数学 - Python - JavaScript - 解析学 - 多変数の関数 - 微分可能性と勾配ベクトル(絶対値、偏微分、グラディエント(gradient)、一般化)

解析入門〈3〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

解析入門〈3〉(松坂 和夫(著)、岩波書店)の第14章(多変数の関数)、14.1(微分可能性と勾配ベクトル)、問題4.を取り組んでみる。

4. 
   
   $\begin{array}{}\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x_i}\\ =g\text{'}\left(r\right)\frac{1}{2}{\left(\underset{i=1}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}{x}_i^2\right)}^{-\frac{1}{2}}·\; <!--\; middle\; dot\; -->2x_i\\ =\frac{g\text{'}\left(r\right)}{\sqrt{\underset{i=1}{\overset{n}{\sum \; <!--\; n-ary\; summation\; -->}}{x}_i^2}}{x}_i\\ =\frac{g\text{'}\left(r\right)}{\left|x\right|}{x}_i\end{array}$

   よって、

   $gradf\left(x\right)=\frac{g\text{'}\left(r\right)}{\left|x\right|}x$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sqrt, log, exp, Derivative, Function, Matrix

n = 2
xs = symbols([f'x{i}' for i in range(1, n + 1)])
g = Function('g')(sqrt(sum([x ** 2 for x in xs])))

grad = [Derivative(g, x, 1) for x in xs]

for t in [grad, [f.doit() for f in grad]]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample4.py
⎡   ⎛ ⎛   ___________⎞⎞     ⎛ ⎛   ___________⎞⎞⎤
⎢ ∂ ⎜ ⎜  ╱   2     2 ⎟⎟   ∂ ⎜ ⎜  ╱   2     2 ⎟⎟⎥
⎢───⎝g⎝╲╱  x₁  + x₂  ⎠⎠, ───⎝g⎝╲╱  x₁  + x₂  ⎠⎠⎥
⎣∂x₁                     ∂x₂                   ⎦

⎡   ⎛ d        ⎞│      ___________     ⎛ d        ⎞│      ___________⎤
⎢x₁⋅⎜───(g(ξ₁))⎟│     ╱   2     2   x₂⋅⎜───(g(ξ₁))⎟│     ╱   2     2 ⎥
⎢   ⎝dξ₁       ⎠│ξ₁=╲╱  x₁  + x₂       ⎝dξ₁       ⎠│ξ₁=╲╱  x₁  + x₂  ⎥
⎢─────────────────────────────────, ─────────────────────────────────⎥
⎢             ___________                        ___________         ⎥
⎢            ╱   2     2                        ╱   2     2          ⎥
⎣          ╲╱  x₁  + x₂                       ╲╱  x₁  + x₂           ⎦

$

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="0.001">
<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">
<br>

<label for="k0">k0 = </label>
<input id="k0" 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="sample4.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_k0 = document.querySelector('#k0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_k0],
    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 g = (x) => x,
    f = (x) => g(Math.abs(x)),
    gradf = (x) => 2 / Math.abs(x) * x;

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 = [],
        fns = [[g, 'red'],
               [f, 'green'],
               [gradf, 'blue']];

    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();

r = dx =
x1 = x2 =
y1 = y2 =
k0 =