function Y = ode2(odefun,tspan,y0,varargin)
%ODE2  Solve differential equations with a non-adaptive method of order 2.
%   Y = ODE2(ODEFUN,TSPAN,Y0) with TSPAN = [T1, T2, T3, ... TN] integrates 
%   the system of differential equations y' = f(t,y) by stepping from T0 to 
%   T1 to TN. Function ODEFUN(T,Y) must return f(t,y) in a column vector.
%   The vector Y0 is the initial conditions at T0. Each row in the solution 
%   array Y corresponds to a time specified in TSPAN.
%
%   Y = ODE2(ODEFUN,TSPAN,Y0,P1,P2...) passes the additional parameters 
%   P1,P2... to the derivative function as ODEFUN(T,Y,P1,P2...). 
%
%   This is a non-adaptive solver. The step sequence is determined by TSPAN
%   but the derivative function ODEFUN is evaluated multiple times per step.
%   The solver implements the improved Euler (Heun's) method of order 2.   
%
%   Example 
%         tspan = 0:0.1:20;
%         y = ode2(@vdp1,tspan,[2 0]);  
%         plot(tspan,y(:,1));
%     solves the system y' = vdp1(t,y) with a constant step size of 0.1, 
%     and plots the first component of the solution.   
%

if ~isnumeric(tspan)
  error('TSPAN should be a vector of integration steps.');
end

if ~isnumeric(y0)
  error('Y0 should be a vector of initial conditions.');
end

h = diff(tspan);
if any(sign(h(1))*h <= 0)
  error('Entries of TSPAN are not in order.') 
end  

try
  f0 = feval(odefun,tspan(1),y0,varargin{:});
catch
  msg = ['Unable to evaluate the ODEFUN at t0,y0. ',lasterr];
  error(msg);  
end  

y0 = y0(:);   % Make a column vector.
if ~isequal(size(y0),size(f0))
  error('Inconsistent sizes of Y0 and f(t0,y0).');
end  

neq = length(y0);
N = length(tspan);
Y = zeros(neq,N);
F = zeros(neq,2);

Y(:,1) = y0;
for i = 2:N
  ti = tspan(i-1);
  hi = h(i-1);
  yi = Y(:,i-1);
  F(:,1) = feval(odefun,ti,yi,varargin{:});
  F(:,2) = feval(odefun,ti+hi,yi+hi*F(:,1),varargin{:});
  Y(:,i) = yi + (hi/2)*(F(:,1) + F(:,2));
end
Y = Y.';
