MATLAB: Nested loops to estimate cos(pi/4)

loopstaylor series

I have written this code that is giving me an error:

true = cos(pi/4);
n = 2;
es = .5^(2-n);
x = pi/4;
ea = 1;
est = 1;
k = 1;
while(1)
  for ea >= es
      est = 1+x^(k*2)/factorial(k*2)*(-1)^k
      ea = abs(true-est)/true*100
  end
  k = k+1;
end

Clearly, I do not understand the relationship of for to while loops. Can anyone help me understand?

Thank you, Mary

Best Answer

You should avoid calling your variable true, as that already has a meaning in MATLAB.
This line of code "for ea >= es" is not valid MATLAB syntax.
Your while loop is an infinite loop; you have not specified a way to break out of it.

Most of your line of code "est = 1+x^(k*2)/factorial(k*2)*(-1)^k" matches the term in the series definition of the cosine function, but the "1+" part doesn't.

Basically you want your code to look something like this pseudocode:

 1) initialize variables
 2) while (the approximation is not close enough to the known value)
 2a) add another term to your approximation

There's a while loop in step 2, but the condition is not 1. There is no for loop in the pseudocode. I can't say much more without simply giving you the code.

In general, if you're still unsure about how while loops work, I think one way you can understand them better is to walk through the first example (calculating factorial(10)) on its documentation page yourself, with you playing the role of MATLAB. Take a piece of paper and a pencil and "execute" each line yourself, recording the values of variables as they change on that paper. [When you reach the end statement, return to the while statement. If the while condition is not satisfied, go to the statement immediately after the end.]

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