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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Heron's method ： ウィキペディア英語版 Methods of computing square roots In numerical analysis, a branch of mathematics, there are several square root algorithms or methods of computing the principal square root of a nonnegative real number. For the square roots of a negative or complex number, see below. Finding $\backslash sqrt$ is the same as solving the equation $f(x)\; =\; x^2\; -\; S\; =\; 0\backslash ,\backslash !$. Therefore, any general numerical root-finding algorithm can be used. Newton's method, for example, reduces in this case to the so-called Babylonian method: :$x\_=x\_n-\backslash frac=x\_n-\backslash frac=\backslash frac\backslash left(x\_n+\backslash frac\backslash right)$ Generally, these methods yield approximate results. To get a higher precision for the root, a higher precision for the square is required and a larger number of steps must be calculated. == Rough estimation == Many square root algorithms require an initial seed value. If the initial seed value is far away from the actual square root, the algorithm will be slowed down. It is therefore useful to have a rough estimate, which may be very inaccurate but easy to calculate. With $S$ expressed in scientific notation as $a\backslash times10^$ where and ''n'' is an integer, the square root $\backslash sqrt\; =\; \backslash sqrt\backslash times10^n$ can be estimated as :$\backslash sqrt\; \backslash approx\; \backslash begin$ 2 \cdot 10^n & \text a < 10, \\ 6 \cdot 10^n & \text a \geq 10. \end The factors two and six are used because they approximate the geometric means of the lowest and highest possible values with the given number of digits: $\backslash sqrt\}\; =\; \backslash sqrt()\; \backslash approx\; 2\; \backslash ,$ and $\backslash sqrt\}\; =\; \backslash sqrt()\; \backslash approx\; 6\; \backslash ,$. For $S\; =\; 125348\; =\; 12.5348\; \backslash times\; 10^4$, the estimate is $\backslash sqrt\; \backslash approx\; 6\; \backslash cdot\; 10^2\; =\; 600$. When working in the binary numeral system (as computers do internally), by expressing $S$ as $a\backslash times2^$ where , the square root $\backslash sqrt\; =\; \backslash sqrt\backslash times2^n$ can be estimated as $\backslash sqrt\; \backslash approx\; 2^n$, since the geometric mean of the lowest and highest possible values is $\backslash sqrt\}\; =\; \backslash sqrt()\; =\; 1$. For $S\; =\; 125348\; =\; 1\backslash ;1110\backslash ;1001\backslash ;1010\backslash ;0100\_2\; =\; 1.1110\backslash ;1001\backslash ;1010\backslash ;0100\_2\backslash times2^\backslash ,$ the binary approximation gives $\backslash sqrt\; \backslash approx\; 2^8\; =\; 1\backslash ;0000\backslash ;0000\_2\; =\; 256\backslash ,\; .$ These approximations are useful to find better seeds for iterative algorithms, which results in faster convergence. == Babylonian method == Perhaps the first algorithm used for approximating is known as the Babylonian method, named after the Babylonians,〔There is no direct evidence showing how the Babylonians computed square roots, although there are informed conjectures. (Square root of 2#Notes gives a summary and references.)〕 or "Hero's method", named after the first-century Greek mathematician Hero of Alexandria who gave the first explicit description of the method. It can be derived from (but predates by 16 centuries) Newton's method. The basic idea is that if is an overestimate to the square root of a non-negative real number then will be an underestimate and so the average of these two numbers may reasonably be expected to provide a better approximation (though the formal proof of that assertion depends on the inequality of arithmetic and geometric means that shows this average is always an overestimate of the square root, as noted in the article on square roots, thus assuring convergence). More precisely, if is our initial guess of and is the error in our estimate such that , then we can expand the binomial and solve for :$e=\backslash frac\; \backslash approx\; \backslash frac,$ since $e\; \backslash ll\; x$. Therefore, we can compensate for the error and update our old estimate as :$x\; :=\; x\; +\; e\; =\; \backslash frac\; =\; \backslash frac\}$ Since the computed error was not exact, this becomes our next best guess. The process of updating is iterated until desired accuracy is obtained. This is a quadratically convergent algorithm, which means that the number of correct digits of the approximation roughly doubles with each iteration. It proceeds as follows: #Begin with an arbitrary positive starting value (the closer to the actual square root of , the better). #Let be the average of and (using the arithmetic mean to approximate the geometric mean). #Repeat step 2 until the desired accuracy is achieved. It can also be represented as: :$x\_0\; \backslash approx\; \backslash sqrt,$ :$x\_\; =\; \backslash frac\; \backslash left(x\_n\; +\; \backslash frac\backslash right),$ :$\backslash sqrt\; S\; =\; \backslash lim\_\; x\_n.$ This algorithm works equally well in the s, but cannot be used to identify real square roots with -adic square roots; one can, for example, construct a sequence of rational numbers by this method that converges to +3 in the reals, but to −3 in the 2-adics. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Methods of computing square roots」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.