Cos Taylor Series script rounds for 4 decimal places but changing to 5 causes it to loop infinitely

I have to write a script to count the nth number of terms it takes to get a 4 decimal match in cos(x) and the taylor series expansion of cos(x). I am using a while loop and round function to do this, and when i have the round set to 4 decimal places, the script runs as expected, but when I change it to 5 decimal places, it seems to run infinitely. I am also not using the built in factorial function, as the professor took points off for using it in a previous assignmet, so that is what the nested for loop is doing, running the factorial, which may be part of the issue

clear, clc

cts = 1; % Initial sum of cos taylor series, and also first term
x = pi/7;
f= 1;
ii = 1;

while round(cts,4) ~= round(cos(x),4)
    ii = ii * 2;
    for jj = 1:ii
        f = f * jj;
    end
    nth_term = ( (-1)^(ii / 2) ) * ( (x^(ii) ) / (f));
    cts = cts + nth_term;
    f = 1;
end

terms = (ii / 2) + 1; % Number of terms (including the first '1' term to
% reach 4 decimal agreement
disp(round(cts,4));
disp(round(cos(x),4));
fprintf('The number of terms needed to reach 4 decimal place is %g.\n', terms)

This returns the expected answers of 0.9010 for both cts and cos(x) and prints 3 terms, but when I change round(..., 4) to round(...,5), it runs the while loop infinitely. Any help would greatly appreciated as to why this happens.

[Walter Roberson]

    for jj = 1:ii
        f = f * jj;
    end

Have you considered using factorial() ?

[Spencer]

I will try it to see if it works but my professor took points off a previous assignment for using factorial() because he “wanted to ensure we understood how for loops worked” (I’m still a little bitter about that because the question didn’t specify we couldn’t use factorial() then either) so I was trying to write it without using it. I’m also curious as to why 4 decimal places works, but 5 doesn’t though.

[Stephen23]

"’m also curious as to why 4 decimal places works, but 5 doesn’t though."

Possibly because ROUND is not a robust approach:

https://www.mathworks.com/matlabcentral/answers/350303-why-0-6-is-not-equal-to-0-2-3#comment_472139

https://www.mathworks.com/matlabcentral/answers/321709-strange-behaviour-of-matlab-find-command#comment_444258

The standard robust approach is to compare the absolute difference against a tolerance:

abs(A-B)<tol

[Moksh]

Hi Spencer,

You can try using the standard tolerance method and try printing the values for 'cts' and 'cos(x)', to get your required stopping criteria.

Sample code for this:

%% Try using different values for tolerance until the required results
while abs(cts - cos(x)) > tol:
    %% Your logic
end

Hope this helps!

Answer 1

[Ayush Anand]

Hi Spencer,

I understand you want to write a script to count the nth number of terms it takes to get a 4 decimal match in cos(x) and the taylor series expansion of cos(x). Setting an equality constraint as the stopping criteria is probably not a good idea, as decimal representation in machine precision works a bit differently than expected.

You can refer to this MATLAB Answer for more understanding on the same:

https://in.mathworks.com/matlabcentral/answers/269201-what-s-going-on-with-the-precision-calculation

So, I would recommend using a tolerance factor instead(you can set this according to your precision standards) and comparing it to the absolute difference between the two values of cos(x). This is a standard way of comparison used in code, and as the difference between the values being compared falls below the tolerance factor, we terminate(assume they are close enough to consider them equal). Here is an example of how you could write this in code:

tol= 1e-4 %Set this value according to your requirement
while abs(cts - cos(x)) > tol:
    %% Your code
end

I hope this helps!