function [tout, yout] = bdode45(ypfun, t0, tfinal, y0, tol, trace)
%BDODE45 is a slight modification of the routine ODE45 supplied with MATLAB, 
%which uses 4th and 5th order Runge-Kutta formulas to integrate a system of 
%ordinary differential equations.  The only difference is that BDODE45 will 
%solve backwards (tfinal < t0) while ODE45 will not.  Usage and variables are 
%exactly the same as for ODE45.  Type "help ode45" for more information.
%                                                           (M. Day 1/6/95)

% The Fehlberg coefficients:
alpha = [1/4  3/8  12/13  1  1/2]';
beta  = [ [    1      0      0     0      0    0]/4
          [    3      9      0     0      0    0]/32
          [ 1932  -7200   7296     0      0    0]/2197
          [ 8341 -32832  29440  -845      0    0]/4104
          [-6080  41040 -28352  9295  -5643    0]/20520 ]';
gamma = [ [902880  0  3953664  3855735  -1371249  277020]/7618050
          [ -2090  0    22528    21970    -15048  -27360]/752400 ]';
pow = 1/5;
if nargin < 5, tol = 1.e-6; end
if nargin < 6, trace = 0; end

% Initialization
t = t0;
hmin=tol;
hmax = (tfinal - t)/16;
dir=sign(hmax);
h = hmax/8;
y = y0(:);
f = zeros(length(y),6);
chunk = 128;
tout = zeros(chunk,1);
yout = zeros(chunk,length(y));
k = 1;
tout(k) = t;
yout(k,:) = y.';

if trace
   clc, t, h, y
end

% The main loop

while (dir*t < dir*tfinal) & (dir*(t + h) > dir*t)
   if dir*(t + h) > dir*tfinal, h = tfinal - t; end

   % Compute the slopes
   temp = feval(ypfun,t,y);
   f(:,1) = temp(:);
   for j = 1:5
      temp = feval(ypfun, t+alpha(j)*h, y+h*f*beta(:,j));
      f(:,j+1) = temp(:);
   end

   % Estimate the error and the acceptable error
   delta = norm(h*f*gamma(:,2),'inf');
   tau = tol*max(norm(y,'inf'),1.0);

   % Update the solution only if the error is acceptable
   if delta <= tau
      t = t + h;
      y = y + h*f*gamma(:,1);
      k = k+1;
      if k > length(tout)
         tout = [tout; zeros(chunk,1)];
         yout = [yout; zeros(chunk,length(y))];
      end
      tout(k) = t;
      yout(k,:) = y.';
   end
   if trace
      home, t, h, y
   end

   % Update the step size
   if delta ~= 0.0
      h = hmax*min(1.0, 0.8*(h/hmax)*(tau/delta)^pow);
   end
   if dir*h<hmin, h=0; end
end

if (dir*t < dir*tfinal)

   disp(' ') 
   disp('Singularity likely at the following t-value:')
   t
end

if (dir==0)
   disp('No integration; interval length 0.')
end

tout = tout(1:k);
yout = yout(1:k,:);