2017年11月4日土曜日

学習環境

ラング線形代数学(上)(S.ラング (著)、芹沢 正三 (翻訳)、ちくま学芸文庫)の2章(ベクトル空間)、3(基底)、練習問題5.を取り組んでみる。


    1. a·1+bt=0 d dt ( a·1+bt )=0 b=0 a=0

    2. at+b t 2 =0 d dt ( at+b t 2 )=0 a+2bt=0 d dt ( a+2bt )=0 2b=0 b=0 a=0

    3. at+b t 4 =0 d dt ( at+b t 4 )=0 a+4b t 3 =0 d dt ( a+4b t 3 )=0 12b t 2 =0 b=0 a=0

    4. a e t +bt=0 d dt ( a e t +bt )=0 a e t +b=0 a e t =0 a=0 b=0

    5. at e t +b e 2t =0 a e t +at e t +2b e 2t =0 e t ( a+at+2b e t )=0 a+at+2b e t =0 a+2b e t =0 2b e t =0 b=0 a=0

    6. asint+bcost=0 asint=bcost a 2 sin 2 t= b 2 cos 2 t a 2 sin 2 t= b 2 ( 1 sin 2 t ) ( a 2 + b 2 ) sin 2 t= b 2 ( a 2 + b 2 )2sintcos=0 ( a 2 + b 2 )sin2t=0 a 2 + b 2 =0 a=b=0

    7. at+bsint=0 a+bcost=0 bsint=0 b=0 a=0

    8. asint+bsin2t=0 asint+2bsintcost=0 sint( a+2bcost )=0 a+2bcost=0 2bsint=0 b=0 a=0

    9. asint+bcos3t=0 asint+b( cos2tcostsin2tsint )=0 asint+b( ( cos 2 t sin 2 t )cost2sintcostsint )=0 asint+bcost( cos 2 t sin 2 t2 sin 2 t )=0 asint+bcost( cos 2 t3 sin 2 t )=0 asint+bcost( cos 2 t3( 1 cos 2 t ) )=0 asint+b( 2 cos 2 t3 )cost=0 1cost1 2 cos 2 t30 a=0,b( 2 cos 2 t3 )=0 b=0

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, exp, sin, cos, solve, plot

print('5.')
t = symbols('t', real=True)
a, b = symbols('a b')

ps = [(1, t),
      (t, t**2),
      (t, t ** 4),
      (exp(t), t),
      (t * exp(t), exp(2 * t)),
      (sin(t), cos(t)),
      (t, sin(t)),
      (sin(t), sin(2 * t)),
      (cos(t), cos(3 * t))]

for i, (f, g) in enumerate(ps):
    c = chr(ord('a') + i)
    print(f'({c})')
    eq = a * f + b * g
    pprint(solve(eq, (a, b)))
    print()
    p = plot(f, g, ylim=(-5, 5), show=False, legend=True)
    for j, color in enumerate(['red', 'green']):
        p[j].line_color = color
    p.save(f'sample5_{c}.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample5.py
5.
(a)
{a: 0, b: 0}

(b)
{a: 0, b: 0}

(c)
{a: 0, b: 0}

(d)
⎡⎧         -t⎫⎤
⎢⎨a: -b⋅t⋅ℯ  ⎬⎥
⎣⎩           ⎭⎦

(e)
⎡⎧       t ⎫⎤
⎢⎪   -b⋅ℯ  ⎪⎥
⎢⎨a: ──────⎬⎥
⎢⎪     t   ⎪⎥
⎣⎩         ⎭⎦

(f)
⎡⎧    -b   ⎫⎤
⎢⎨a: ──────⎬⎥
⎣⎩   tan(t)⎭⎦

(g)
⎡⎧   -b⋅sin(t) ⎫⎤
⎢⎨a: ──────────⎬⎥
⎣⎩       t     ⎭⎦

(h)
[{a: -2⋅b⋅cos(t)}]

(i)
⎡⎧   -b⋅cos(3⋅t) ⎫⎤
⎢⎨a: ────────────⎬⎥
⎣⎩      cos(t)   ⎭⎦

$

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="a0">a = </label>
<input id="a0" type="number" value="2">
<label for="b0">b = </label>
<input id="b0" type="number" value="3">

<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,
    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_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_b0],
    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) => Math.sin(x),
    g = (x) => Math.cos(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),
        a0 = parseFloat(input_a0.value),
        b0 = parseFloat(input_b0.value);

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

    let points = [],
        lines = [],
        h = (x) => a0 * f(x) + b0 * g(x),
        fns = [[f, 'red'],
               [g, 'green'],
               [h, 'blue']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

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

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








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