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、指数関数、対数関数、偏微分、微分鎖律、グラディエント(gradient))

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

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3.を取り組んでみる。

1. $\begin{array}{l} g r a d f (x) \\ = g r a d {(\sum_{i = 1}^{n} x_{i}^{2})}^{\frac{k}{2}} \\ = (\frac{k}{2} {(\sum_{i = 1}^{n} x_{i}^{2})}^{\frac{k}{2} - 1} 2 x_{1}, \dots, \frac{k}{2} {(\sum_{i = 1}^{n} x_{i}^{2})}^{\frac{k}{2} - 1} 2 x_{n}) \\ = (k {|x|}^{k - 2} x_{1}, \dots, k {|x|}^{k - 2} x_{n}) \\ = k {|x|}^{k - 2} x \end{array}$
2. $\begin{array}{l} g r a d f (x) \\ = g r a d {(\sum_{i = 1}^{n} x_{i}^{2})}^{- \frac{k}{2}} \\ = (\begin{matrix} - \frac{k}{2} - 1 & - \frac{k}{2} - 1 \\ \frac{- k}{2} (\sum_{i = 1}^{n} x_{i}^{2}) 2 x_{1}, \dots, \frac{- k}{2} (\sum_{i = 1}^{n} x_{i}^{2}) & 2 x_{n} \end{matrix}) \end{array} = (- k {|x|}^{- k - 2} x_{1}, \dots, - k {|x|}^{- k - 2} - x_{n}) = - k {|x|}^{- k - 2} x$
3. $\begin{array}{l} \frac{\partial f}{\partial x_{i}} \\ = \frac{1}{|x|} \cdot \frac{1}{2 |x|} \cdot 2 x_{i} \\ = \frac{x_{i}}{{|x|}^{2}} \\ g r a d f (x) = \frac{x}{{|x|}^{2}} \end{array}$
4. $\begin{array}{l} \frac{\partial f}{\partial x_{i}} \\ = e^{- {|x|}^{2}} \cdot (- 2 |x|) \cdot \frac{1}{2 |x|} \cdot 2 x_{i} \\ = - 2 e^{- {|x|}^{2}} x_{i} \\ g r a d f (x) = - 2 e^{- {|x|}^{2}} x \end{array}$

コード(Emacs)

Python 3

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

n = 2
xs = symbols([f'x{i}' for i in range(1, n + 1)])
k = symbols('k', positive=True)
r = sqrt(sum([xi ** 2 for xi in xs]))
fs = [r ** k, 1 / r ** k, log(r), exp(-r ** 2)]

for i, f in enumerate(fs):
    print(f'({chr(ord("a") + i)})')
    gradf = [Derivative(f, xn, 1) for xn in xs]
    for t in [gradf, [D.doit() for D in gradf]]:
        pprint(t)
        print()
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample3.py
(a)
⎡   ⎛           k⎞     ⎛           k⎞⎤
⎢   ⎜           ─⎟     ⎜           ─⎟⎥
⎢   ⎜           2⎟     ⎜           2⎟⎥
⎢ ∂ ⎜⎛  2     2⎞ ⎟   ∂ ⎜⎛  2     2⎞ ⎟⎥
⎢───⎝⎝x₁  + x₂ ⎠ ⎠, ───⎝⎝x₁  + x₂ ⎠ ⎠⎥
⎣∂x₁                ∂x₂              ⎦

⎡                k                  k⎤
⎢                ─                  ─⎥
⎢                2                  2⎥
⎢     ⎛  2     2⎞        ⎛  2     2⎞ ⎥
⎢k⋅x₁⋅⎝x₁  + x₂ ⎠   k⋅x₂⋅⎝x₁  + x₂ ⎠ ⎥
⎢─────────────────, ─────────────────⎥
⎢      2     2            2     2    ⎥
⎣    x₁  + x₂           x₁  + x₂     ⎦


(b)
⎡   ⎛           -k ⎞     ⎛           -k ⎞⎤
⎢   ⎜           ───⎟     ⎜           ───⎟⎥
⎢   ⎜            2 ⎟     ⎜            2 ⎟⎥
⎢ ∂ ⎜⎛  2     2⎞   ⎟   ∂ ⎜⎛  2     2⎞   ⎟⎥
⎢───⎝⎝x₁  + x₂ ⎠   ⎠, ───⎝⎝x₁  + x₂ ⎠   ⎠⎥
⎣∂x₁                  ∂x₂                ⎦

⎡                 -k                     -k  ⎤
⎢                 ───                    ─── ⎥
⎢                  2                      2  ⎥
⎢      ⎛  2     2⎞            ⎛  2     2⎞    ⎥
⎢-k⋅x₁⋅⎝x₁  + x₂ ⎠      -k⋅x₂⋅⎝x₁  + x₂ ⎠    ⎥
⎢─────────────────────, ─────────────────────⎥
⎢        2     2                2     2      ⎥
⎣      x₁  + x₂               x₁  + x₂       ⎦


(c)
⎡   ⎛   ⎛   ___________⎞⎞     ⎛   ⎛   ___________⎞⎞⎤
⎢ ∂ ⎜   ⎜  ╱   2     2 ⎟⎟   ∂ ⎜   ⎜  ╱   2     2 ⎟⎟⎥
⎢───⎝log⎝╲╱  x₁  + x₂  ⎠⎠, ───⎝log⎝╲╱  x₁  + x₂  ⎠⎠⎥
⎣∂x₁                       ∂x₂                     ⎦

⎡    x₁         x₂   ⎤
⎢─────────, ─────────⎥
⎢  2     2    2     2⎥
⎣x₁  + x₂   x₁  + x₂ ⎦


(d)
⎡   ⎛     2     2⎞     ⎛     2     2⎞⎤
⎢ ∂ ⎜ - x₁  - x₂ ⎟   ∂ ⎜ - x₁  - x₂ ⎟⎥
⎢───⎝ℯ           ⎠, ───⎝ℯ           ⎠⎥
⎣∂x₁                ∂x₂              ⎦

⎡           2     2             2     2⎤
⎢       - x₁  - x₂          - x₁  - x₂ ⎥
⎣-2⋅x₁⋅ℯ           , -2⋅x₂⋅ℯ           ⎦


$

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="sample3.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 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),
        k0 = parseInt(input_k0.value, 10);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [],
        fa = (x) => Math.abs(x) ** k0,
        fb = (x) => 1 / Math.abs(x) ** k0,
        fc = (x) => Math.log(Math.abs(x)),
        fd = (x) => Math.exp(- (Math.abs(x) ** 2)),
        fns = [[fa, 'red'],
               [fb, 'green'],
               [fc, 'blue'],
               [fd, 'brown']];

    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 =

Kamimura's blog

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2018年1月10日水曜日

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、指数関数、対数関数、偏微分、微分鎖律、グラディエント(gradient))

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