2017年7月31日月曜日

学習環境

解析入門〈2〉(松坂 和夫(著)、岩波書店)の第8章(積分の計算)、8.2(定積分の計算)、問題4.を取り組んでみる。


  1. f( x )= a 0 + a 1 x+ a 2 x 2 + a 3 x 3 a b f( x )dx = [ a 0 x+ a 1 2 x 2 + a 2 3 x 3 + a 3 4 x 4 ] a b =( a 0 b+ a 1 2 b 2 + a 2 3 b 3 + a 3 4 b 4 )( a 0 a+ a 1 2 a 2 + a 2 3 a 3 + a 3 4 a 4 ) ba 6 ( f( a )+4f( a+b 2 )+f( b ) ) = ba 6 ( a 0 + a 1 a+ a 2 a 2 + a 3 a 3 +4( a 0 + a 1 ( a+b ) 2 + a 2 ( a 2 +2ab+ b 2 ) 4 + a 3 ( a 3 +3 a 2 b+3a b 2 + b 3 ) 8 )+ a 0 + a 1 b+ a 2 b 2 + a 3 b 3 ) = ba 6 ( 6 a 0 +( a+2( a+b )+b ) a 1 +( a 2 + a 2 +2ab+ b 2 + b 2 ) a 2 +( a 3 + a 3 +3 a 2 b+3a b 2 + b 3 2 + b 3 ) a 3 ) =( ba ) a 0 + b 2 a 2 2 a 1 + ( ba )( a 2 +ab+ b 2 ) 3 a 2 + ( ba )( a 3 +3 a 2 b+3a b 2 + b 3 ) 4 a 3 =( ba ) a 0 + b 2 a 2 2 a 1 + b 3 a 3 3 a 2 + b 4 a 4 4 a 3 =( b a 0 + b 2 2 a 1 + b 3 3 a 2 + b 4 4 a 3 )( a a 0 + a 2 2 a 1 + a 3 3 a 2 + a 4 4 a 3 ) a b f( x )dx = ba 6 ( f( a )+4f( a+b 2 )+f( b ) )

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Integral

print('4.')
x, a, b = symbols('x a b')
a0, a1, a2, a3 = symbols('a0 a1 a2 a3', integer=True, nonnegative=True)
f = a0 + a1 * x + a2 * x ** 2 + a3 * x ** 3

pprint(f)

I = Integral(f, (x, a, b))
pprint(I)
g = I.doit()
pprint(g)

h = (b - a) / 6 * (f.subs({x: a}) + 4 *
                   f.subs({x: (a + b) / 2}) + f.subs({x: b}))
pprint(h)

print(g.expand() == h.expand())

入出力結果(Terminal, IPython)

$ ./sample4.py
4.
                2       3
a₀ + a₁⋅x + a₂⋅x  + a₃⋅x 
b                               
⌠                               
⎮ ⎛                2       3⎞   
⎮ ⎝a₀ + a₁⋅x + a₂⋅x  + a₃⋅x ⎠ dx
⌡                               
a                               
   4       3       2                        2       3       4
  a ⋅a₃   a ⋅a₂   a ⋅a₁                 a₁⋅b    a₂⋅b    a₃⋅b 
- ───── - ───── - ───── - a⋅a₀ + a₀⋅b + ───── + ───── + ─────
    4       3       2                     2       3       4  
          ⎛                                                                   
⎛  a   b⎞ ⎜ 3       2                                ⎛a   b⎞       2        ⎛a
⎜- ─ + ─⎟⋅⎜a ⋅a₃ + a ⋅a₂ + a⋅a₁ + 6⋅a₀ + a₁⋅b + 4⋅a₁⋅⎜─ + ─⎟ + a₂⋅b  + 4⋅a₂⋅⎜─
⎝  6   6⎠ ⎝                                          ⎝2   2⎠                ⎝2

     2                       3⎞
   b⎞        3        ⎛a   b⎞ ⎟
 + ─⎟  + a₃⋅b  + 4⋅a₃⋅⎜─ + ─⎟ ⎟
   2⎠                 ⎝2   2⎠ ⎠
True
$

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="a01">a = </label>
<input id="a01" type="number" value="-1">
<label for="b01">b = </label>
<input id="b01" type="number" value="1">

<label for="a0">a0 = </label>
<input id="a0" type="number" value="1">
<label for="a1">a1 = </label>
<input id="a1" type="number" value="1">
<label for="a2">a2 = </label>
<input id="a2" type="number" value="1">
<label for="a3">a3 = </label>
<input id="a3" type="number" 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="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_a01 = document.querySelector('#a01'),
    input_b01 = document.querySelector('#b01'),
    input_a0 = document.querySelector('#a0'),
    input_a1 = document.querySelector('#a1'),
    input_a2 = document.querySelector('#a2'),
    input_a3 = document.querySelector('#a3'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_a1, input_a2, input_a3],
    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),
        a = parseFloat(input_a01.value),
        b = parseFloat(input_b01.value),
        a0 = parseFloat(input_a0.value),
        a1 = parseFloat(input_a1.value),
        a2 = parseFloat(input_a2.value),
        a3 = parseFloat(input_a3.value);
        
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        f = (x) => a0 + a1 * x + a2 * x ** 2 + a3 * x ** 3,
        g = (x) => (b - a) / 6 * (f(a) + 4 * f((a + b) / 2) + f(b)),
        lines = [[a, y1, a, y2, 'red'],
                 [b, y1, b, y2, 'red']],
        fns = [[f, 'green'],
               [g, 'orange']];

    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])
        .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('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();








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