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<title>16. NumPy (1): ベクトル — Python早見帳</title>
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基礎
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発展
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科学技術計算とデータ分析
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16. NumPy (1): ベクトル
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16.1. 作成
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16.2. スライス
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16.3. 算術演算
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16.4. 様々な作成方法
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16.5. 数学関数
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16.6. ユニバーサル関数
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<h1>NumPy (1): ベクトル</h1>
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16.2. スライス
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16.5. 数学関数
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<div class="tex2jax_ignore mathjax_ignore section" id="numpy-1">
<h1><span class="section-number">16. </span>NumPy (1): ベクトル<a class="headerlink" href="#numpy-1" title="このヘッドラインへのパーマリンク">#</a></h1>
<p>このノートブックでは<code class="docutils literal notranslate"><span class="pre">numpy</span></code>モジュールのエイリアス（短縮表記）として<code class="docutils literal notranslate"><span class="pre">np</span></code>を用いる。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
</pre></div>
</div>
</div>
</div>
<div class="section" id="id1">
<h2><span class="section-number">16.1. </span>作成<a class="headerlink" href="#id1" title="このヘッドラインへのパーマリンク">#</a></h2>
<p>NumPyのベクトルを<a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.array.html">np.array</a>で作成する。以下のコードは2次元ベクトル</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{x} = \left(\begin{array}{c}
x_0 \\
x_1 \\
x_2
\end{array}\right) = \left(\begin{array}{c}
1 \\
2 \\
3
\end{array}\right)
\end{split}\]</div>
<p>を作成し、変数<code class="docutils literal notranslate"><span class="pre">x</span></code>に代入する。</p>
<div class="warning admonition">
<p class="admonition-title">インデックス番号</p>
<p>数学において先頭要素のインデックス番号は<span class="math notranslate nohighlight">\(1\)</span>とすることが多いが、Pythonでは先頭要素のインデックス番号が<span class="math notranslate nohighlight">\(0\)</span>である。両方が混在すると紛らわしいので、この資料でベクトルや行列を数式で表現するときも、先頭要素のインデックス番号は<span class="math notranslate nohighlight">\(0\)</span>とする。</p>
</div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span>
</pre></div>
</div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([1, 2, 3, 4, 5])
</pre></div>
</div>
</div>
</div>
<p>作成されたオブジェクトの型は<a class="reference external" href="https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html">numpy.ndarray</a>。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">type</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
</pre></div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>numpy.ndarray
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</div>
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</div>
<p>ベクトルの次元数（軸の数）。<code class="docutils literal notranslate"><span class="pre">x</span></code>は1次元ベクトル。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span><span class="o">.</span><span class="n">ndim</span>
</pre></div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>1
</pre></div>
</div>
</div>
</div>
<p>各次元の要素数を表すタプル。<code class="docutils literal notranslate"><span class="pre">x</span></code>の1次元目には<span class="math notranslate nohighlight">\(5\)</span>つの要素が格納されている。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span><span class="o">.</span><span class="n">shape</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(5,)
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<p>オブジェクトに含まれる要素数。<code class="docutils literal notranslate"><span class="pre">x</span></code>は１次元ベクトルなので１次元目の要素数に等しい。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span><span class="o">.</span><span class="n">size</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>5
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<p>各要素の型。<code class="docutils literal notranslate"><span class="pre">x</span></code>は整数を格納するベクトルとして定義されている。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span><span class="o">.</span><span class="n">dtype</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>dtype('int64')
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</div>
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</div>
<p>ベクトルを作成するときに要素の型を指定できる。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="s1">'float'</span><span class="p">)</span>
<span class="n">x</span>
</pre></div>
</div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([1., 2., 3., 4., 5.])
