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Asked by Mafia on 13 Dec 2012

My script is missing something. Perhaps I do not understand the meaning of 'while' or 'for' correctly. But here is my code:

{% this script approximates the value of pi n=0; x=0; while n<100000 % loop ends at 10^6 n=n+2; % n will always be a multiple of 2 while x<3.14 x=4*((-1).^n/(2*n+1)) +x; % the summation formula end disp(x) end }

I want it such that my value of pi in this case 'x' is no greater than 1e-6 away from the actual value of pi. So when it satisfies this condition the script will cease. However my script gives me endless loop of 3.2 and it pissing me off, all day on this crap :S . Why is matlab so hard?????

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Answer by Roger Stafford on 13 Dec 2012

Accepted answer

Adam, your while-within-a-while construct is certainly not going to work for you. The terms "4*(-1).^n/(2*n+1)" are supposed to be added once for each successive value of n, starting with n = 0, but, as it is, your code will add this term for an unchanging n a number of times before going on to the next n because of your inner while-loop arrangement. Also the n = n+2 is wrong. It should be for each successive n, n = n + 1, not every multiple of 2.

This approximation is based on the infinite series expansion of arctan:

arctan(t) = t - 1/3*t^3 + 1/5*t^5 - 1/7*t^7 + ...

with t set to 1 giving

pi/4 = arctan(1) = 1 - 1/3 + 1/5 - 1/7 + ...

With t equal to 1 this is a very, very slowly converging series, and you should have a single while-loop with the condition that 1/(2*n+1) < 1e-6 to achieve the accuracy you desire, which will require a great many loops. You should start with n = 0 and compute your first x before adding 1 to n. (Your present code erroneously uses n = 2 for its first term.)

Note that if you were to use t = 1/sqrt(2) instead of t = 1, your series would converge much, much faster, but the added terms would have to be changed to 6*(-1)^n/(2*n+1)*t^(2*n+1), since arctan(1/sqrt(2)) = pi/6.

Note 2: Sean is teasing you. He starts with pi itself and his loop never executes.

Roger Stafford

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bym on 15 Dec 2012

@Adam - you might want to consider that

(-1).^n/(2*n+1)

can be positive or negative; but you are only interested in the magnitude. Also consider the comparison to 1e-6, I believe you mean greater than (>)

Mafia on 16 Dec 2012

%e n=0; x=0; k=4*((-1)^n/(2*n+1)); while k>1e-6 n= n+1; x=x+k; k=4*((-1)^n/(2*n+1)); end x This doesn't work either and i only want n to increase by 2 each time through the loop. It still won't converge anywhere near pi... I don't get what i'm missing.

Answer by Sean de Wolski on 13 Dec 2012

function myPi = piEst()

myPi = pi; for ii = 1:0 myPi = myPi+ii; end end

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Sean de Wolski on 13 Dec 2012

I used a loop!

3.141592653589793238466 is still an *approximation* to **actual** pi. So my code meets all of your constraints!

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