forward_euler.m

   1 function df = forward_euler(integer_order, h, f, x)
   2   ##
   3   ## Use the forward Euler method to compute the derivative of `f` at
   4   ## a point `x`.
   5   ##
   6   ## INPUTS:
   7   ##
   8   ##   * ``integer_order`` - The order of the derivative.
   9   ##
  10   ##   * ``h`` - The step size.
  11   ##
  12   ##   * ``f`` - The function whose derivative we're computing.
  13   ##
  14   ##   * ``x`` - The point at which to compute the derivative.
  15   ##
  16
  17   if (integer_order == 0)
  18     df = x;
  19     return;
  20   end
  21
  22   ## We need a few points around `x` to compute the derivative at `x`.
  23   ## The number of points depends on the order.
  24   if (even(integer_order))
  25     offset_b = integer_order / 2;
  26     offset_f = offset_b;
  27   else
  28     ## When the order is odd, we need one more "forward" point than we
  29     ## do "backward" points.
  30     offset_b = (integer_order - 1) / 2;
  31     offset_f = offset_b + 1;
  32   end
  33
  34   backward_xs = [x-(offset_b*h) : h : x];
  35
  36   ## We'll always have at least one forward point, so start this vector
  37   ## from (x + h) and include `x` itself in the backward points.
  38   forward_xs = [x+h : h : x+(offset_f*h)];
  39
  40   xs = horzcat(backward_xs, forward_xs);
  41
  42   ## Now that we have all of the points that we need in xs, we can use
  43   ## the divided_difference function to compute the derivative.
  44   df = divided_difference(f, xs);
  45 end