Two optimal square-rooting algorithms ussing iterative multiplication

With a view to the computational speed, it is obviously advantageous to perform the frequently used arithmetic operations in the hardware rather than in software, especially in scientifically oriented real-time applications. Here we present two optimal square-rooting algorithms for hardware implementation which utilize high-speed-multiplication of sophisticated scientific computers and no division. One of them is the well-known one, titled here as Algorithm N, which was derived on the mathematical basis of NEWTON iteration scheme;, while the other is a new algorithm, titled as Algorithm G, derived on the basis of Normalization Technique. Algorithm G is compared with Algorithm N in various aspects since Algorithm N has been believed so far to be the most efficient among this class of algorithms. Both possess the property of quadratic convergence, an important property as far as speed is concerned. However, Algorithm G turns out to be faster with no more hardware than Algorithm N, since it requires less time for each iteration, analogous to the higher speed of Goldschmidt\´ s division algorithm over original NEWTON iteration scheme for division. It is shown that the higher speed of Algorithm G purposed purely for hardware implementation is primarily due to the inherent parallelism which makes full use of characteristics of hardware implementation. Furthermore, in case where Goldschmidt" s division algorithm is implemented as in IBM 91/360, Algorithm G has more advantages because of possibility to share most of the required hardware with division unit including one for initial approximation. Therefore, it is believed that at present Algorithm G has the most promising advantages for hardware implementation.