coding such a logistic recursion

Axel Vogt — Sat, 28 Aug 2010 12:59:49 Z

I do not think the approach to the problem makes sense beyond examples,
but here is a way:

Your recursion is x(n+1) = a(n)*(1-x(n))*x(n), a(n+1) = w+a(n) and you
let it start in 0.

Then you have initial values a(0) = alpha, x(0) = xi and some w.

The recursion for a is shifting by w and has a closed form

  rsolve({a(n+1) = w+a(n), a(0)=alpha}, {a}); 
  simplify(%);
 
                         {a(n) = alpha + w n}

and therefore your recursion is x(n+1) = (alpha + w*n)*(1-x(n))*x(n),
which can be written as procedure to give you results after n steps:

For saver coding first express x(n) in terms of x(n-1)

  x(n+1) = (alpha + w*n)*(1-x(n))*x(n);
  eval(%, n=n-1);

          x(n) = ((n - 1) w + alpha) (1 - x(n - 1)) x(n - 1)

and then write the recursion down as

  R:= proc(xi,alpha, w , n::nonnegint)
  local x0, x1;
  option remember;                # look up the help for this option
  if n=0 then return xi end if;   # initial value
  x0:= R(xi,alpha, w, n-1);       # this stands for x(n-1);
  x1:= ((n-1)*w+alpha)*(1-x0)*x0; # we just did that
  return x1; 
  end proc;

Do a check for the first few steps, to check for coding errors by

  R(xi,alpha, w , 1);
                          alpha (1 - xi) xi

  R(xi,alpha, w , 2);

        (w + alpha) (1 - alpha (1 - xi) xi) alpha (1 - xi) xi

But be aware, that it will 'explode' in size very soon, for = 20:

  'R(xi,alpha, w , 20)';
  length(%);             # length in characters
  evalf[2](%/80/60);     # ~size in printed pages

                                5900.

This says: after already 20 steps the expression would be 5900 pages
long, covering several books. And trying to handle it with Maple is
likely to crash it soon, for n=25 you get 190000 pages. Forget n=100.
You need Math for that (do not ask me), not a computer system.

thanks axel

haider — Sat, 28 Aug 2010 14:36:05 Z

writing a code wasnt the problem, the main issue was how to get rid of such a huge expression if any? Does maple provide any method to carry out such a calculation?

hyder

MaplePrimes - answers and comments on Question, How do I write efficient codes using symbolic computation?

coding such a logistic recursion

thanks axel