While loop runs infinitely

I'm trying to create a function that takes two parameters: x and threshold. X is the angle and the threshold is the percent error I need to get when approximating sine with the taylor series. However my while loop runs infinitely and I'm very confused as to how do go about fixing it.

Here's what I have so far.

   function [approx, terms] = approx_sine(x, threshold)
approx = x; % Initial approximation 
terms = 0; % Number of additional terms added to improve the approximation;
% Write your code here using a while loop to improve the above approximation
while threshold <= abs((sin(x)-approx)/sin(x))
   terms = terms + 2;
   approx = (-1)^(terms+1)*(x.^terms)/ factorial(terms)+x;
end
end

Answer 1

[Thorsten]

The code for approximation is wrong. There are some errors in your formula: term takes values 0, 2, 4, ... but should take values 1,3,5,7; the (-1)^n changes sign with every new term, but since your term is always even, (-1)^(term+1) is always odd; and finally you add the n'th approximation term to the initial approximation x, instead of adding it to the most recent approximation.

This works

function [approx, n] = approxsine(x, threshold)
approx = x; % Initial approximation 
n = 0; % Number of additional terms added to improve the approximation;
% Write your code here using a while loop to improve the above approximation
while threshold <= abs((sin(x)-approx)/sin(x))
   n = n + 1;
   approx = approx + (-1)^(n)*x.^(2*n+1)/factorial(2*n+1);
end
end

[Image Analyst]

And I'm a very strong advocate of failsafes, especially in while loops to prevent just this problem of infinite loops. Check the loop counter and if it's more than you ever realistically expect, bail out with an error message:

n = 1;
nMax = 1000000
while threshold <= abs((sin(x)-approx)/sin(x)) && n < nMax
   n = n + 1;
   approx = approx + (-1)^(n)*x.^(2*n+1)/factorial(2*n+1);
end
if n >= nMax
    errorMessage = sprintf('No solution found after %d iterations!', n-1);
    uiwait(errordlg(errorMessage));
end

Note the additional check of "&& n < nMax" I added to the while line.

[Torsten]

Since sin(x) < = 1, better use absolute instead of relative error within error estimate:

while threshold <= abs(sin(x)-approx)

Best wishes

Torsten.

Answer 2

[Walter Roberson]

Remove the line

   threshold = abs((sin(x)-approx)/sin(x))

It is overwriting the threshold that you passed in.

[Anthony Siddique]

I'm still having the same problem. The while loop won't resolve.

[Walter Roberson]

>> limit((-1)^(terms+1)*x^terms/factorial(terms), terms = infinity);
                             0

Your approx is (-1)^(terms+1)*x^terms/factorial(terms) + x . As I indicate above, as terms increases, the first part of that tends to 0 for all x. With the first part tending to 0, only the second part will start to matter, the "+ x" part. Your approx goes to x.

Your termination test then becomes

threshold <= abs((sin(x)-x)/sin(x))

or threshold <= abs(1 - x/sin(x))

For any given threshold there is a limited range of x that can satisfy this, and that range of x does not increase smoothly with the threshold.