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<main>
<h1 id="线性代数"><a class="header" href="#线性代数">线性代数</a></h1>
<h3 id="矩阵相加"><a class="header" href="#矩阵相加">矩阵相加</a></h3>
<p>使用 <a href="https://docs.rs/ndarray/*/ndarray/fn.arr2.html">ndarray::arr2</a> 可以创建二阶矩阵，并计算它们的和。</p>
<pre><pre class="playground"><code class="language-rust editable edition2021">use ndarray::arr2;
fn main() {
let a = arr2(&amp;[[1, 2, 3],
[4, 5, 6]]);
let b = arr2(&amp;[[6, 5, 4],
[3, 2, 1]]);
// 借用 a 和 b，求和后生成新的矩阵 sum
let sum = &amp;a + &amp;b;
println!(&quot;{}&quot;, a);
println!(&quot;+&quot;);
println!(&quot;{}&quot;, b);
println!(&quot;=&quot;);
println!(&quot;{}&quot;, sum);
}
</code></pre></pre>
<h3 id="矩阵相乘"><a class="header" href="#矩阵相乘">矩阵相乘</a></h3>
<p><a href="https://docs.rs/ndarray/0.15.4/ndarray/struct.ArrayBase.html#method.dot-1">ndarray::ArrayBase::dot</a> 可以用于计算矩阵乘法。</p>
<pre><pre class="playground"><code class="language-rust editable edition2021">use ndarray::arr2;
fn main() {
let a = arr2(&amp;[[1, 2, 3],
[4, 5, 6]]);
let b = arr2(&amp;[[6, 3],
[5, 2],
[4, 1]]);
println!(&quot;{}&quot;, a.dot(&amp;b));
}
</code></pre></pre>
<h3 id="标量向量矩阵相乘"><a class="header" href="#标量向量矩阵相乘">标量、向量、矩阵相乘</a></h3>
<p>在 <code>ndarry</code>中，1 阶数组根据上下文既可以作为行向量也可以作为列向量。如果对你来说，这个行或列的方向很重要，可以考虑使用一行或一列的 2 阶数组来表示。</p>
<p>在下面例子中，由于 1 阶数组处于乘号的右边位置，因此 <code>dot</code> 会把它当成列向量来处理。</p>
<pre><pre class="playground"><code class="language-rust editable edition2021">use ndarray::{arr1, arr2, Array1};
fn main() {
let scalar = 4;
let vector = arr1(&amp;[1, 2, 3]);
let matrix = arr2(&amp;[[4, 5, 6],
[7, 8, 9]]);
let new_vector: Array1&lt;_&gt; = scalar * vector;
println!(&quot;{}&quot;, new_vector);
let new_matrix = matrix.dot(&amp;new_vector);
println!(&quot;{}&quot;, new_matrix);
}
</code></pre></pre>
<h3 id="向量比较"><a class="header" href="#向量比较">向量比较</a></h3>
<p>浮点数通常是不精确的，因此比较浮点数不是一件简单的事。<a href="https://docs.rs/approx/*/approx/index.html">approx</a> 提供的 <a href="https://docs.rs/approx/0.5.1/approx/macro.assert_abs_diff_eq.html">assert_abs_diff_eq!</a> 宏提供了方便的按元素比较的方式。为了使用 <code>approx</code> ，你需要在 <code>ndarray</code> 的依赖中开启相应的 feature：例如，在 <code>Cargo.toml</code> 中修改 <code>ndarray</code> 的依赖引入为 <code>ndarray = { version = &quot;0.13&quot;, features = [&quot;approx&quot;] }</code>。</p>
<pre><pre class="playground"><code class="language-rust editable edition2021">use approx::assert_abs_diff_eq;
use ndarray::Array;
fn main() {
let a = Array::from(vec![1., 2., 3., 4., 5.]);
let b = Array::from(vec![5., 4., 3., 2., 1.]);
let mut c = Array::from(vec![1., 2., 3., 4., 5.]);
let mut d = Array::from(vec![5., 4., 3., 2., 1.]);
