function Y = ode4(odefun,tspan,y0,varargin)

%ODE4  Solve differential equations with a non-adaptive method of order 4.

%   Y = ODE4(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 = ODE4(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 classical Runge-Kutta method of order 4.   

%

%   Example 

%         tspan = 0:0.1:20;

%         y = ode4(@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,4);

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+0.5*hi,yi+0.5*hi*F(:,1),varargin{:});

  F(:,3) = feval(odefun,ti+0.5*hi,yi+0.5*hi*F(:,2),varargin{:});  

  F(:,4) = feval(odefun,tspan(i),yi+hi*F(:,3),varargin{:});

  Y(:,i) = yi + (hi/6)*(F(:,1) + 2*F(:,2) + 2*F(:,3) + F(:,4));

end

Y = Y.';