2018年6月14日木曜日

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7章(スカラー積と直交性)、2(正値スカラー積)、練習問題4.を取り組んでみる。


  1. 直交基底。

    t t 2 - t 2 , t t , t t = t 2 - t 0 1 t 3 dt 0 1 t 2 dt = t 2 - t · 1 4 1 3 = t 2 - 3 4 t

    正規直交基底。

    t < t , t > = t 0 1 t 2 dt = t 1 3 = 3 t t 2 - 3 4 t t 2 - 3 4 t , t 2 - 3 4 t = t 2 - 3 4 t 0 1 t 4 - 3 2 t 3 + 9 16 t 2 dt = t 2 - 3 4 t 1 5 - 3 8 + 3 16 = t 2 - 3 4 t 16 - 30 + 15 80 = 80 t 2 - 3 4 t = 4 5 t 2 - 3 4 t = 4 5 t 2 - 3 5 t

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sqrt, Integral, plot

print('4.')
t = symbols('t')


def sp(f, g):
    return Integral(f * g, (t, 0, 1)).doit()


def norm(f):
    return sqrt(sp(f, f))


f = t
g = t ** 2

v1 = f
v2 = g - sp(g, v1) / sp(v1, v1) * v1

for v in [v1, v2]:
    for s in [v, (v / norm(v)).expand()]:
        pprint(s)
        print()
    print()
p = plot(f, g, f * g, ylim=(-10, 10), legend=True, show=False)
colors = ['red', 'green', 'blue']
for i, color in enumerate(colors):
    p[i].line_color = color

p.save('sample4.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample4.py
4.
t

√3⋅t


 2   3⋅t
t  - ───
      4 

      2         
4⋅√5⋅t  - 3⋅√5⋅t


$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.005">
<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">

<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,
    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'),
    input_a1 = document.querySelector('#a1'),
    input_b1 = document.querySelector('#b1'),
    input_a2 = document.querySelector('#a2'),
    input_b2 = document.querySelector('#b2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let f = (x) => x,
    g = (x) => x ** 2,
    h = (x) => f(x) * g(x),
    fns = [[f, 'red'],
           [g, 'green'],
           [h, 'blue']];

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 = [[0, y1, 0, y2, 'pink'],
                 [1, y1, 1, y2, 'skyblue']];

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

            for (let x0 = x1; x0 <= x2; x0 += dx) {
                points.push([x0, f(x0), 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);

    p(fns.join('\n'));
};

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







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