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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.cubic_roots"></a><a class="link" href="cubic_roots.html" title="Roots of Cubic Polynomials">Roots of Cubic Polynomials</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.cubic_roots.h0"></a>
<span class="phrase"><a name="math_toolkit.cubic_roots.synopsis"></a></span><a class="link" href="cubic_roots.html#math_toolkit.cubic_roots.synopsis">Synopsis</a>
</h4>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">cubic_roots</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span> <span class="special">{</span>
<span class="comment">// Solves ax³ + bx² + cx + d = 0.</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span><span class="number">3</span><span class="special">></span> <span class="identifier">cubic_roots</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">c</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">d</span><span class="special">);</span>
<span class="comment">// Computes the empirical residual p(r) (first element) and expected residual ε|rṗ(r)| (second element) for a root.</span>
<span class="comment">// Recall that for a numerically computed root r satisfying r = r⁎(1+ε) for the exact root r⁎ of a function p, |p(r)| ≲ ε|rṗ(r)|.</span>
<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">></span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">2</span><span class="special">></span> <span class="identifier">cubic_root_residual</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">c</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">d</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">root</span><span class="special">);</span>
<span class="comment">// Computes the condition number of rootfinding. Computed via Corless, A Graduate Introduction to Numerical Methods, Section 3.2.1:</span>
<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">></span>
<span class="identifier">Real</span> <span class="identifier">cubic_root_condition_number</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">c</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">d</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">root</span><span class="special">);</span>
<span class="special">}</span>
</pre>
<h4>
<a name="math_toolkit.cubic_roots.h1"></a>
<span class="phrase"><a name="math_toolkit.cubic_roots.background"></a></span><a class="link" href="cubic_roots.html#math_toolkit.cubic_roots.background">Background</a>
</h4>
<p>
The <code class="computeroutput"><span class="identifier">cubic_roots</span></code> function extracts
all real roots of a cubic polynomial ax³ + bx² + cx + d. The result is a
<code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">></span></code>, which has length three, irrespective of
whether there are 3 real roots. There is always 1 real root, and hence (barring
overflow or other exceptional circumstances), the first element of the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span></code>
is always populated. If there is only one real root of the polynomial, then
the second and third elements are set to <code class="computeroutput"><span class="identifier">nans</span></code>.
The roots are returned in nondecreasing order.
</p>
<p>
Be careful with double roots. First, if you have a real double root, it is
numerically indistinguishable from a complex conjugate pair of roots, where
the complex part is tiny. Second, the condition number of rootfinding is infinite
at a double root, so even changes as subtle as different instruction generation
can change the outcome. We have some heuristics in place to detect double roots,
but these should be regarded with suspicion.
</p>
<h4>
<a name="math_toolkit.cubic_roots.h2"></a>
<span class="phrase"><a name="math_toolkit.cubic_roots.example"></a></span><a class="link" href="cubic_roots.html#math_toolkit.cubic_roots.example">Example</a>
</h4>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span>
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">sstream</span><span class="special">></span>
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">cubic_roots</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">cubic_roots</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">cubic_root_residual</span><span class="special">;</span>
<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">></span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">string</span> <span class="identifier">print_roots</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="number">3</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">roots</span><span class="special">)</span> <span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">ostringstream</span> <span class="identifier">out</span><span class="special">;</span>
<span class="identifier">out</span> <span class="special"><<</span> <span class="string">"{"</span> <span class="special"><<</span> <span class="identifier">roots</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">roots</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">roots</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special"><<</span> <span class="string">"}"</span><span class="special">;</span>
<span class="keyword">return</span> <span class="identifier">out</span><span class="special">.</span><span class="identifier">str</span><span class="special">();</span>
<span class="special">}</span>
<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> <span class="special">{</span>
<span class="comment">// Solves x³ - 6x² + 11x - 6 = (x-1)(x-2)(x-3).</span>
<span class="keyword">auto</span> <span class="identifier">roots</span> <span class="special">=</span> <span class="identifier">cubic_roots</span><span class="special">(</span><span class="number">1.0</span><span class="special">,</span> <span class="special">-</span><span class="number">6.0</span><span class="special">,</span> <span class="number">11.0</span><span class="special">,</span> <span class="special">-</span><span class="number">6.0</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"The roots of x³ - 6x² + 11x - 6 are "</span> <span class="special"><<</span> <span class="identifier">print_roots</span><span class="special">(</span><span class="identifier">roots</span><span class="special">)</span> <span class="special"><<</span> <span class="string">".\n"</span><span class="special">;</span>