// 消耗 a 和 b 的所有权
let z = a + b;
// 借用 c 和 d
let w = &amp;c + &amp;d;
assert_abs_diff_eq!(z, Array::from(vec![6., 6., 6., 6., 6.]));
println!(&quot;c = {}&quot;, c);
c[0] = 10.;
d[1] = 10.;
assert_abs_diff_eq!(w, Array::from(vec![6., 6., 6., 6., 6.]));
}
</code></pre></pre>
<h3 id="向量范数-norm-"><a class="header" href="#向量范数-norm-">向量范数( norm )</a></h3>
<p>需要注意的是 <code>Array</code> 和 <code>ArrayView</code> 都是 <code>ArrayBase</code> 的别名。因此一个更通用的参数应该是 <code>&amp;ArrayBase&lt;S, Ix1&gt; where S: Data</code>，特别是在你提供一个公共 API 给其它用户时，但由于咱们是内部使用，因此更精准的 <code>ArrayView1&lt;f64&gt;</code> 会更适合。</p>
<pre><pre class="playground"><code class="language-rust editable edition2021">use ndarray::{array, Array1, ArrayView1};
fn l1_norm(x: ArrayView1&lt;f64&gt;) -&gt; f64 {
x.fold(0., |acc, elem| acc + elem.abs())
}
fn l2_norm(x: ArrayView1&lt;f64&gt;) -&gt; f64 {
x.dot(&amp;x).sqrt()
}
fn normalize(mut x: Array1&lt;f64&gt;) -&gt; Array1&lt;f64&gt; {
let norm = l2_norm(x.view());
x.mapv_inplace(|e| e/norm);
x
}
fn main() {
let x = array![1., 2., 3., 4., 5.];
println!(&quot;||x||_2 = {}&quot;, l2_norm(x.view()));
println!(&quot;||x||_1 = {}&quot;, l1_norm(x.view()));
println!(&quot;Normalizing x yields {:?}&quot;, normalize(x));
}
</code></pre></pre>
<h3 id="矩阵的逆变换"><a class="header" href="#矩阵的逆变换">矩阵的逆变换</a></h3>
<p>例子中使用 <a href="https://docs.rs/nalgebra/*/nalgebra/base/type.Matrix3.html">nalgebra::Matrix3</a> 创建一个 3x3 的矩阵，然后尝试对其进行逆变换，获取一个逆矩阵。</p>
<pre><pre class="playground"><code class="language-rust editable edition2021">use nalgebra::Matrix3;
fn main() {
let m1 = Matrix3::new(2.0, 1.0, 1.0, 3.0, 2.0, 1.0, 2.0, 1.0, 2.0);
println!(&quot;m1 = {}&quot;, m1);
match m1.try_inverse() {
Some(inv) =&gt; {
println!(&quot;The inverse of m1 is: {}&quot;, inv);
}
None =&gt; {
println!(&quot;m1 is not invertible!&quot;);
}
}
}
</code></pre></pre>
<h3 id="序列反序列化一个矩阵"><a class="header" href="#序列反序列化一个矩阵">序列/反序列化一个矩阵</a></h3>
<p>下面将展示如何将矩阵序列化为 JSON ，然后再反序列化为原矩阵。</p>
<pre><pre class="playground"><code class="language-rust editable edition2021">extern crate nalgebra;
extern crate serde_json;
use nalgebra::DMatrix;
fn main() -&gt; Result&lt;(), std::io::Error&gt; {
let row_slice: Vec&lt;i32&gt; = (1..5001).collect();
let matrix = DMatrix::from_row_slice(50, 100, &amp;row_slice);
// 序列化矩阵
let serialized_matrix = serde_json::to_string(&amp;matrix)?;
// 反序列化
let deserialized_matrix: DMatrix&lt;i32&gt; = serde_json::from_str(&amp;serialized_matrix)?;
// 验证反序列化后的矩阵跟原始矩阵相等
assert!(deserialized_matrix == matrix);
Ok(())
}
</code></pre></pre>
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