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大家知道,計算三次方程的根虽然找到了著名的卡旦公式,可是这个公式还有这样一个缺点,是用包含虚数的立方根来表示方程的实根,并且我們不能用代数的方法去掉这个虛数,加上式子的计算冗繁,因而并不常用于求三次方程的根,而用近似解来替代。近似解法很多,本文介紹一个逐次逼近法。这个方法,虽然課本上从未讲过,但由于它計算簡单,容易掌握,尚有实用价值。我們先看这样一个数列它有二个单調的子数列:{Q_(2n-1)},{Q_(2n)},且有上界a+b/a~2与下界a。根据极限存在的基本定理知道,子数列:{Q_(2n-1)},{Q_(2n)}各自收斂于确定的极限S_1和S_2。
Everyone knows that although the famous Kadan formula was found at the root of the Cubic Equation, this formula has the disadvantage of using a cube root containing imaginary numbers to represent the real roots of the equation, and we cannot use algebraic methods to remove this imaginary number. With the computational tediousness of the formula, it is not often used to find the root of a cubic equation, instead it is replaced by an approximate solution. There are many approximate solutions. This article describes a successive approximation method. This method, although it has never been mentioned in textbooks, is practically valuable because it is simple and easy to master. Let’s look at such a sequence. It has two monotonous sub-series: {Q_(2n-1)}, {Q_(2n)}, with the upper bound a+b/a~2 and the lower bound a. According to the basic theorem of the existence of the limit, it is known that the sub-series of numbers: {Q_(2n-1)}, {Q_(2n)} converge to the determined limits S_1 and S_2, respectively.