</pre></div>
</div>
</div>
</div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span><span class="o">.</span><span class="n">dtype</span>
</pre></div>
</div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>dtype('float64')
</pre></div>
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</div>
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</div>
<div class="section" id="id2">
<h2><span class="section-number">16.2. </span>スライス<a class="headerlink" href="#id2" title="このヘッドラインへのパーマリンク">#</a></h2>
<p>ベクトル<code class="docutils literal notranslate"><span class="pre">x</span></code>の要素<span class="math notranslate nohighlight">\(x_1\)</span></p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
</pre></div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>2.0
</pre></div>
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</div>
</div>
<p>ベクトル<code class="docutils literal notranslate"><span class="pre">x</span></code>の<span class="math notranslate nohighlight">\(0\)</span>番目から<span class="math notranslate nohighlight">\(1\)</span>番目までの要素を取り出したベクトル<span class="math notranslate nohighlight">\(\pmb{x}_{0:2}\)</span>（スライス）。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">2</span><span class="p">]</span>
</pre></div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([1., 2.])
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</div>
</div>
</div>
<p>スライスの開始位置が<span class="math notranslate nohighlight">\(0\)</span>の場合は省略できる（リストと同様）。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span>
</pre></div>
</div>
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</div>
</div>
<p>要素を逆順に並べる。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
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</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([5., 4., 3., 2., 1.])
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</div>
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</div>
<p>スライスで取り出す要素をリストで指定してもよい。</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span><span class="p">[[</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">]]</span>
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</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([2., 4.])
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</div>
</div>
</div>
<p>要素に値を代入する。<span class="math notranslate nohighlight">\(x_1 \leftarrow -1\)</span></p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
<span class="n">x</span>
</pre></div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([ 1., -1., 3., 4., 5.])
</pre></div>
</div>
</div>
</div>
<p>ベクトルのスライスに値を代入する。<span class="math notranslate nohighlight">\((x_0, x_1) \leftarrow (2, 2)\)</span></p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="mf">2.</span>
<span class="n">x</span>
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</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([2., 2., 3., 4., 5.])
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="id3">
<h2><span class="section-number">16.3. </span>算術演算<a class="headerlink" href="#id3" title="このヘッドラインへのパーマリンク">#</a></h2>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{x} = \left(\begin{array}{c}
1 \\
2 \\
3
\end{array}\right),
\boldsymbol{y} = \left(\begin{array}{c}
6 \\
4 \\
2
\end{array}\right)
\end{split}\]</div>
<p>に対して、様々な演算を紹介する。</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
</pre></div>
</div>
</div>
</div>
<p>ベクトルの和</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{x} + \boldsymbol{y} = \left(\begin{array}{c}
1 \\
2 \\
3
\end{array}\right) + \left(\begin{array}{c}
6 \\
4 \\
2
\end{array}\right)
= \left(\begin{array}{c}
1+6 \\
2+4 \\
3+2
\end{array}\right)
= \left(\begin{array}{c}
7 \\
6 \\
5
\end{array}\right)
\end{split}\]</div>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span>
</pre></div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([7, 6, 5])
</pre></div>
</div>
</div>
</div>
<p>ベクトルの差</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{x} - \boldsymbol{y} = \left(\begin{array}{c}
1 \\
2 \\
3
\end{array}\right) - \left(\begin{array}{c}
6 \\
4 \\
2
\end{array}\right)
= \left(\begin{array}{c}
1-6 \\
2-4 \\
3-2
\end{array}\right)
= \left(\begin{array}{c}
-5 \\
-2 \\
1
\end{array}\right)
\end{split}\]</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([-5, -2, 1])
</pre></div>
</div>
</div>
</div>
<p>ベクトルとベクトルの要素積（アダマール積）</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{x} \odot \boldsymbol{y} = \left(\begin{array}{c}
1 \\
2 \\
3
\end{array}\right) \odot \left(\begin{array}{c}
6 \\
4 \\
2
\end{array}\right)
= \left(\begin{array}{c}
1 \times 6 \\
2 \times 4 \\
3 \times 2
\end{array}\right)
= \left(\begin{array}{c}
6 \\
8 \\
6
\end{array}\right)
\end{split}\]</div>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">*</span> <span class="n">y</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([6, 8, 6])
</pre></div>
</div>
</div>
</div>
<p>ベクトルとベクトルの内積</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{x} \cdot \boldsymbol{y} = \left(\begin{array}{c}
1 \\