<span class="comment">// Double root; YMMV:</span>
<span class="comment">// (x+1)²(x-2) = x³ - 3x - 2:</span>
<span class="identifier">roots</span> <span class="special">=</span> <span class="identifier">cubic_roots</span><span class="special">(</span><span class="number">1.0</span><span class="special">,</span> <span class="number">0.0</span><span class="special">,</span> <span class="special">-</span><span class="number">3.0</span><span class="special">,</span> <span class="special">-</span><span class="number">2.0</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"The roots of x³ - 3x - 2 are "</span> <span class="special"><<</span> <span class="identifier">print_roots</span><span class="special">(</span><span class="identifier">roots</span><span class="special">)</span> <span class="special"><<</span> <span class="string">".\n"</span><span class="special">;</span>
<span class="comment">// Single root: (x-i)(x+i)(x-3) = x³ - 3x² + x - 3:</span>
<span class="identifier">roots</span> <span class="special">=</span> <span class="identifier">cubic_roots</span><span class="special">(</span><span class="number">1.0</span><span class="special">,</span> <span class="special">-</span><span class="number">3.0</span><span class="special">,</span> <span class="number">1.0</span><span class="special">,</span> <span class="special">-</span><span class="number">3.0</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"The real roots of x³ - 3x² + x - 3 are "</span> <span class="special"><<</span> <span class="identifier">print_roots</span><span class="special">(</span><span class="identifier">roots</span><span class="special">)</span> <span class="special"><<</span> <span class="string">".\n"</span><span class="special">;</span>
<span class="comment">// I don't know the roots of x³ + 6.28x² + 2.3x + 3.6;</span>
<span class="comment">// how can I see if they've been computed correctly?</span>
<span class="identifier">roots</span> <span class="special">=</span> <span class="identifier">cubic_roots</span><span class="special">(</span><span class="number">1.0</span><span class="special">,</span> <span class="number">6.28</span><span class="special">,</span> <span class="number">2.3</span><span class="special">,</span> <span class="number">3.6</span><span class="special">);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"The real root of x³ +6.28x² + 2.3x + 3.6 is "</span> <span class="special"><<</span> <span class="identifier">roots</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special"><<</span> <span class="string">".\n"</span><span class="special">;</span>
<span class="keyword">auto</span> <span class="identifier">res</span> <span class="special">=</span> <span class="identifier">cubic_root_residual</span><span class="special">(</span><span class="number">1.0</span><span class="special">,</span> <span class="number">6.28</span><span class="special">,</span> <span class="number">2.3</span><span class="special">,</span> <span class="number">3.6</span><span class="special">,</span> <span class="identifier">roots</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"The residual is "</span> <span class="special"><<</span> <span class="identifier">res</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special"><<</span> <span class="string">", and the expected residual is "</span> <span class="special"><<</span> <span class="identifier">res</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special"><<</span> <span class="string">". "</span><span class="special">;</span>
<span class="keyword">if</span> <span class="special">(</span><span class="identifier">abs</span><span class="special">(</span><span class="identifier">res</span><span class="special">[</span><span class="number">0</span><span class="special">])</span> <span class="special"><=</span> <span class="identifier">res</span><span class="special">[</span><span class="number">1</span><span class="special">])</span> <span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"This is an expected accuracy.\n"</span><span class="special">;</span>
<span class="special">}</span> <span class="keyword">else</span> <span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cerr</span> <span class="special"><<</span> <span class="string">"The residual is unexpectedly large.\n"</span><span class="special">;</span>
<span class="special">}</span>
<span class="special">}</span>
</pre>
<p>
This prints:
</p>
<pre class="programlisting">The roots of x<sup>3</sup> - 6x<sup>2</sup> + 11x - 6 are {1, 2, 3}.
The roots of x<sup>3</sup> - 3x - 2 are {-1, -1, 2}.
The real roots of x<sup>3</sup> - 3x<sup>2</sup> + x - 3 are {3, nan, nan}.
The real root of x<sup>3</sup> +6.28x<sup>2</sup> + 2.3x + 3.6 is -5.99656.
The residual is -1.56586e-14, and the expected residual is 4.64155e-14. This is an expected accuracy.
</pre>
<h4>
<a name="math_toolkit.cubic_roots.h3"></a>
<span class="phrase"><a name="math_toolkit.cubic_roots.performance_and_accuracy"></a></span><a class="link" href="cubic_roots.html#math_toolkit.cubic_roots.performance_and_accuracy">Performance
and Accuracy</a>
</h4>
<p>
On an Intel laptop chip running at 2700 MHz, we observe 3 roots taking ~90ns
to compute. If the polynomial only possesses a single real root, it takes ~50ns.
See <code class="computeroutput"><span class="identifier">reporting</span><span class="special">/</span><span class="identifier">performance</span><span class="special">/</span><span class="identifier">cubic_roots_performance</span><span class="special">.</span><span class="identifier">cpp</span></code>.
</p>
<p>
The forward error cannot be effectively bounded. However, for an approximate
root r returned by this routine, the residuals should be constrained by |p(r)|
≲ ε|rṗ(r)|, where '≲' should be interpreted as an order of magnitude
estimate.
</p>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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