heun.m

% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
function [tout, yout] = heun(ode_function, tspan, y0, h)
% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%{ 
  This function uses the predictor-corrector method to integrate a system
  of first-order differential equations dy/dt = f(t,y).

  y             - column vector of solutions
  f             - column vector of the derivatives dy/dt
  ode_function  - handle for the user M-function in which the derivatives
                  f are computed
  t             - time
  t0            - initial time
  tf            - final time
  tspan         - the vector [t0 tf] giving the time interval for the
                  solution
  h             - time step
  y0            - column vector of initial values of the vector y
  tout          - column vector of the times at which y was evaluated
  yout          - a matrix, each row of which contains the components of y
                  evaluated at the correponding time in tout
  feval         - a built-in MATLAB function which executes 'ode_function'
                  at the arguments t and y
  tol           - Maximum allowable relative error for determining
                  convergence of the corrector
  itermax       - maximum allowable number of iterations for corrector
                  convergence
  iter          - iteration number in the corrector convergence loop
  t1            - time at the beginning of a time step
  y1            - value of y at the beginning of a time step
  f1            - derivative of y at the beginning of a time step
  f2            - derivative of y at the end of a time step
  favg          - average of f1 and f2
  y2p           - predicted value of y at the end of a time step
  y2            - corrected value of y at the end of a time step
  err           - maximum relative error (for all components) between y2p
                  and y2 for given iteration
  eps           - unit roundoff error (the smallest number for which
                  1 + eps > 1). Used to avoid a zero denominator.
 
  User M-function required: ode_function
%}
% ------------------------------------------------------------------------

tol      = 1.e-6;
itermax  = 100;

t0       = tspan(1);
tf       = tspan(2);
t        = t0;
y        = y0;
tout     = t;
yout     = y';

while t < tf
    h    = min(h, tf-t);
    t1   = t;
    y1   = y;
    f1   = feval(ode_function, t1, y1);
    y2   = y1 + f1*h;
    t2   = t1 + h;
    err  = tol + 1;
    iter = 0;
    while err > tol && iter <= itermax
        y2p  = y2;
        f2   = feval(ode_function, t2, y2p); 
        favg = (f1 + f2)/2;
        y2   = y1 + favg*h;
        err  = max(abs((y2 - y2p)./(y2 + eps)));
        iter = iter + 1;
    end
    
    if iter > itermax
        fprintf('\n Maximum no. of iterations (%g)',itermax)
        fprintf('\n exceeded at time = %g',t)
        fprintf('\n in function ''heun.''\n\n')
        return
    end
    
    t    = t + h;
    y    = y2;
    tout = [tout;t];  % adds t to the bottom of the column vector tout
    yout = [yout;y']; % adds y' to the bottom of the matrix yout
end
% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~