数学 - 基本理論(Basic Theory) - 微分(Differentiation) - 関数のグラフを描く - 高次代数方程式の根 - オイラーの贈物

オイラーの贈物―人類の至宝eiπ=-1を学ぶ
(東海大学出版会)
吉田 武 (著)

学習環境

• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

オイラーの贈物―人類の至宝eiπ=-1を学ぶ (吉田 武(著)、東海大学出版会)の第Ⅰ部(基礎理論(Basic Theory))、3章(微分(Differentiation))、3.8(関数のグラフを描く)、問題5.を解いてみる。

問題5.

$\begin{array}{l}f\left(x\right)=x^3+x^2-4x+1\\ f^{\left(1\right)}\left(x\right)=3x^2+2x-4\\ =\frac{1}{3}{\left(x+3\right)}^2-7\\ f^{\left(1\right)}\left(x\right)=0\\ x=\frac{-1\pm \sqrt{13}}{3}\\ f^{\left(2\right)}\left(x\right)=6x+2=6\left(x+\frac{1}{3}\right)\\ f^{\left(2\right)}\left(x\right)=0\\ x=-\frac{1}{3}\\ \frac{-1-\sqrt{13}}{3}<-\frac{1}{3}<\frac{-1+\sqrt{13}}{3}\\ f\left(-\frac{1}{3}\right)=-\frac{1}{27}+\frac{1}{9}+\frac{4}{3}+1=\frac{-1+3+36+27}{27}=\frac{65}{27}\\ f\left(\frac{-1+\sqrt{13}}{3}\right)=\frac{-1+3\sqrt{13}-39+13\sqrt{13}}{27}+\frac{14-2\sqrt{13}}{9}+\frac{4-4\sqrt{13}}{3}+1\\ =\frac{-40+16\sqrt{13}+42-6\sqrt{13}+36-36\sqrt{13}+27}{27}\\ =\frac{65-26\sqrt{13}}{27}<0\\ f\left(\frac{-1-\sqrt{13}}{3}\right)=\frac{65+26\sqrt{13}}{27}\\ x_{n+1}=x_n-\frac{f\left(x_n\right)}{f^{\left(1\right)}\left(x_n\right)}\\ =x_n-\frac{x_n^3+x_n{}^2-4x_n+1}{3x_n^2+2x_n-4}\\ =\frac{2x_n^3+x_n^2-1}{3x_n^2+2x_n-4}\\ \\ -\frac{5}{3}<\frac{-1-\sqrt{13}}{3}\\ x_0=-\frac{5}{3}\\ x_1=\frac{-\frac{250}{27}+\frac{25}{9}-1}{\frac{25}{3}-\frac{10}{3}-4}=\frac{-250+75-27}{27}=\frac{-202}{27}(=-7.481···)\\ 以下、pythonを使って計算。\\ x_2=\frac{\text{-15402791}}{\text{2931876}}\left(=\text{-5}\text{.253561542166176}\right)\\ x_3=\frac{\text{-3319063268830266108581}}{\text{860558640294071840166}}\left(=\text{-3}\text{.856870541321936}\right)\\ x_4=\text{-3}\text{.0647804042831877}\\ x_5=\text{-2}\text{.7248676719102827}\\ x_6=\text{-2}\text{.654106846739904}\\ x_7=\text{-2}\text{.6510987533831787}\\ x_8=\text{-2}\text{.651093408954031}\\ x_9=\text{-2}\text{.651093408937175}\end{array}$

コード(Emacs)

#!/usr/bin/env python3
# -*- coding: utf-8 -*_

import fractions

def newton(x, r=10):
    inner = lambda x:fractions.Fraction(2 * x ** 3 + x ** 2 - 1,
                                        3 * x ** 2 + 2 * x - 4)
    for i in range(r):
        print('x_{0} = {1}'.format(i, float(x)))
        x = inner(x)
    print()

x = fractions.Fraction(-5, 3)
newton(x)

x = 0
newton(x)

x = fractions.Fraction(5, 3)
newton(x)

入出力結果(Terminal, IPython)

$ ./sample5.py
x_0 = -1.6666666666666667
x_1 = -7.481481481481482
x_2 = -5.253561542166176
x_3 = -3.856870541321936
x_4 = -3.0647804042831877
x_5 = -2.7248676719102827
x_6 = -2.654106846739904
x_7 = -2.6510987533831787
x_8 = -2.651093408954031
x_9 = -2.651093408937175

x_0 = 0.0
x_1 = 0.25
x_2 = 0.27358490566037735
x_3 = 0.2738905022655762
x_4 = 0.27389055496421605
x_5 = 0.2738905549642176
x_6 = 0.2738905549642176
x_7 = 0.2738905549642176
x_8 = 0.2738905549642176
x_9 = 0.2738905549642176

x_0 = 1.6666666666666667
x_1 = 1.4396135265700483
x_2 = 1.3812200268652726
x_3 = 1.3772213440738694
x_4 = 1.3772028543676846
x_5 = 1.3772028539729577
x_6 = 1.3772028539729577
x_7 = 1.3772028539729577
x_8 = 1.3772028539729577
x_9 = 1.3772028539729577

$