2017年6月13日火曜日

学習環境

解析入門〈2〉(松坂 和夫(著)、岩波書店)の第7章(積分法)、7.1(リーマン積分)、問題1.を取り組んでみる。


  1. x i =a+ i( ba ) n P=( x 0 , x 1 ,, x n ) lim n ba n =0

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Limit, summation, Integral, Function, S
a, b, i, n, x = symbols('a b i n x')
f = Function('f')
l = Limit(summation(f(a + i * (b - a) / n), (i, 1, n))
          * (b - a) / n, n, S.Infinity)
pprint(l)
pprint(l.doit())

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

print(l.doit() == expr.doit())

入出力結果(Terminal, IPython)

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

from sympy import pprint, symbols, Limit, summation, Integral, Function, S
a, b, i, n, x = symbols('a b i n x')
f = Function('f')
l = Limit(summation(f(a + i * (b - a) / n), (i, 1, n))
          * (b - a) / n, n, S.Infinity)
pprint(l)
pprint(l.doit())

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

print(l.doit() == expr.doit())

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="n0">n = </label>
<input id="n0" min="1" step="1" type="number" value="10">
<label for="a0">a = </label>
<input id="a0" type="number" value="-2">
<label for="b0">b = </label>
<input id="b0" type="number" 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="sample1.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_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_n0, 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);

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),
        a = parseFloat(input_a0.value),
        b = parseFloat(input_b0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [];

    [[f, 'green']]
        .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 dx0 = (b - a) / n;
    for (let i = 0; i < n; i += 1) {
        let x = a + i * dx0,
            y = f(x);
        
        lines.push([x, y, x + dx0, y, 'red'],
                   [x, 0, x, y, 'red'],
                   [x + dx0, 0, x + dx0, y, 'red']);
    }
    for (let i = 0; i < n; i += 1) {
        let x = a + i * dx0,
            y = f(x + dx0);
        
        lines.push([x, y, x + dx0, y, 'blue'],
                   [x, 0, x, y, 'blue'],
                   [x + dx0, 0, x + dx0, y, 'blue']);
    }
    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);

};

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








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