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  26 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  27 <a name="math_toolkit.oct_value_ops"></a><a class="link" href="oct_value_ops.html" title="Octonion Value Operations">Octonion Value Operations</a>
  28 </h2></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.oct_value_ops.h0"></a>
  31       <span class="phrase"><a name="math_toolkit.oct_value_ops.real_and_unreal"></a></span><a class="link" href="oct_value_ops.html#math_toolkit.oct_value_ops.real_and_unreal">Real
  32       and Unreal</a>
  33     </h5>
  34 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">T</span>  <span class="identifier">real</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">o</span><span class="special">);</span>
  35 <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">unreal</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">o</span><span class="special">);</span>
  36 </pre>
  37 <p>
  38       These return <code class="computeroutput"><span class="identifier">o</span><span class="special">.</span><span class="identifier">real</span><span class="special">()</span></code> and
  39       <code class="computeroutput"><span class="identifier">o</span><span class="special">.</span><span class="identifier">unreal</span><span class="special">()</span></code>
  40       respectively.
  41     </p>
  42 <h5>
  43 <a name="math_toolkit.oct_value_ops.h1"></a>
  44       <span class="phrase"><a name="math_toolkit.oct_value_ops.conj"></a></span><a class="link" href="oct_value_ops.html#math_toolkit.oct_value_ops.conj">conj</a>
  45     </h5>
  46 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">conj</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">o</span><span class="special">);</span>
  47 </pre>
  48 <p>
  49       This returns the conjugate of the octonion.
  50     </p>
  51 <h5>
  52 <a name="math_toolkit.oct_value_ops.h2"></a>
  53       <span class="phrase"><a name="math_toolkit.oct_value_ops.sup"></a></span><a class="link" href="oct_value_ops.html#math_toolkit.oct_value_ops.sup">sup</a>
  54     </h5>
  55 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">T</span>   <span class="identifier">sup</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">o</span><span class="special">);</span>
  56 </pre>
  57 <p>
  58       This return the sup norm (the greatest among <code class="computeroutput"><span class="identifier">abs</span><span class="special">(</span><span class="identifier">o</span><span class="special">.</span><span class="identifier">R_component_1</span><span class="special">())...</span><span class="identifier">abs</span><span class="special">(</span><span class="identifier">o</span><span class="special">.</span><span class="identifier">R_component_8</span><span class="special">()))</span></code> of the octonion.
  59     </p>
  60 <h5>
  61 <a name="math_toolkit.oct_value_ops.h3"></a>
  62       <span class="phrase"><a name="math_toolkit.oct_value_ops.l1"></a></span><a class="link" href="oct_value_ops.html#math_toolkit.oct_value_ops.l1">l1</a>
  63     </h5>
  64 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">T</span>   <span class="identifier">l1</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">o</span><span class="special">);</span>
  65 </pre>
  66 <p>
  67       This return the l1 norm (<code class="computeroutput"><span class="identifier">abs</span><span class="special">(</span><span class="identifier">o</span><span class="special">.</span><span class="identifier">R_component_1</span><span class="special">())+...+</span><span class="identifier">abs</span><span class="special">(</span><span class="identifier">o</span><span class="special">.</span><span class="identifier">R_component_8</span><span class="special">())</span></code>) of the octonion.
  68     </p>
  69 <h5>
  70 <a name="math_toolkit.oct_value_ops.h4"></a>
  71       <span class="phrase"><a name="math_toolkit.oct_value_ops.abs"></a></span><a class="link" href="oct_value_ops.html#math_toolkit.oct_value_ops.abs">abs</a>
  72     </h5>
  73 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">T</span>   <span class="identifier">abs</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">o</span><span class="special">);</span>
  74 </pre>
  75 <p>
  76       This return the magnitude (Euclidian norm) of the octonion.
  77     </p>
  78 <h5>
  79 <a name="math_toolkit.oct_value_ops.h5"></a>
  80       <span class="phrase"><a name="math_toolkit.oct_value_ops.norm"></a></span><a class="link" href="oct_value_ops.html#math_toolkit.oct_value_ops.norm">norm</a>
  81     </h5>
  82 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">T</span>  <span class="identifier">norm</span><span class="special">(</span><span class="identifier">octonion</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span><span class="keyword">const</span>  <span class="special">&amp;</span> <span class="identifier">o</span><span class="special">);</span>
  83 </pre>
  84 <p>
  85       This return the (Cayley) norm of the octonion. The term "norm" might
  86       be confusing, as most people associate it with the Euclidian norm (and quadratic
  87       functionals). For this version of (the mathematical objects known as) octonions,
  88       the Euclidian norm (also known as magnitude) is the square root of the Cayley
  89       norm.
  90     </p>
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