How to make a function that calculate appoximate value of pi

Hi every one; I am going to make a function called a_pi that uses the following approximations of pi

http://www.mathworks.com/matlabcentral/answers/uploaded_files/31727/pi_appoximate.jpg

but that function should have following specifications Instead of going to infinity, the function stops at the smallest k for which the approximation differs from pi (i.e., the value returned MATLAB’s built-­‐in function) by no more than the positive scalar, delta, which is the only input argument. The first output of the function is the approximate value of π, while the second is k. (Note: if your program or the grader takes a long time, you may have created an infinite loop and need to hit Ctrl-­‐C on your keyboard.) How to deal with that question.Thanks in advance for assistance..

[Muhammad Usman Saleem]

formaula is given in the image attached in the the post

[James Tursa]

Please post what you have done so far so we can comment on it and make suggestions.

Answer 1

[Andrei Bobrov]

function [pi_here,k1] = a_pi(delta)
    f = @(k)sqrt(12)* 1./(  (2*k+1).*(-3).^k  )  ;
    k1 = ceil(fzero(@(k)abs(f(k)) - delta,1)); 
    pi_here = sum(f(0:k1));
end

or

function [pi_here,k1] = a_pi(delta)
    f = @(k)sqrt(12)* sum(1./(  (2*k+1).*(-3).^k  ))  ;
    k1 = ceil(fzero(@(k)abs(f(0:k) - pi) - delta,1)); 
    pi_here = f(0:k1);
end

or

function [pi_here,k1] = a_pi(delta)
    k = 0;
    pi_here = 0;
    while  abs(pi_here - pi) > delta
        pi_here = pi_here +  sqrt(12)./(  (2*k+1).*(-3).^k  );
        k = k + 1;
    end
    k1= k - 1;
end

[Muhammad Usman Saleem]

@Andrei Bobrov thank you very much.Actually this and another question of my work was the only questions which i unable to understand or solve... thanks again for your assistance..

[Cedric]

 function main

    tol = 1e-15 ;
    n   = 1e3 ;

    tic ;
    for k = 1 : n
        a_pi1( tol ) ;
    end
    toc

    tic ;
    for k = 1 : n
        a_pi2( tol ) ;
    end
    toc

    tic ;
    for k = 1 : n
        a_pi3( tol ) ;
    end
    toc

 end

 function [pi_here,k1] = a_pi1(delta)
    f = @(k)sqrt(12)* 1./(  (2*k+1).*(-3).^k  )  ;
    k1 = ceil(fzero(@(k)abs(f(k)) - delta,1)); 
    pi_here = sum(f(0:k1));
 end

 function [pi_here,k1] = a_pi2(delta)
    f = @(k)sqrt(12)* sum(1./(  (2*k+1).*(-3).^k  ))  ;
    k1 = ceil(fzero(@(k)abs(f(0:k) - pi) - delta,1)); 
    pi_here = f(0:k1);
 end

 function [pi_here,k1] = a_pi3(delta)
    k = 0;
    pi_here = 0;
    while  abs(pi_here - pi) > delta
        pi_here = pi_here +  sqrt(12)./(  (2*k+1).*(-3).^k  );
        k = k + 1;
    end
    k1= k - 1;
 end

Gives

 >> main
 Elapsed time is 0.731927 seconds.
 Elapsed time is 2.314194 seconds.
 Elapsed time is 0.007396 seconds.

[Muhammad Usman Saleem]

Thanks, its lengthy approach but will be fine.