17 lines
821 B
Text
17 lines
821 B
Text
[http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Almkvist-Berndt585-608.pdf Almkvist Berndt 1988] begins with an investigation of why the agm is such an efficient algorithm, and proves that it converges quadratically. This is an efficient method to calculate <math>\pi</math>.
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With the same notations used in [[Arithmetic-geometric mean]], we can summarize the paper by writing:
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<math>\pi =
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\frac{4\; \mathrm{agm}(1, 1/\sqrt{2})^2}
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{1 - \sum\limits_{n=1}^{\infty} 2^{n+1}(a_n^2-g_n^2)}
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</math>
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This allows you to make the approximation, for any large '''N''':
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<math>\pi \approx
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\frac{4\; a_N^2}
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{1 - \sum\limits_{k=1}^N 2^{k+1}(a_k^2-g_k^2)}
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</math>
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The purpose of this task is to demonstrate how to use this approximation in order to compute a large number of decimals of <math>\pi</math>.
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