3, 0, -1, -2, -3, 0, 1, 2];
%! sy = [3, 3, 3, 3, -3, -3, -3, -3];
%! h = streamline (x, y, u, v, sx, sy, 0.05);
%! set (h, "color", "r");
%! hold on;
%! quiver (x, y, u, v);
%! scatter (sx(:), sy(:), 20, "filled", "o", "markerfacecolor", "r");
%! grid on;
%! title ("Asymptotically Stable Equilibrium");
%! axis equal;

%!test
%! xy = stream2 ([1,1,1;2,2,2;3,3,3], [1,1,1;2,2,2;3,3,3], 1, 1, [0.01,5]);
%! assert (numel (xy{:}), 10);

## Test input validation
%!error <Invalid call> stream2 ()
%!error <Invalid call> stream2 (1)
%!error <Invalid call> stream2 (1,2)
%!error <Invalid call> stream2 (1,2,3)
%!error <OPTIONS must be a 1- or 2-element> stream2 (1,2,3,4, [1,2,3])
%!error <STEPSIZE must be a real scalar != 0> stream2 (1,2,3,4, [1i])
%!error <STEPSIZE must be a real scalar != 0> stream2 (1,2,3,4, [0])
%!error <MAX_VERTICES must be an integer> stream2 (1,2,3,4, [1, 1i])
%!error <MAX_VERTICES must be an integer> stream2 (1,2,3,4, [1, 0])
%!error <matrix dimensions must match> stream2 ([1 1],2,3,4)
%!error <matrix dimensions must match> stream2 (1,[2 2],3,4)
%!error <matrix dimensions must match> stream2 (1,2,[3 3],4)
%!error <matrix dimensions must match> stream2 (1,2,3,[4 4])
%!error <all inputs must be real-valued> stream2 (1i,2,3,4)
%!error <all inputs must be real-valued> stream2 (1,2i,3,4)
%!error <all inputs must be real-valued> stream2 (1,2,3i,4)
%!error <all inputs must be real-valued> stream2 (1,2,3,4i)
%!error <non-monotonically increasing or NaN values found in mesh>
%! stream2 ([2 1], [1 2], [1 1], [2 2], [3 3], [4 4]);
%!error <non-monotonically increasing or NaN values found in mesh>
%! stream2 ([1 NaN], [1 2], [1 1], [2 2], [3 3], [4 4]);
## FIXME: vectors representing x, y mesh are not accepted.
%#!error <non-monotonically increasing or NaN values found in mesh>
%! stream2 ([1 2], [2 1], [1 1], [2 2], [3 3], [4 4]);
%#!error <non-monotonically increasing or NaN values found in mesh>
%! stream2 ([1 2], [1 NaN], [1 1], [2 2], [3 3], [4 4]);

