Borwein's algorithm
Borwein's algorithm is an algorithm devised by Jonathan[?] and Peter Borwein[?] to calculate the value of 1/π.It works as follows:
- Start out by setting
- <math>a_0 = 6 - 4\sqrt{2}</math>
- <math>y_0 = \sqrt{2} - 1</math>
- Then iterate
- <math>y_{k+1} = \frac{1-(1-y_k^4)^{1/4}}{1+(1-y_k^4)^{1/4}}</math>
- <math>a_{k+1} = a_k(1+y_{k+1})^4 - 2^{2k+3} y_{k+1} (1 + y_{k+1} + y_{k+1}^2)</math>
Then ak converges quartically against 1/π; that is, each iteration approximately quadruples the number of correct digits.
See also
- Borwein's algorithm (others) for an explanation of other algorithms by Jonathan and Peter Borwein to determine the digits of π.
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PANDULPH. Hail, noble prince of France!
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Between this chastis'd kingdom and myself
And now 'tis far too huge to be blown out
You taught me how to know the face of right,
Yea, thrust this enterprise into my heart;
His peace with Rome? What is that peace to me?
After young Arthur, claim this land for mine;
Because that John hath made his peace with Rome?
What men provided, what munition sent,
That undergo this charge? Who else but I,
Sweat in this business and maintain this war?
'Vive le roi!' as I have bank'd their towns?
To will this easy match, play'd for a crown?
No, no, on my soul, it never shall be said.
LEWIS. Outside or inside, I will not return
As to my ample hope was promised
And cull'd these fiery spirits from the world
Even in the jaws of danger and of death.
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Let me have audience: I am sent to speak.
I come, to learn how you have dealt for him;
And warrant limited unto my tongue.
And will not temporize with my entreaties;
BASTARD. By all the blood that ever fury.
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