Nested loops to estimate cos(pi/4)

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

Answer 1

[Steven Lord]

1. You should avoid calling your variable true, as that already has a meaning in MATLAB.
2. This line of code "for ea >= es" is not valid MATLAB syntax.
3. 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.]

[Mary Jeppson]

Thank you for your thorough and conceptual answer. I appreciate it!

Answer 2

[Image Analyst]

You don't need both. You need just one of the other. the for loop is most straightforward in this case:

theta = pi;
trueValue = cos(theta);
numberOfTerms = 10; % Whatever.
theSum = 0;
for k = 0 : numberOfTerms-1
  thisTerm = (-1)^k * theta ^ (2*k) / factorial(2*k);
  theSum(k+2) = theSum(k+1) + thisTerm;
  theError(k+1) = theSum(k+2) - trueValue;
end
% Make fancy plots.
subplot(2, 1, 1);
plot(theSum, 'b*-', 'LineWidth', 2);
ylabel('The Taylor Sum', 'FontSize', 20);
xlabel('Number of Terms', 'FontSize', 20);
grid on;
subplot(2, 1, 2);
plot(theError, 'b*-', 'LineWidth', 2);
ylabel('The Error from True Value', 'FontSize', 20);
xlabel('Number of Terms', 'FontSize', 20);
grid on;
% Set up figure properties:
% Enlarge figure to full screen.
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
% Get rid of tool bar and pulldown menus that are along top of figure.
set(gcf, 'Toolbar', 'none', 'Menu', 'none');
% Give a name to the title bar.
set(gcf, 'Name', 'Demo by ImageAnalyst', 'NumberTitle', 'Off')

[Mary Jeppson]

Thank you for this! I am looking closely at it and learning from it. Thank you!