2018年3月23日金曜日

学習環境

解析入門〈3〉(松坂 和夫(著)、岩波書店)の第14章(多変数の関数)、14.3(極値問題)、問題1-(e).を取り組んでみる。



    1. D 1 f x , y = 2 x - 2 y 2 D 2 f x , y = - 4 x y + 1 - 4 y 3
      D 1 f x , y = 0 D 2 f x , y = 0
      2 x - 2 y 2 = 0 - 4 x y + 1 - 4 y 3 = 0
      x = y 2 - 4 y 3 + 1 - 4 y 3 = 0 y = 1 2 x = 1 4

      よって臨界点は、

      1 4 , 1 2

      第2次導関数の計算。

      D 1 2 f x , y = 2 D 2 2 f x , y = - 4 x - 12 y 2 D 1 D 2 f x , y = - 4 y

      よって、

      Δ 1 4 , 1 2 = - 4 · 1 2 2 - 2 - 4 · 1 4 - 12 · 1 4 = 4 - 2 - 1 - 3 = 4 + 8 > 0

      ゆえに、極値点ではない。

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, exp, solve

x, y = symbols('x, y')
f = x ** 2 - 2 * x * y ** 2 + y - y ** 4

for t in [x, y]:
    for n in range(1, 3):
        D = Derivative(f, t, n)
        for s in [D, D.doit()]:
            pprint(s)
            print()
Dxy = Derivative(Derivative(f, x, 1).doit(), y, 1).doit()
pprint(Dxy)

delta = Dxy - Derivative(f, x, 2).doit() * Derivative(f, y, 2).doit()

critical_point = solve(
    (Derivative(f, x, 1).doit(), Derivative(f, y, 1).doit()), dict=True)

for t in [delta, critical_point]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample1.py
∂ ⎛ 2        2    4    ⎞
──⎝x  - 2⋅x⋅y  - y  + y⎠
∂x                      

         2
2⋅x - 2⋅y 

  2                      
 ∂ ⎛ 2        2    4    ⎞
───⎝x  - 2⋅x⋅y  - y  + y⎠
  2                      
∂x                       

2

∂ ⎛ 2        2    4    ⎞
──⎝x  - 2⋅x⋅y  - y  + y⎠
∂y                      

            3    
-4⋅x⋅y - 4⋅y  + 1

  2                      
 ∂ ⎛ 2        2    4    ⎞
───⎝x  - 2⋅x⋅y  - y  + y⎠
  2                      
∂y                       

   ⎛       2⎞
-4⋅⎝x + 3⋅y ⎠

-4⋅y
         ⎛       2⎞
-4⋅y + 8⋅⎝x + 3⋅y ⎠

⎡                  ⎧     1   √3⋅ⅈ       1   √3⋅ⅈ⎫  ⎧     1   √3⋅ⅈ       1   √3
⎢{x: 1/4, y: 1/2}, ⎨x: - ─ - ────, y: - ─ + ────⎬, ⎨x: - ─ + ────, y: - ─ - ──
⎣                  ⎩     8    8         4    4  ⎭  ⎩     8    8         4    4

⋅ⅈ⎫⎤
──⎬⎥
  ⎭⎦

$

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="0.25">
<label for="y0">y0 = </label>
<input id="y0" type="number" value="0.5">

<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="sample1.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'),
    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) => x ** 2 - 2 * x * y ** 2 + y - y ** 4;

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])
        .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('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();








0 コメント:

コメントを投稿

関連コンテンツ