## 2018年1月11日木曜日

### 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、偏微分、グラディエント(gradient)、一般化)

1. $\begin{array}{}\frac{\partial f}{\partial {x}_{i}}\\ =g\text{'}\left(r\right)\frac{1}{2}{\left(\sum _{i=1}^{n}{x}_{i}^{2}\right)}^{-\frac{1}{2}}·2{x}_{i}\\ =\frac{g\text{'}\left(r\right)}{\sqrt{\sum _{i=1}^{n}{x}_{i}^{2}}}{x}_{i}\\ =\frac{g\text{'}\left(r\right)}{\left|x\right|}{x}_{i}\end{array}$

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]

pprint(t)
print()


$./sample4.py ⎡ ⎛ ⎛ ___________⎞⎞ ⎛ ⎛ ___________⎞⎞⎤ ⎢ ∂ ⎜ ⎜ ╱ 2 2 ⎟⎟ ∂ ⎜ ⎜ ╱ 2 2 ⎟⎟⎥ ⎢───⎝g⎝╲╱ x₁ + x₂ ⎠⎠, ───⎝g⎝╲╱ x₁ + x₂ ⎠⎠⎥ ⎣∂x₁ ∂x₂ ⎦ ⎡ ⎛ d ⎞│ ___________ ⎛ d ⎞│ ___________⎤ ⎢x₁⋅⎜───(g(ξ₁))⎟│ ╱ 2 2 x₂⋅⎜───(g(ξ₁))⎟│ ╱ 2 2 ⎥ ⎢ ⎝dξ₁ ⎠│ξ₁=╲╱ x₁ + x₂ ⎝dξ₁ ⎠│ξ₁=╲╱ x₁ + x₂ ⎥ ⎢─────────────────────────────────, ─────────────────────────────────⎥ ⎢ ___________ ___________ ⎥ ⎢ ╱ 2 2 ╱ 2 2 ⎥ ⎣ ╲╱ x₁ + x₂ ╲╱ x₁ + x₂ ⎦$


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,
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'],

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();