parallel computation in matlab

How can i use parallel computation in this Function ?

function [x,y]=euler_backward(f,xinit,yinit,xfinal,n)
% calculate h
h=(xfinal-xinit)/n;
% Initialize x and y as column vectors
x=[xinit zeros(1,n)];
y=[yinit zeros(1,n)];
% Calculate of x and y
for i=1:n
x(i+1)=x(i)+h;
ynew=y(i)+h*(f(x(i),y(i)));
y(i+1)=y(i)+h*f(x(i+1),ynew);
end
end

[Matt J]

You can't do it in that function, but you might be able to do it in f() depending on what it looks like.

[Walter Roberson]

Computing in parallel (or vectorizing) f will not help for any given iteration, as the second parameter passed in to f() in the second call depends upon the value returned by the first call to f().

[Matt J]

@Walter Roberson I'm not sure I follow. Parallelization of the computations within f() could allow each individual call to f() to go faster. If so, then the total time for the loop should decrease as well.

It might also be worth pointing out that the x(i) can all be pre-computed and the loop reduced as follows

x=linspace(xinit,xfinal,n+1);
h=x(2)-x(1);

for i=1:n

 ynew=y(i)+h*(f(x(i),y(i)));
 y(i+1)=y(i)+h*f(x(i+1),ynew);

end

Therefore, if for example f() looks something like f(a,b)=p(a)+q(a,b) where p() is an expensive function but q() is simple, then the loop can be accelerated with the following strategy:

x=linspace(xinit,xfinal,n+1);
h=x(2)-x(1);

parfor i=1:n+1
    px=p(x(i));  
end

for i=1:n

  ynew=y(i)+h*(  px(i) + q(x(i),y(i))  );
  y(i+1)=y(i)+h*( px(i+1) + q(x(i),ynew) );

end

[Walter Roberson]

The large majority of the ode functions I see people posting have code that ignore the first parameter (such as "time") and depend only on the second parameter (current boundary conditions). I do see the occasional toy example that ignores the boundary conditions... usually in the context of people being asked to program Euler method.