数学 – Python - JavaScript - 数学はここから始まる - 数 – 高次方程式 – 高次方程式の解法(3次式の解(根)、実数解、虚数解)

学習環境

数学読本〈1〉数・式の計算/方程式/不等式 (松坂和夫(著)、岩波書店)の第3章(数学の威力を発揮する - 方程式)、3.3(高次方程式)、高次方程式の解法の問24-(1)、(2)、(3).を取り組んでみる。

1. $\begin{array}{l} x^{3} - 8 = 0 \\ (x - 2) (x^{2} + 2 x + 4) = 0 \\ x^{2} + 2 x + 4 = 0 \\ x = - 1 \pm \sqrt{1 - 4} \\ = - 1 \pm \sqrt{3} i \\ x = 2, - 1 \pm \sqrt{3} i \end{array}$
2. $\begin{array}{l} x^{3} + 1 = 0 \\ (x + 1) (x^{2} - x + 1) = 0 \\ x = - 1, \frac{1 \pm \sqrt{3} i}{2} \end{array}$
3. $\begin{array}{l} x^{3} + 8 = 0 \\ (x + 2) (x^{2} - 2 x + 4) = 0 \\ x = - 2, 1 \pm \sqrt{3} i \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, solve, plot

print('24.')

x = symbols('x')
eqs = [x ** 3 - 8,
       x ** 3 + 1,
       x ** 3 + 8]

for i, eq in enumerate(eqs, 1):
    print(f'({i})')
    pprint(solve(eq, x))
    print()

p = plot(*eqs, legend=True, show=False)
colors = ['red', 'green', 'blue']

for i, color in enumerate(colors):
    p[i].line_color = color

p.save('sample24.png')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample24.py
24.
(1)
[2, -1 - √3⋅ⅈ, -1 + √3⋅ⅈ]

(2)
⎡    1   √3⋅ⅈ  1   √3⋅ⅈ⎤
⎢-1, ─ - ────, ─ + ────⎥
⎣    2    2    2    2  ⎦

(3)
[-2, 1 - √3⋅ⅈ, 1 + √3⋅ⅈ]

$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample24.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let fns = [[x => x ** 3 - 8, 'red'],
           [x => x ** 3 + 1, 'green'],
           [x => x ** 3 + 8, 'blue']]
    
let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [[2, y1, 2, y2, 'brown'],
                 [-1, y1, -1, y2, 'orange'],
                 [-2, y1, -2, y2, 'purple']];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    [fns].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =

Kamimura's blog

2018年12月5日水曜日

数学 – Python - JavaScript - 数学はここから始まる - 数 – 高次方程式 – 高次方程式の解法(3次式の解(根)、実数解、虚数解)

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