## 2018年1月12日金曜日

### 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(和、スカラー倍、積、逆数、グラディエント(gradient))

1. $\begin{array}{}\frac{\partial }{\partial {x}_{i}}\left(f+g\right)=\frac{\partial f}{\partial {x}_{i}}+\frac{\partial g}{\partial {x}_{i}}\\ grad\left(f+g\right)=gradf+gra\mathrm{dg}\end{array}$
$\begin{array}{}\frac{\partial }{\partial {x}_{i}}\left(cf\right)=c\frac{\partial f}{\partial {x}_{i}}\\ grad\left(cf\right)=\mathrm{cg}radf\end{array}$
$\begin{array}{}\frac{\partial }{\partial {x}_{i}}\left(fg\right)=\frac{\partial f}{\partial {x}_{i}}g+f·\frac{\partial g}{\partial {x}_{i}}\\ grad\left(fg\right)=g\left(gradf\right)+f\left(gra\mathrm{dg}\right)=f\left(gra\mathrm{dg}\right)+g\left(gradf\right)\end{array}$
$\begin{array}{}\frac{\partial }{\partial {x}_{i}}\left(\frac{1}{f}\right)=-\frac{1}{{f}^{2}}·\frac{\partial f}{\partial {x}_{i}}\\ grad\left(\frac{1}{f}\right)=-\frac{1}{{f}^{2}}gradf\end{array}$

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)])
f = Function('f')(sum([x for x in xs]))
g = Function('g')(sum([2 * x for x in xs]))
c = symbols('c')
grad1 = [Derivative(f + g, x, 1) for x in xs]
grad2 = [Derivative(c * f, x, 1) for x in xs]
grad3 = [Derivative(f * g, x, 1) for x in xs]
grad4 = [Derivative(1 / f, x, 1) for x in xs]

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


$./sample5.py ⎡ ∂ ∂ ⎤ ⎢───(f(x₁ + x₂) + g(2⋅x₁ + 2⋅x₂)), ───(f(x₁ + x₂) + g(2⋅x₁ + 2⋅x₂))⎥ ⎣∂x₁ ∂x₂ ⎦ ⎡⎛ d ⎞│ ⎛ d ⎞│ ⎛ d ⎞│ ⎢⎜───(f(ξ₁))⎟│ + 2⋅⎜───(g(ξ₁))⎟│ , ⎜───(f(ξ₁))⎟│ ⎣⎝dξ₁ ⎠│ξ₁=x₁ + x₂ ⎝dξ₁ ⎠│ξ₁=2⋅x₁ + 2⋅x₂ ⎝dξ₁ ⎠│ξ₁=x₁ + ⎛ d ⎞│ ⎤ + 2⋅⎜───(g(ξ₁))⎟│ ⎥ x₂ ⎝dξ₁ ⎠│ξ₁=2⋅x₁ + 2⋅x₂⎦ ⎡ ∂ ∂ ⎤ ⎢───(c⋅f(x₁ + x₂)), ───(c⋅f(x₁ + x₂))⎥ ⎣∂x₁ ∂x₂ ⎦ ⎡ ⎛ d ⎞│ ⎛ d ⎞│ ⎤ ⎢c⋅⎜───(f(ξ₁))⎟│ , c⋅⎜───(f(ξ₁))⎟│ ⎥ ⎣ ⎝dξ₁ ⎠│ξ₁=x₁ + x₂ ⎝dξ₁ ⎠│ξ₁=x₁ + x₂⎦ ⎡ ∂ ∂ ⎤ ⎢───(f(x₁ + x₂)⋅g(2⋅x₁ + 2⋅x₂)), ───(f(x₁ + x₂)⋅g(2⋅x₁ + 2⋅x₂))⎥ ⎣∂x₁ ∂x₂ ⎦ ⎡ ⎛ d ⎞│ ⎛ d ⎞│ ⎢2⋅f(x₁ + x₂)⋅⎜───(g(ξ₁))⎟│ + g(2⋅x₁ + 2⋅x₂)⋅⎜───(f(ξ₁))⎟│ ⎣ ⎝dξ₁ ⎠│ξ₁=2⋅x₁ + 2⋅x₂ ⎝dξ₁ ⎠│ξ₁=x₁ ⎛ d ⎞│ ⎛ d ⎞│ , 2⋅f(x₁ + x₂)⋅⎜───(g(ξ₁))⎟│ + g(2⋅x₁ + 2⋅x₂)⋅⎜───(f(ξ₁))⎟│ + x₂ ⎝dξ₁ ⎠│ξ₁=2⋅x₁ + 2⋅x₂ ⎝dξ₁ ⎠│ξ ⎤ ⎥ ₁=x₁ + x₂⎦ ⎡ ∂ ∂ ⎤ ⎢───(f(x₁ + x₂)⋅g(2⋅x₁ + 2⋅x₂)), ───(f(x₁ + x₂)⋅g(2⋅x₁ + 2⋅x₂))⎥ ⎣∂x₁ ∂x₂ ⎦ ⎡ ⎛ d ⎞│ ⎛ d ⎞│ ⎢2⋅f(x₁ + x₂)⋅⎜───(g(ξ₁))⎟│ + g(2⋅x₁ + 2⋅x₂)⋅⎜───(f(ξ₁))⎟│ ⎣ ⎝dξ₁ ⎠│ξ₁=2⋅x₁ + 2⋅x₂ ⎝dξ₁ ⎠│ξ₁=x₁ ⎛ d ⎞│ ⎛ d ⎞│ , 2⋅f(x₁ + x₂)⋅⎜───(g(ξ₁))⎟│ + g(2⋅x₁ + 2⋅x₂)⋅⎜───(f(ξ₁))⎟│ + x₂ ⎝dξ₁ ⎠│ξ₁=2⋅x₁ + 2⋅x₂ ⎝dξ₁ ⎠│ξ ⎤ ⎥ ₁=x₁ + x₂⎦$


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="sample5.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 c = 5,
f = (x) => x,
g = (x) => 2 * x,
h1 = (x) => f(x) + g(x),
h2 = (x) => c * f(x),
h3 = (x) => f(x) * g(x),
h4 = (x) => 1 / f(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 = [[f, 'red'],
[g, 'green'],
[h1, 'blue'],
[h2, 'brown'],
[h3, 'orange'],
[h4, 'purple']];

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();