## 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))

1. $\begin{array}{}gradf\left(x\right)\\ =grad{\left(\sum _{i=1}^{n}{x}_{i}^{2}\right)}^{\frac{k}{2}}\\ =\left(\frac{k}{2}{\left(\sum _{i=1}^{n}{x}_{i}^{2}\right)}^{\frac{k}{2}-1}2{x}_{1},\dots ,\frac{k}{2}{\left(\sum _{i=1}^{n}{x}_{i}^{2}\right)}^{\frac{k}{2}-1}2{x}_{n}\right)\\ =\left(k{\left|x\right|}^{k-2}{x}_{1},\dots ,k{\left|x\right|}^{k-2}{x}_{n}\right)\\ =k{\left|x\right|}^{k-2}x\end{array}$

2. $\begin{array}{}gradf\left(x\right)\\ =grad{\left(\sum _{i=1}^{n}{x}_{i}^{2}\right)}^{-\frac{k}{2}}\\ =\left(\begin{array}{cc}-\frac{k}{2}-1& -\frac{k}{2}-1\\ \frac{-k}{2}\left(\sum _{i=1}^{n}{x}_{i}^{2}\right)2{x}_{1},\dots ,\frac{-k}{2}\left(\sum _{i=1}^{n}{x}_{i}^{2}\right)& 2{x}_{n}\end{array}\right)\end{array}=\left(-k{\left|x\right|}^{-k-2}{x}_{1},\dots ,-k{\left|x\right|}^{-k-2}-{x}_{n}\right)\\ =-k{\left|x\right|}^{-k-2}x$

3. $\begin{array}{}\frac{\partial f}{\partial {x}_{i}}\\ =\frac{1}{\left|x\right|}·\frac{1}{2\left|x\right|}·2{x}_{i}\\ =\frac{{x}_{i}}{{\left|x\right|}^{2}}\\ gradf\left(x\right)=\frac{x}{{\left|x\right|}^{2}}\end{array}$

4. $\begin{array}{}\frac{\partial f}{\partial {x}_{i}}\\ ={e}^{-{\left|x\right|}^{2}}·\left(-2\left|x\right|\right)·\frac{1}{2\left|x\right|}·2{x}_{i}\\ =-2{e}^{-{\left|x\right|}^{2}}{x}_{i}\\ gradf\left(x\right)=-2{e}^{-{\left|x\right|}^{2}}x\end{array}$

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]
pprint(t)
print()
print()


$./sample3.py (a) ⎡ ⎛ k⎞ ⎛ k⎞⎤ ⎢ ⎜ ─⎟ ⎜ ─⎟⎥ ⎢ ⎜ 2⎟ ⎜ 2⎟⎥ ⎢ ∂ ⎜⎛ 2 2⎞ ⎟ ∂ ⎜⎛ 2 2⎞ ⎟⎥ ⎢───⎝⎝x₁ + x₂ ⎠ ⎠, ───⎝⎝x₁ + x₂ ⎠ ⎠⎥ ⎣∂x₁ ∂x₂ ⎦ ⎡ k k⎤ ⎢ ─ ─⎥ ⎢ 2 2⎥ ⎢ ⎛ 2 2⎞ ⎛ 2 2⎞ ⎥ ⎢k⋅x₁⋅⎝x₁ + x₂ ⎠ k⋅x₂⋅⎝x₁ + x₂ ⎠ ⎥ ⎢─────────────────, ─────────────────⎥ ⎢ 2 2 2 2 ⎥ ⎣ x₁ + x₂ x₁ + x₂ ⎦ (b) ⎡ ⎛ -k ⎞ ⎛ -k ⎞⎤ ⎢ ⎜ ───⎟ ⎜ ───⎟⎥ ⎢ ⎜ 2 ⎟ ⎜ 2 ⎟⎥ ⎢ ∂ ⎜⎛ 2 2⎞ ⎟ ∂ ⎜⎛ 2 2⎞ ⎟⎥ ⎢───⎝⎝x₁ + x₂ ⎠ ⎠, ───⎝⎝x₁ + x₂ ⎠ ⎠⎥ ⎣∂x₁ ∂x₂ ⎦ ⎡ -k -k ⎤ ⎢ ─── ─── ⎥ ⎢ 2 2 ⎥ ⎢ ⎛ 2 2⎞ ⎛ 2 2⎞ ⎥ ⎢-k⋅x₁⋅⎝x₁ + x₂ ⎠ -k⋅x₂⋅⎝x₁ + x₂ ⎠ ⎥ ⎢─────────────────────, ─────────────────────⎥ ⎢ 2 2 2 2 ⎥ ⎣ x₁ + x₂ x₁ + x₂ ⎦ (c) ⎡ ⎛ ⎛ ___________⎞⎞ ⎛ ⎛ ___________⎞⎞⎤ ⎢ ∂ ⎜ ⎜ ╱ 2 2 ⎟⎟ ∂ ⎜ ⎜ ╱ 2 2 ⎟⎟⎥ ⎢───⎝log⎝╲╱ x₁ + x₂ ⎠⎠, ───⎝log⎝╲╱ x₁ + x₂ ⎠⎠⎥ ⎣∂x₁ ∂x₂ ⎦ ⎡ x₁ x₂ ⎤ ⎢─────────, ─────────⎥ ⎢ 2 2 2 2⎥ ⎣x₁ + x₂ x₁ + x₂ ⎦ (d) ⎡ ⎛ 2 2⎞ ⎛ 2 2⎞⎤ ⎢ ∂ ⎜ - x₁ - x₂ ⎟ ∂ ⎜ - x₁ - x₂ ⎟⎥ ⎢───⎝ℯ ⎠, ───⎝ℯ ⎠⎥ ⎣∂x₁ ∂x₂ ⎦ ⎡ 2 2 2 2⎤ ⎢ - x₁ - x₂ - x₁ - x₂ ⎥ ⎣-2⋅x₁⋅ℯ , -2⋅x₂⋅ℯ ⎦$


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,
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])

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();