Function inside function - generating Random Walks

I'm trying to create a random walk in an arbitrary number of dimensions (with the goal to estimate the number of self-avoiding walks of length n in d dimensions).

To do this I need two external functions. The first checks if any given coordinate has already been crossed during previous steps of a walk.

The second one returns a list of directions possible to walk in (since we want to use importance sampling with resampling to create a walk which is not allowed to go to a coordinate where it has already been). For example, if I have a path in d = 2 with coordinates (0,0) (0,1) (-1,1) (-1, 0), and thus the walker is now standing at point (-1, 0), the function should return a list of two directions (representing either +1/-1 steps in either x or y direction) since the only free neighbouring points would be (-2, 0) or (-1, -1).

I have named these two functions free.m and possible_dir.m respectively, and they look as follows:

 function [p] = free(coord, path, n)
% Checks if a coordinate is free 
%   Checks if the input coordinate is in the path or not, i.e. if it has
%   already been crossed by our walker. If coord is not in path, returns TRUE (=1)
if n>0
    if isequal(coord, path(n,:))
        p=false;
    else 
        free(coord, path, n-1)
    end
    else
        p=true;
end
end

and:

function [dir] = possible_dir(current_coord, path, n, dims, direction)
% Give us vector with possible directions to walk in
% Possible directions rank from 1 to 2*d, with each direction being given
% the corresponding number
dir_list = zeros(1, 2*dims);
for i=1:2*dims
  if free((current_coord + direction(i,:)), path, n)
      dir_list(i) = i;
  else
      dir_list(i) = 0;
  end
end
dir = dir_list(dir_list>0); 
end

Now take this 2-dimensional walk of currently three steps as an example:

test_path = zeros(3, 2); 
test_path(1,:)=[0 0];
test_path(2,:)=[1 0];
test_path(3,:)=[1 1];

Inserting the coordinate (1 2) into the free-function returns TRUE, and inserting the coordinate (1 1) returns FALSE, as expected.

Now, inserting coordinate (1 1) into the possible_dir-function just gives me the error message:

Output argument "p" (and maybe others) not assigned during call to "free".

Error in possible_dir (line 7) if free((current_coord + direction(i,:)), path, n)

Error in proj3 (line 682) possible_dir(cc, test_path, 3, 2, DIR)

Why is this error occurring?

[John D'Errico]

How high a dimension? You could just use an n-d boolean matrix to see if you have ever passed through a point. As long as n is not too large, that will work. Make it a logical array, so it does not take much space.

The set of possible directions would then also be easy to identify, since you just check each dimension in the array.

Answer 1

[John D'Errico]

As far as why that error occurs, LOOK AT YOUR CODE! READ THE ERROR MESSAGE!

"Output argument "p" (and maybe others) not assigned during call to "free"."

Your code:

if isequal(coord, path(n,:))
    p=false;
else 
    free(coord, path, n-1)
end

Are there any circumstances where p will not be defined?

OF COURSE! What happens if the second branch is taken in that if? Is p defined?

If you call a function, even if the call is a recursive ne, and you do NT define the output variable, then p is never returned.

[Jan Novotny]

Thank you so so much!!! I'm pretty green when it comes to programming in general, and MATLAB in particular. Since the free-function always worked on all kinds of examples when I tested it independently of the possible_dir-function, I didn't think that the problem lied in the free-function. I've change it so that it's defined and it seems to work!

However, for each number of dimensions, there is a threshold which n cannot exceed, or the program returns "index exceeds matrix dimensions" (when d=2, the threshold is n=8, when d=3, the threshold is n=36, and so on). Is this perhaps connected to what you mentioned in the comment about using an n-d boolean matrix as long as n was not too large?