Matlab Nested Radicals - matlab

Question

Working on an assignment in MATLAB and I can't seem to figure this problem out due to the arithmetic, I've been trying it for about 6 hours now
I need to create a loop that accepts user input > 1 (done) and loops through the following (m is input)
t1 = sqrt(m);
t2 = sqrt(m-sqrt(m));
t3 = sqrt(m-sqrt(m+sqrt(m)))
t4 = sqrt(m-sqrt(m+sqrt(m-sqrt(m))))
t5 = sqrt(m-sqrt(m+sqrt(m-sqrt(m+sqrt(m)))))
and so on until the new t value minus the old t value is < 1e-12
My current code is as follows
%Nested Radicals
clear all;
clc;
%User input for m
m = input('Please enter a value for m: ');
%Error message if m is less than 1
if m <= 1
fprintf('ERROR: m must be greater than 1\n')
m = input('Please enter a value for m: ');
end
%Error message if m is not an integer
if mod(m,1) ~= 0
fprintf('m must be an integer\n')
m = input('Please enter a value for m: \n');
end
%Nested things
t_old = m;
t_new = sqrt(m);
varsign = -1;
index = 1;
loop = true;
endResult = 1e-12;
sqrts = [sqrt(m), sqrt(m-sqrt(m)), sqrt(m-sqrt(m+sqrt(m))), sqrt(m-sqrt(m+sqrt(m-sqrt(m)))), sqrt(m-sqrt(m+sqrt(m-sqrt(m+sqrt(m)))))];
fprintf('m = %d\n',m)
fprintf('t1 = %14.13f\n', t_new')
while loop
if index ~= 1
curResult = abs(sqrts(1,index) - sqrts(1, index-1));
else
curResult = abs(sqrts(1, index));
end
if curResult > endResult
if index < 5
t_new = sqrts(1, index+1);
else
t_new = sqrts(1, index);
loop=false;
end
if index
fprintf('t%d = %14.13f\n', index, t_new)
end
else
fprintf('t%d = %14.13f\n', index, t_new);
break;
end
index = index + 1;
if index > 50
fprintf('t%d = %14.13f\n', index, t_new);
break;
end
end

Answer 1

Unless I'm very much mistaken, you can write the expression for t(n) as follows:
t(n) = sqrt(m-sqrt(m+t(n-2));
which makes it a lot easier to loop:
%Nested Radicals
clear all;
clc;
%User input for m
m = input('Please enter a value for m: ');
%Error message if m is less than 1
if m <= 1
fprintf('ERROR: m must be greater than 1\n')
m = input('Please enter a value for m:');
end
%Error message if m is not an integer
if mod(m,1) ~= 0
fprintf('m must be an integer\n')
m = input('Please enter a value for m:');
end
%Nested things
t_old = sqrt(m);
t_new = sqrt(m-sqrt(m));
threshold = 1e-12;
k = 3;
while abs(t_new - t_old) >= threshold
temp = sqrt(m-sqrt(m+t_old));
t_old = t_new;
t_new = temp;
k = k+1;
end
fprintf('t%d = %14.13f\n', k-2, t_old);
fprintf('t%d = %14.13f\n', k-1, t_new);
fprintf('t%d - t%d = %14.13f\n', k-2, k-1, t_old - t_new);
which gives for m=9 for example:
Please enter a value for m: 9
t17 = 2.3722813232696
t18 = 2.3722813232691
t17 - t18 = 0.0000000000005

Answer 2

I'm not sure what you're trying to do with the sqrts variable, you should be calculating each step on the fly in your loop, since you can't possibly know how deep you need to go
m = 5; % Get m however you want to
n = 0; % Iteration counter
tol = 1e-12 % Tolerance at which to stop
dt = 1; % initialise to some value greater than 'tol' so we can start the loop
% Loop until tn is less than tolerance. Would be sensible to add a condition on n,
% like "while tn > tol && n < 1000", so the loop doesn't go on for years if the
% condition takes a trillion loops to be satisfied
while dt > tol
% Set the value of the deepest nested expression
tn = sqrt(m);
% We know how many times take sqrt, so for loop our way out of the nested function
% Initially we want the sign to be -1, then +1, -1, ...
% This is achieved using ((-1)^ii)
for ii = 1:n
tn = sqrt(m + ((-1)^ii)*tn); % Calculate next nested function out
end
% Increment iteration number
n = n + 1;
dt = abs( t_old - tn );
t_old = tn;
end
I've not done any analysis on your function, so have no idea if it's guaranteed to converge to some value <1e-12. If it isn't then you definitely need to add some maximum iteration condition as I suggest in the comments above.