## 2018年3月15日木曜日

### 数学 - Python - JavaScript - 解析学 - 多変数の関数 - 極値問題(極値点と臨界点、鞍点)

1. $\begin{array}{}{D}_{1}f\left(x,y\right)\\ =-2x\left(y-2{x}^{2}\right)+\left(y-{x}^{2}\right)\left(-4x\right)\\ =-2x\left(y-2{x}^{2}+2y-2{x}^{2}\right)\\ =-2x\left(3y-4{x}^{2}\right)\end{array}$
$\begin{array}{}{D}_{2}f\left(x,y\right)\\ =\left(y-2{x}^{2}\right)+\left(y-{x}^{2}\right)\\ =2y-3{x}^{2}\end{array}$
$\begin{array}{}2y-3{x}^{2}=0\\ y=\frac{3}{2}{x}^{2}\end{array}$
$\begin{array}{}-2x\left(\frac{9}{2}{x}^{2}-4{x}^{2}\right)=0\\ {x}^{3}=0\\ x=0\end{array}$

よって、

$gradf\left(0,0\right)=\left(0,0\right)$

ゆえに、臨界点は、

$\left(0,0\right)$

macOS High Sierraの標準搭載されているグラフ作成ソフト、Grapher で作成。

コード(Emacs)

Python 3

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

x, y = symbols('x, y')
f = (y - x ** 2) * (y - 2 * x ** 2)
D = [Derivative(f, t, 1) for t in [x, y]]
gradf = Matrix([Dt.doit() for Dt in D])
pprint(t)
print()


$./sample0.py ⎡∂ ⎛⎛ 2 ⎞ ⎛ 2 ⎞⎞ ∂ ⎛⎛ 2 ⎞ ⎛ 2 ⎞⎞⎤ ⎢──⎝⎝- 2⋅x + y⎠⋅⎝- x + y⎠⎠, ──⎝⎝- 2⋅x + y⎠⋅⎝- x + y⎠⎠⎥ ⎣∂x ∂y ⎦ ⎡ ⎛ 2 ⎞ ⎛ 2 ⎞⎤ ⎢- 2⋅x⋅⎝- 2⋅x + y⎠ - 4⋅x⋅⎝- x + y⎠⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎣ - 3⋅x + 2⋅y ⎦ [{x: 0, y: 0}]$


HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" 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="x0">x0 = </label>
<input id="x0" type="number" value="0">
<label for="y0">y0 = </label>
<input id="y0" type="number" value="0">

<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="sample0.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_x0 = document.querySelector('#x0'),
input_y0 = document.querySelector('#y0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_x0, input_y0],
p = (x) => pre0.textContent += x + '\n',
range = (start, end, step=1) => {
let res = [];
for (let i = start; i < end; i += step) {
res.push(i);
}
return res;
};

let f = (x, y) => (y - x ** 2) * (y - 2 * x ** 2);

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),
x0 = parseFloat(input_x0.value);
y0 = parseFloat(input_y0.value);

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

let points = [],
g = (x) => f(x, y0),
h = (y) => f(x0, y),
lines = [],
fns = [[g, 'red'],
[h, 'green']];
fns
.forEach((o) => {
let [fn, color] = o;

for (let x = x1; x <= x2; x += dx) {
let y = fn(x);

if (Math.abs(y) < Infinity) {
points.push([x, y, 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('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.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.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();