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

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Anastasia Kyriakou 2020-2-22

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回答： Pravin Jagtap 2020-2-25

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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

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Anastasia Kyriakou 2020-2-23

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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 2020-2-23

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.

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回答（1 个）

Pravin Jagtap 2020-2-25

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/506975-algorithm-for-computing-is-due-to-archimedes-how-steps-can-be-repeated#answer_417211

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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