function [ a, f ] = assemble_heat_sparse ( node_num, node_xy, node_condition, ...
  element_order, element_num, element_node, quad_num, nz_num, time )

%% ASSEMBLE_HEAT_SPARSE assembles the finite element system for the heat equation.
%
%  Discussion:
%
%    MATLAB's built-in sparse matrix storage is used.
%
%  Modified:
%
%    07 January 2007
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer NODE_NUM, the number of nodes.
%
%    Input, real NODE_XY(2,NODE_NUM), the coordinates of nodes.
%
%    Input, integer NODE_CONDITION(NODE_NUM), reports the condition
%    used to set the unknown associated with the node.
%    0, unknown.
%    1, finite element equation.
%    2, Dirichlet condition;
%    3, Neumann condition.
%
%    Input, integer ELEMENT_ORDER, the order of the elements.
%
%    Input, integer ELEMENT_NUM, the number of elements.
%
%    Input, integer ELEMENT_NODE(ELEMENT_ORDER,ELEMENT_NUM);
%    ELEMENT_NODE(I,J) is the global index of local node I in element J.
%
%    Input, integer QUAD_NUM, the number of quadrature points used in assembly.
%
%    Input, integer NZ_NUM, the (maximum) number of nonzeros in the matrix.
%    If set to 0 on input, we hope MATLAB's sparse utility will be able
%    to take over the task of reallocating space as necessary.
%
%    Input, real TIME, the current time.
%
%    Output, real sparse A, the NODE_NUM by NODE_NUM coefficient matrix, 
%    stored in MATLAB's sparse format.
%
%    Output, real F(NODE_NUM), the right hand side.
%
%  Local parameters:
%
%    Local, real BI, DBIDX, DBIDY, the value of some basis function
%    and its first derivatives at a quadrature point.
%
%    Local, real BJ, DBJDX, DBJDY, the value of another basis
%    function and its first derivatives at a quadrature point.
%

%
%  Initialize the arrays to zero.
%
  f(1:node_num) = 0.0;

  fprintf ( 1, '\n' );
  fprintf ( 1, 'ASSEMBLE_HEAT_SPARSE:\n' );
  fprintf ( 1, '  Setting up sparse heat matrix with NZ_NUM = %d\n', nz_num );

  a = sparse ( [], [], [], node_num, node_num, nz_num );
%
%  Get the quadrature weights and nodes.
%
  [ quad_w, quad_xy ] = quad_rule ( quad_num );
%
%  Add up all quantities associated with the TRIANGLE-th triangle.
%
  for element = 1 : element_num
%
%  Make a copy of the triangle.
%
    t3(1:2,1:3) = node_xy(1:2,element_node(1:3,element));
    t6(1:2,1:6) = node_xy(1:2,element_node(1:6,element));
%
%  Map the quadrature points QUAD_XY to points XY in the physical triangle.
%
    xy(1:2,1:quad_num) = reference_to_physical_t3 ( t3, quad_num, quad_xy );

    area = abs ( triangle_area_2d ( t3 ) );

    w(1:quad_num) = area * quad_w(1:quad_num);

    k_quad = k_coef ( quad_num, xy, time );

    rhs_quad = rhs  ( quad_num, xy, time );
%
%  Consider the QUAD-th quadrature point.
%
    for quad = 1 : quad_num
%
%  Consider the TEST-th test function.
%
%  We generate an integral for every node associated with an unknown.
%
      for test = 1 : element_order

        i = element_node(test,element);

        [ bi, dbidx, dbidy ] = basis_11_t6 ( t6, test, xy(1:2,quad) );

        f(i) = f(i) + w(quad) * rhs_quad(quad) * bi;
%
%  Consider the BASIS-th basis function, which is used to form the
%  value of the solution function.
%
        for basis = 1 : element_order

          j = element_node(basis,element);

          [ bj, dbjdx, dbjdy ] = basis_11_t6 ( t6, basis, xy(1:2,quad) );

          a(i,j) = a(i,j) ...
            + w(quad) * ( dbidx * dbjdx + dbidy * dbjdy + k_quad(quad) * bj * bi );

        end

      end

    end

  end