2017年6月9日金曜日

学習環境

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第4章(正弦と余弦)、3(加法公式)、練習問題7、8.を取り組んでみる。


  1. cosmxcosnx+sinmxsinnxcosmxcosnx+sinmxsinnx 2 =sinmxsinnx sinmxcosnx+cosmxsinnx+sinmxcosnxcosmxsinnx 2 =sinmxcosnx cosmxcosnxsinmxsinnx+cosmxcosnx+sinmxsinnx 2 =cosmxcosnx

  2. cos2x= cos 2 x sin 2 x cos2x=12 sin 2 x sin 2 x= 1cos2x 2 sin 3 x= 1 2 sinx 1 2 cos2xsinx = 1 2 sinx 1 2·2 2cos2xsinx sin( 2x+x )=sin2xcosx+cos2xsinx sin( 2xx )=sin2xcosxcos2xsinx sin3xsinx=2cos2xsinx sin 3 x= 1 2 sinx 1 4 ( sin3xsinx ) sin 3 x= 1 4 sinx 1 2 sin3x sin k+1 x= sin k xsinx a n sinxcosnx = 1 2 ( sin( 1+n )x+sin( 1n )x ) = 1 2 ( 1+n )x 1 2 sin( n1 )x b n sinnxsinx = 1 2 ( cos( n1 )xcos( n+1 )x ) = 1 2 cos( n1 )x 1 2 cos( n+1 )x cos k+1 x= cos k xcosx a n sinnxcosx= a n 1 2 ( sin( n+1 )x+sin( n1 )x ) a n cosnxcosx= a n 1 2 ( cos( n+1 )x+cos( n1 )x )

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, sin, cos, plot, solve, Rational
import random
x = symbols('x')
n, m = symbols('n m', integer=True)
eqs = [
    sin(m * x) * sin(n * x) - Rational(1, 2) *
    (cos((m - n) * x) - cos((m + n) * x)),
    sin(m * x) * cos(n * x) - Rational(1, 2) *
    (sin((m + n) * x) + sin((m - n) * x)),
    cos(m * x) * cos(n * x) - Rational(1, 2) *
    (cos((m + n) * x) + cos((m - n) * x))
]

for eq in eqs:
    pprint(eq == 0)

m0 = random.randrange(1, 10)
n0 = random.randrange(1, 10)
p = plot(*map(lambda f: f.subs({m: m0, n: n0}), eqs), show=False, legend=True)
for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color

p.save('sample7.svg')

入出力結果(Terminal, IPython)

$ ./sample7.py
False
False
False
$

0になって欲しいところが、SymPyの問題で誤差で上手くいってないかも?

JavaScriptではグラフ上で0になった。(計算結果の出力では誤差有り。)

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="-2">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="2">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-2">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="2">
<br>
<label for="n0">n = </label>
<input id="n0" type="number" min="0" step="1" value="5">
<label for="m0">m = </label>
<input id="m0" type="number" min="0" step="1" value="10">

<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="sample7.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_n0 = document.querySelector('#n0'),
    input_m0 = document.querySelector('#m0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_n0, input_m0],    
    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),
        n = parseInt(input_n0.value, 10),
        m = parseInt(input_m0.value, 10);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;n
    }

    let points = [],
        lines = [],
        f1 = (x) => Math.sin(m * x) * Math.sin(n * x),
        f2 = (x) => 1 / 2 * (Math.cos((m - n) * x) - Math.cos((m + n) * x)),
        f = (x) => f1(x) - f2(x),
        g1 = (x) => Math.sin(m * x) * Math.cos(n * x),
        g2 = (x) => 1 / 2 * (Math.sin((m + n) * x) + Math.sin((m - n) * x)),
        g = (x) => g1(x) - g2(x),
        h1 = (x) => Math.cos(m * x) * Math.cos(n * x),
        h2 = (x) => 1 / 2 * (Math.cos((m + n) * x) + Math.cos((m - n) * x)),
        h = (x) => h1(x) - h2(x);
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f1(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t1 = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        let y = g1(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t2 = points.length;

    for (let x = x1; x <= x2; x += dx) {
        let y = h1(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t3 = points.length;

    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    for (let x = x1; x <= x2; x += dx) {
        let y = g(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    for (let x = x1; x <= x2; x += dx) {
        let y = h(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    
    
    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('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', '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, i) =>
              i < t1 ? 'red' :
              i < t2 ? 'green' :
              i < t3 ? 'blue' : 'brown');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    [f, g, h].forEach((fn) => p(fn(Math.random())));
};

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








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