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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 Solving quadratic equations with continued fractions ： ウィキペディア英語版 Solving quadratic equations with continued fractions In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is :$ax^2+bx+c=0,\backslash ,\backslash !$ where ''a'' ≠ 0. The quadratic equation can be solved using the well-known quadratic formula, which can be derived by completing the square. That formula always gives the roots of the quadratic equation, but the solutions are expressed in a form that often involves a quadratic irrational number, which is an algebraic fraction that can be evaluated as a decimal fraction only by applying an additional root extraction algorithm. If the roots are real, there is an alternative technique that obtains a rational approximation to one of the roots by manipulating the equation directly. The method works in many cases, and long ago it stimulated further development of the analytical theory of continued fractions. ==A simple example== Here is a simple example to illustrate the solution of a quadratic equation using continued fractions. We begin with the equation :$$ x^2 = 2\, and manipulate it directly. Subtracting one from both sides we obtain :$$ x^2 - 1 = 1.\, This is easily factored into :$$ (x+1)(x-1) = 1\, from which we obtain :$$ (x-1) = \frac\, and finally :$$ x = 1+\frac.\, Now comes the crucial step. We substitute this expression for ''x'' back into itself, recursively, to obtain :$$ x = 1+\cfrac\right)} = 1+\cfrac}.\, But now we can make the same recursive substitution again, and again, and again, pushing the unknown quantity ''x'' as far down and to the right as we please, and obtaining in the limit the infinite continued fraction :$$ x = 1+\cfrac }}}} = \sqrt.\, By applying the fundamental recurrence formulas we may easily compute the successive convergents of this continued fraction to be 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, ..., where each successive convergent is formed by taking the numerator plus the denominator of the preceding term as the denominator in the next term, then adding in the preceding denominator to form the new numerator. This sequence of denominators is a particular Lucas sequence known as the Pell numbers. 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「Solving quadratic equations with continued fractions」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.