円周率公式の改良と高速多倍長計算の実装

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  • Improvement of the π Calculation Algorithm and Implementation of Fast Multiple-Precision Computation
  • エンシュウリツ コウシキ ノ カイリョウ ト コウソク タバイチョウ ケイサン ノ ジッソウ

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In this paper, we derive efficient quadratic and quartic iteration algorithms from the improvement of Gauss'arithmetic-geometric mean (AGM) algorithm. The number of multiplications in the improved quadratic algorithm is only half the number of the original algorithm, but the number of the square root operations in the improved AGM iterations is equal to the number of the original algorithm. So we derive an efficient simultaneous Newton iteration for the square root calculation. Next, weimplement a fast multiple-precision computation for the proposed algorithms and estimate the number of floating point operations and the execution time to compute the AGM iterations.

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