Fast multiple-precision calculation of square root

In this paper a family of higher-order converging algorithms is derived for the multiple-precision calculation of square roots. This family includes the Newton and the Bailey iterations for the approximation to the square root and is globally convergent. The iteration functions of these algorithms are rational functions. To reduce the time-complexities of these algorithms, the author proposes the various methods for the actual computations of the rational functions. The time-complexity analysis of the methods shows that in the fixed-length arithmetic the 5th order converging algorithm by square factor decomposition is the fastest, and that in the variable-length arithmetic the reverse square root algorithm of order 2 is the fastest. In any of the algorithms and the computational methods, it is proved that using variable-length arithmetic the computation is terminated at most the time required for two iterations of the fixed-length arithmetic. The present results are independent of the methods of the multiple-precision multiplications. [In Japanese.]