2017年5月22日月曜日

学習環境

解析入門〈1〉(松坂 和夫(著)、岩波書店)の第5章(各種の初等関数)、5.4(三角関数(続き)、逆三角関数)、問題10.を取り組んでみる。


  1. d dx f 1 ( x )=1= g 1 ( x ) d dx f n+1 ( x )= g n ( x )+ d dx ( 1 ) n x 2n+1 ( 2n1 )! = g n ( x )+ ( 1 ) n x 2n ( 2n )! = g n+1 d dx f n ( x )= g n ( x ) d dx g 2 ( x )=x= f 21 ( x ) d dx g n+1 ( x )= f n1 + d dx ( 1 ) n x 2n ( 2n )! = f n1 ( 1 ) n1 x 2n1 ( 2n1 )! = f n ( x ) d dx g n ( x )= f n1 ( x ) d dx ( f 2n1 ( x )sinx )= g 2n1 ( x )cosx d dx ( sinx f 2n ( x ) )=cosx g 2n ( x ) d dx ( g 2n+1 ( x )cosx )= f 2n ( x )+sinx d dx ( cosx g 2n ( x ) )=sinx+ f 2n1 ( x ) 0<x<2π g 1 ( x )=1>cosx sinx< f 1 ( x ) g 2 ( x )<cosx f 2 ( x )<sinx cosx< g 3 ( x ) g 2 ( x )<cosx< g 1 ( x ) f 2 ( x )<sinx< f 1 ( x ) g 2n ( x )<cosx< g 2n1 ( x ) f 2n ( x )<sinx< f 2n1 ( x ) g 2n+2 ( x )<cosx< g 2n+1 ( x ) f 2n+2 ( x )<sinx< f 2n+1 ( x ) f 2n ( x )<sinx< f 2n1 ( x ) g 2n ( x )<cosx< g 2n1 ( x )

コード(Emacs)

Python 3

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

from sympy import Symbol, symbols, sin, cos, factorial, summation, solve, pprint
import random

x = Symbol('x', positive=True)
n, i = symbols('n i', integer=True, positive=True)


expr1 = summation((-1) ** (i - 1) * x ** (2 * i - 1) / factorial(2 * i - 1),
                  (i, 1, n))


expr2 = summation((-1) ** (i - 1) * x ** (2 * i - 2) / factorial(2 * i - 2),
                  (i, 1, n))
pprint(expr1)
pprint(expr2)

try:
    print(expr1.subs({n: 2 * n}) < sin(x) < expr1.subs({n: 2 * n - 1}))
    print(expr2.subs({n: 2 * n}) < cos(x) < expr2.subs({n: 2 * n - 1}))
except Exception as err:
    print(type(err), err)

入出力結果(Terminal, IPython)

$ ./sample10.py
                        ⎛               │   2 ⎞         
    n + 1  2⋅n + 2  ┌─  ⎜      1        │ -x  ⎟         
(-1)     ⋅x       ⋅ ├─  ⎜               │ ────⎟         
                   1╵ 2 ⎝n + 1, n + 3/2 │  4  ⎠         
─────────────────────────────────────────────── + sin(x)
                  x⋅(2⋅n + 1)!                          
                        ⎛               │   2 ⎞         
    n + 1  2⋅n + 2  ┌─  ⎜      1        │ -x  ⎟         
(-1)     ⋅x       ⋅ ├─  ⎜               │ ────⎟         
                   1╵ 2 ⎝n + 1/2, n + 1 │  4  ⎠         
─────────────────────────────────────────────── + cos(x)
                    2                                   
                   x ⋅(2⋅n)!                            
<class 'TypeError'> cannot determine truth value of Relational
$

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="0.01">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-1.5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1.5">
<br>
<label for="n0">n = </label>
<input id="n0" type="number" min="1" step="1" 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="sample10.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_n = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_n],
    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 factorial = (n) => range(1, n + 1).reduce((x, y) => x * y, 1);

let f = (n, x) =>
    range(1, n + 1)
    .map((i) => (-1) ** (i - 1) * x ** (2 * i - 1) / factorial(2 * i - 1))
    .reduce((x, y) => x + y, 0),
    g = (n, x) =>
    range(1, n + 1)
    .map((i) => (-1) ** (i - 1) * x ** (2 * i - 2) / factorial(2 * i - 2))
    .reduce((x, y) => x + y, 0);
    

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_n.value, 10);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        f0 = (x) => f(2 * n, x),
        f1 = (x) => f(2 * n - 1, x),
        g0 = (x) => g(2 * n, x),
        g1 = (x) => g(2 * n - 1, x);
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f0(x);
        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t1 = points.length;

    for (let x = x1; x <= x2; x += dx) {
        let y = f1(x);
        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t2 = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        let y = g0(x);
        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t3 = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        let y = g1(x);
        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t4 = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        let y = Math.sin(x);
        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t5 = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        let y = Math.cos(x);
        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);

    let ys = points.map((a) => a[1]);
    y1 = Math.min(y1, ...ys);
    y2 = Math.max(y2, ...ys);
    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, i) =>
              i < t1 ? 'red':
              i < t2 ? 'green' :
              i < t3 ? 'blue' :
              i < t4 ? 'orange' :
              i < t5 ? 'brown': 'purple');

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]])
        .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.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();








0 コメント:

コメントを投稿

Comments on Google+: