Continuous fraction approximation of pi - matlab

Question

While studying for Matlab, I came up with this problem: given an integer 1<d<15 find the smallest p,q (positive integers) such that abs(p/q-pi)<=10^-d.
So here's my attempt: I first thought that I need to bound p,q in order to create loops, so I put as input data some M,Nupper bounds for p,q respectively. So here is my algorithm:
M=input('Give an upper bound for p:')
N=input('Give an upper bound for q:')
d=input('Give a positive integer between 1 and 15:')
for q=1:N
for p=1:M
if abs(pi-p/q)<=10^(-d)
break
end
end
end
What is wrong about it?
Thank you in advance.

Answer 1

The problem lies in the the way you chose to terminate the for loops: break only stops the inner loop. Try this code instead, and check the value of p and q at which the execution stops:
M=input('Give an upper bound for p:')
N=input('Give an upper bound for q:')
d=input('Give a positive integer between 1 and 15:')
for q=1:N
for p=1:M
if abs(pi-p/q)<=10^(-d)
[p,q]
error('Found!') % Arrest the program when condition is met
end
end
end
which gives the following output:
Of course, there are better ways of capturing all the possible pairs of integers that meet the condition (e.g. by using disp instead of error). That goes beyond the scope of this answer, but I shall provide a couple of examples here:
clear; clc;
M=input('Give an upper bound for p:')
N=input('Give an upper bound for q:')
d=input('Give a positive integer between 1 and 15:')
pairs = []; % p,q pairs are stored here
k = 1; % counter
for q=1:N
for p=1:M
if abs(pi-p/q)<=10^(-d)
pairs(k,1) = p;
pairs(k,2) = q;
k = k + 1; % increment the counter
end
end
end
The script above will end quietly: the (p,q) pairs will be stored in the pair matrix.
The following one will print directly the pairs:
clear; clc;
M=input('Give an upper bound for p:')
N=input('Give an upper bound for q:')
d=input('Give a positive integer between 1 and 15:')
for q=1:N
for p=1:M
if abs(pi-p/q)<=10^(-d)
out = sprintf("(%d, %d)", p,q);
disp(out);
end
end
end
For the sake of the experiment, and following up on #Cris Luengo's comment, here's a slightly more elaborate version of the script: the for loops are encapsulated in a dedicated function and a while loop keeps good track of the progress and populates the res matrix with the (p,q) pairs:
clear; clc;
M=input('Give an upper bound for p:');
N=input('Give an upper bound for q:');
d=input('Give a positive integer between 1 and 15:');
close_to_pi = #(px,qx) abs(pi-px/qx)<=10^(-d); % returns true/false
p = 1; q = 1;
count = 0;
res = nan(N*M,2) ; % (p,q) pairs are stored here; preallocate for speed!
tic % starts the timer
while (q <= N)
[p,q, found] = approx_pi(p, q, N, M, close_to_pi);
if found
count = count + 1;
res(count,:) = [p,q]; % populates output var
end
if p<M % we aren't done with p
p = p + 1;
else % reset p and increment q
p = 1;
q = q + 1;
end
end
res(isnan(res(:,1)),:) = []; % gets rid of the empty elements
disp(count)
toc % stops the timer and prints elapsed time
function [p, q, found] = approx_pi(p0, q0, N, M, fun)
for q=q0:N
for p=p0:M
if fun(p,q)
found = 1;
return
end % if
end % for p
p0 = 1;
end % for q
found = 0;
end % approx_pi
If you are interested in continuous fraction approximations of pi, try rat(pi, Tol) where Tol is the tolerance. Further details here.