2018年8月29日水曜日

数学 - Python - JavaScript - 線形代数学 - 多項式と素因子分解 – 最大公約因子(整除)

ラング線形代数学(下)(S.ラング (著)、芹沢 正三 (翻訳)、ちくま学芸文庫)の12章(多項式と素因子分解)、2(最大公約因子)、練習問題1.を取り組んでみる。

1. $\begin{array}{}{t}^{n}-1\\ =\left(t-1\right)\left({t}^{n-1}+{t}^{n-2}+{t}^{n-3}+\dots +{t}^{n-\left(n-1\right)}+1\right)\end{array}$

コード(Emacs)

Python 3

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

print('1.')

t = symbols('t')
n, k = symbols('n, k', integer=True)
eq = t ** n - 1

for m in range(10):
eqm = eq.subs({n: m})
eq1 = (t - 1) * summation(t ** k, (k, 0, m - 1))
for s in [eqm, eqm.factor(), eq1, eq1.expand()]:
pprint(s)
print()


$./sample1.py 1. 0 0 0 0 t - 1 t - 1 t - 1 t - 1 2 t - 1 (t - 1)⋅(t + 1) (t - 1)⋅(t + 1) 2 t - 1 3 t - 1 ⎛ 2 ⎞ (t - 1)⋅⎝t + t + 1⎠ ⎛ 2 ⎞ (t - 1)⋅⎝t + t + 1⎠ 3 t - 1 4 t - 1 ⎛ 2 ⎞ (t - 1)⋅(t + 1)⋅⎝t + 1⎠ ⎛ 3 2 ⎞ (t - 1)⋅⎝t + t + t + 1⎠ 4 t - 1 5 t - 1 ⎛ 4 3 2 ⎞ (t - 1)⋅⎝t + t + t + t + 1⎠ ⎛ 4 3 2 ⎞ (t - 1)⋅⎝t + t + t + t + 1⎠ 5 t - 1 6 t - 1 ⎛ 2 ⎞ ⎛ 2 ⎞ (t - 1)⋅(t + 1)⋅⎝t - t + 1⎠⋅⎝t + t + 1⎠ ⎛ 5 4 3 2 ⎞ (t - 1)⋅⎝t + t + t + t + t + 1⎠ 6 t - 1 7 t - 1 ⎛ 6 5 4 3 2 ⎞ (t - 1)⋅⎝t + t + t + t + t + t + 1⎠ ⎛ 6 5 4 3 2 ⎞ (t - 1)⋅⎝t + t + t + t + t + t + 1⎠ 7 t - 1 8 t - 1 ⎛ 2 ⎞ ⎛ 4 ⎞ (t - 1)⋅(t + 1)⋅⎝t + 1⎠⋅⎝t + 1⎠ ⎛ 7 6 5 4 3 2 ⎞ (t - 1)⋅⎝t + t + t + t + t + t + t + 1⎠ 8 t - 1 9 t - 1 ⎛ 2 ⎞ ⎛ 6 3 ⎞ (t - 1)⋅⎝t + t + 1⎠⋅⎝t + t + 1⎠ ⎛ 8 7 6 5 4 3 2 ⎞ (t - 1)⋅⎝t + t + t + t + t + t + t + t + 1⎠ 9 t - 1$


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="0.005">
<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">
<br>
<label for="n0">n = </label>
<input id="n0" type="number" min="0" step="1" value="1">

<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="sample1.js"></script>


JavaScript

let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
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'),
input_a0 = document.querySelector('#a0'),
input_b0 = document.querySelector('#b0'),
input_a1 = document.querySelector('#a1'),
input_b1 = document.querySelector('#b1'),
input_a2 = document.querySelector('#a2'),
input_b2 = document.querySelector('#b2'),
input_n0 = document.querySelector('#n0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_n0],
p = (x) => pre0.textContent += x + '\n';

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),
n0 = parseInt(input_n0.value, 10);

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

let points = [],
fns = [[(x) => x ** n0 - 1, 'green']],
lines = [[1, y1, 1, y2, 'red']];

fns
.forEach((o) => {
let [f, color] = o;

for (let x0 = x1; x0 <= x2; x0 += dx) {
points.push([x0, f(x0), color]);
}
});

let xscale = d3.scaleLinear()
.domain([x1, x2])
let yscale = d3.scaleLinear()
.domain([y1, y2])

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);

p(fns.join('\n'));
};

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