2018年2月14日水曜日

学習環境

解析入門〈3〉(松坂 和夫(著)、岩波書店)の第14章(多変数の関数)、14.2(高次偏導関数、テイラーの定理)、問題10-(c).を取り組んでみる。


    1. f x , y = 1 + x 1 + y - 1 2 D 1 f x , y = - 1 2 1 + x 1 + y - 3 2 1 + y D 2 f x , y = - 1 2 1 + x 1 + y - 3 2 1 + x D 1 2 f x , y = 3 4 1 + x 1 + y - 5 2 1 + y 2 D 1 D 2 f x , y = 3 4 1 + x 1 + y - 5 2 1 + x 1 + y - 1 2 1 + x 1 + y - 3 2 D 2 2 f x , y = 3 4 1 + x 1 + y - 5 2 1 + x 2 D 1 3 f x , y = - 15 8 1 + x 1 + y - 7 2 1 + y 3 D 1 2 D 2 f x , y = - 15 8 1 + x 1 + y - 7 2 1 + x 1 + y 2 + 3 4 1 + x 1 + y - 5 2 1 + y 2 D 1 D 2 2 f x , y = - 15 8 1 + x 1 + y - 7 2 1 + y 2 1 + y + 3 4 1 + x 1 + y - 5 2 1 + x 2 D 2 3 f x , y = - 15 8 1 + x 1 + y - 7 2 1 + x 3

      よって、 求める3次のテイラー多項式は、

      f 0 , 0 + D 1 f 0 , 0 x + D 2 f 0 , 0 y + 1 2 ! D 1 2 f 0 , 0 x 2 + 2 D 1 D 2 f 0 , 0 x y + D 2 2 f 0 , 0 y 2 + 1 3 ! D 1 3 f 0 , 0 x 3 + 3 D 1 2 D 2 f 0 , 0 x 2 y + 3 D 1 D 2 2 f 0 , 0 x y 2 + D 2 3 f 0 , 0 y 3 = 1 + - 1 2 x - 1 2 y + 1 2 3 4 x 2 + 2 3 4 - 1 2 x y + 3 4 y 2 + 1 6 - 15 8 x 3 + 3 - 15 8 + 3 2 x 2 y + 3 - 15 8 + 3 2 x y 2 - 15 8 y 3 = 1 - 1 2 x + y + 1 8 3 x 2 + 2 x y + 3 y 2 - 1 16 5 x 3 + 3 x 2 y + 3 x y 2 + 5 y 3

macOS High Sierraの標準搭載されているグラフ作成ソフト、Grapher で作成。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sqrt, Derivative, factorial

a, b = symbols('a, b', nonzero=True)
x, y = symbols('x, y')
f = 1 / sqrt((1 + x) * (1 + y))
d = {x: 0, y: 0}
Dx = Derivative(f, x, 1)
Dy = Derivative(f, y, 1)
Dxx = Derivative(f, x, 2)
Dyy = Derivative(f, y, 2)
Dxy = Derivative(Dx, y, 1)
Dxxx = Derivative(f, x, 3)
Dyyy = Derivative(f, y, 3)
Dxxy = Derivative(Dxx, y, 1)
Dxyy = Derivative(Dyy, x, 1)
expr = f.subs(d) + (Dx.subs(d) * x + Dy.subs(d) * y) + 1 / factorial(2) * (Dxx.subs(d) * x ** 2 + 2 * Dxy.subs(d) * x * y + Dyy.subs(d) * y ** 2) + \
    1 / factorial(3) * (Dxxx.subs(d) * x ** 3 + 3 * Dxxy.subs(d) * x ** 2 *
                        y + 3 * Dxyy.subs(d) * x * y ** 2 + Dyyy.subs(d) * y ** 3)


for t in [f, expr, expr.doit()]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample10.py
         1         
───────────────────
  _________________
╲╱ (x + 1)⋅(y + 1) 

   ⎛  3           ⎞│                  ⎛  2           ⎞│                       
 3 ⎜ d ⎛    1    ⎞⎟│                2 ⎜ d ⎛    1    ⎞⎟│                       
x ⋅⎜───⎜─────────⎟⎟│               x ⋅⎜───⎜─────────⎟⎟│                       
   ⎜  3⎜  _______⎟⎟│         2        ⎜  2⎜  _______⎟⎟│           2           
   ⎝dx ⎝╲╱ x + 1 ⎠⎠│x=0   3⋅x ⋅y      ⎝dx ⎝╲╱ x + 1 ⎠⎠│x=0   3⋅x⋅y    x⋅y     
─────────────────────── - ────── + ─────────────────────── - ────── + ─── + x⋅
           6                16                2                16      4      
                                                                              

                         ⎛  3           ⎞│         ⎛  2           ⎞│          
                       3 ⎜ d ⎛    1    ⎞⎟│       2 ⎜ d ⎛    1    ⎞⎟│          
                      y ⋅⎜───⎜─────────⎟⎟│      y ⋅⎜───⎜─────────⎟⎟│          
                         ⎜  3⎜  _______⎟⎟│         ⎜  2⎜  _______⎟⎟│          
⎛d ⎛    1    ⎞⎞│         ⎝dy ⎝╲╱ y + 1 ⎠⎠│y=0      ⎝dy ⎝╲╱ y + 1 ⎠⎠│y=0     ⎛d
⎜──⎜─────────⎟⎟│    + ─────────────────────── + ─────────────────────── + y⋅⎜─
⎜dx⎜  _______⎟⎟│                 6                         2                ⎜d
⎝  ⎝╲╱ x + 1 ⎠⎠│x=0                                                         ⎝ 

                     
                     
                     
                     
 ⎛    1    ⎞⎞│       
─⎜─────────⎟⎟│    + 1
y⎜  _______⎟⎟│       
 ⎝╲╱ y + 1 ⎠⎠│y=0    

     3      2        2        2                3      2        
  5⋅x    3⋅x ⋅y   3⋅x    3⋅x⋅y    x⋅y   x   5⋅y    3⋅y    y    
- ──── - ────── + ──── - ────── + ─── - ─ - ──── + ──── - ─ + 1
   16      16      8       16      4    2    16     8     2    

$

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="0.005">
<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">
<br>
<label for="y0">y0 = </label>
<input id="y0" 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="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_y0 = document.querySelector('#y0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_y0],
    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, y) => 1 / Math.sqrt((1 + x) * (1 + y)),
    g = (x, y) => 1 - (x + y) / 2 + (3 * x ** 2 + 2 * x * y + 3 * y ** 2) / 8 -
    (5 * x ** 3 + 3 * x ** 2 * y + 3 * x * y ** 2 + 5 * y ** 3) / 16;

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),
        y0 = parseFloat(input_y0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [],
        fns = [[(x) => f(x, y0), 'red'],
               [(x) => g(x, y0), 'green']];
    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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