## 2018年1月18日木曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - 積分法 - 上方和および下方和(逆数、区間、n分割)

1. 上方和。

$\begin{array}{}\frac{1}{2}=\frac{1}{1+\frac{k}{n}}\\ \frac{1}{2}=\frac{n}{n+k}\\ 2n=n+k\\ n=k\\ \frac{1}{n}\left(\frac{1}{1+\frac{0}{n}}+\frac{1}{1+\frac{1}{n}}+\dots +\frac{1}{1+\frac{n-1}{n}}\right)\\ =\frac{1}{n}\left(\frac{n}{n+0}+\frac{n}{n+1}+\dots +\frac{n}{n+\left(n-1\right)}\right)\\ =\sum _{i=1}^{n-1}\frac{1}{n+i}\end{array}$

下方和。

$\sum _{i=1}^{n}\frac{1}{n+i}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, summation

i, n = symbols('i, n', integer=True)
x = symbols('x')
f = 1 / x
u = 1 / n * summation(f.subs({x: 1 + i / n}), (i, 0, n - 1))
l = 1 / n * summation(f.subs({x: 1 + i / n}), (i, 1, n))

for t in [u, l]:
pprint(t.factor())
print()


$./sample5.py n - 1 ____ ╲ ╲ 1 ╲ ───── ╱ i + n ╱ ╱ ‾‾‾‾ i = 0 n ____ ╲ ╲ 1 ╲ ───── ╱ i + n ╱ ╱ ‾‾‾‾ i = 1$


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="0.001" 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="n0">n = </label>
<input id="n0" type="number" min="1" step="1" value="2">

<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_n0 = document.querySelector('#n0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_n0],
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 f = (x) => 1 / x,
u = (n) => range(0, n).map((i) => 1 / (n + i)).reduce((x, y) => x + y),
l = (n) => range(1, n + 1).map((i) => 1 / (n + i)).reduce((x, y) => x + y);

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),
n0 = parseInt(input_n0.value, 10);

if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}

let points = [],
lines = [],
fns = [[f, 'green']],
fns1 = [],
fns2 = [];

for (let i = 1; i <= 2; i += 1 / n0) {
lines.push([i, y1, i, y2, 'red']);
}
fns
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = f(x);

points.push([x, y, color]);
}
});

fns1
.forEach((o) => {
let [f, color] = o;

lines.push([x1, f(x1), x2, f(x2), color]);
});

fns2
.forEach((o) => {
let [f, color] = o;
for (let x = x1; x <= x2; x += dx0) {
let g = f(x);
lines.push([x1, g(x1), x2, g(x2), 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('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.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.append('g')
.attr('transform', translate(0, ${height - padding})) .call(xaxis); svg.append('g') .attr('transform', translate(${padding}, 0))
.call(yaxis);

[fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
[u, l].forEach((s) => p(s(n0)));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();