algorithm for computing ? is due to Archimedes: how steps can be repeated

The following algorithm for computing ? is due to Archimedes:

1. Start with ? = 1 and ? = 6.

2. Replace ? by 2?.

3. Replace ? by sqrt(2 − sqrt(4 − ?))

4. Let ? = ??/2.

5. Let ? = na/2

6. Let ? = (? + ?)/2 (estimate of ?)

7. Let ? = (? − ?)/2 (estimate of error)

8. Repeat steps 2 – 7 until ? becomes smaller than a given tolerance ???.

9. Output ? and ?

Write a function that implements this algorithm. Use your function to determine ? and ? for ??? = 10^(=k)

, ? = 2, 3, … , 10.

I have written a code but i do not know how to write it so that if the value of e is larger than tol then steps 2-7 repeated. I have written this :

```

function [p,e] = algorithmPi(tol)
a = 1;
n = 6;
e = inf;            

    n = 2*n;
    a = sqrt(2-sqrt(4-(a^2)));      
    l = (n*a)/2;
    u = l/(sqrt((1-(a)^2)/2));
    p = (u+l)/2;                % estimate of pi
    e = (u-l)/2;                % estimate of error
    if e < tol
        done = true;
        disp('Complete: Error below tolerance')
    end
end

[Les Beckham]

To repeat steps 2-7 until a tolerance is reached, wrap them in a while loop with an if test inside that issues a break command when the tolerance is reached. Read the documentation for while, if, and break. Try starting here: https://www.mathworks.com/help/matlab/matlab_prog/loop-control-statements.html

I'm pretty sure that there are serious problems with the algorithm as you describe it, however. For example, items 4 and 5 show u and l being defined as the same (n*a/2). Your code does not match the outlined algorithm, either. The code defining a has an extra square root or power of 1/2 and lots of extra and questionably placed parentheses. It definitely doesn't match the definition in item 3 which is relatively simple.

Read the docs for looping help and double check the definition of your algorithm. Respond with a comment if you still have questions.

[Rik]

It is also good practice to tag your homework questions as such.

[Anastasia Kyriakou]

i have edited it..is it correct now?

[Rik]

You forgot to put in the while loop, so steps 2-7 don't repeat in your code.

Personally I would chose to make the breaking condition obvious:

while e<tol

It is fairly easy to test your code: run in. You know the true value of pi, so you can run your function and see if pi comes out. I would also suggest you take another look at how you define your tolerance. To me, the assignment looks as if the input is k and you need to calculate the value of tol.

[Anastasia Kyriakou]

i have to run it with the value tol=0.01, and some more,,,

Also, when you say i have to write the while, do u mean i have to delete ''if'' and write ''while''? will not work

[Rik]

Have you read the documentation for while? It will repeat a block of code while some condition is true. That means you can do two thing:

while true
    %code to be repeated
    if cond
        break
    end
end

or

while cond
    %code to be repeated
end

Personally I think the second is easier to understand. The indentation of your code suggests you already had a while in your function, but removed it before posting here. I would suggest putting the while there and see what happens.

[Anastasia Kyriakou]

Where exactly i have to write this? '' while cond %code to be repeated end ? In which line ?

[Anastasia Kyriakou]

function [p,e] = algorithmPi(tol)
a = 1;
n = 6;
e = inf;            

   
    n = 2*n;
    a = sqrt(2-sqrt(4-(a^2)));      
    l = (n*a)/2;
    u = l/(sqrt((1-(a)^2)/2));
    p = (u+l)/2;                % estimate of pi
    e = (u-l)/2;                % estimate of error
    if e < tol
         
        done = true;
        disp('Complete: Error below tolerance')
        while cond
    %code to be repeated
end
    end
end
% end 

Like that?

[Rik]

What code do you want to repeat? Make sure that is inside the while loop.

I would strongly urge you to read the documentation for the functions you're using if you don't understand what they do.

Answer 1

[Pravin Jagtap]

Hello Anastasia,

As mentioned in the above comments, I would recommend you to follow the documentation for understanding the loops. Refer to the following template which will help you to implement the algorithm:

function [p,e] = algorithmPi(tol)
    % step 1 - Initialize a, n and e(to inf)  
    
    % step 2 - Iterative process from 2 -7
    while (e > tol)
        % step 3 - Replace n and a 
        
        % step 4 - Compute l, u, p and e           
    end
end