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

%% ASSEMBLE_HEAT assembles the finite element system for the heat equation.
%
%  Discussion:
%
%    The matrix is known to be banded.  A special matrix storage format
%    is used to reduce the space required.  Details of this format are
%    discussed in the routine DGB_FA.
%
%  Modified:
%
%    22 July 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 IB, the half-bandwidth of the matrix.
%
%    Input, real TIME, the current time.
%
%    Output, real A(3*IB+1,NODE_NUM), the NODE_NUM by NODE_NUM
%    coefficient matrix, stored in a compressed 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;
  a(1:3*ib+1,1:node_num) = 0.0;
%
%  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 PHYS_XY in the physical triangle.
%
    phys_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);
%
%  Consider the QUAD-th quadrature point.
%
    for quad = 1 : quad_num

      k_value = k_coef ( 1, phys_xy(1:2,quad), time );

      rhs_value = rhs  ( 1, phys_xy(1:2,quad), time );
%
%  Consider the TEST-th test function.
%
%  We generate an integral for every node associated with an unknown.
%  But if a node is associated with a boundary condition, we do nothing.
%
      for test = 1 : element_order

        i = element_node(test,element);

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

        f(i) = f(i) + w(quad) * rhs_value * 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, phys_xy(1:2,quad) );

          a(i-j+2*ib+1,j) = a(i-j+2*ib+1,j) ...
            + w(quad) * ( dbidx * dbjdx + dbidy * dbjdy + k_value * bj * bi );

        end

      end

    end

  end