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[/
Copyright (c) 2021 Nick Thompson
Use, modification and distribution are subject to the
Boost Software License, Version 1.0. (See accompanying file
LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
]
[section:quartic_roots Roots of Quartic Polynomials]
[heading Synopsis]
```
#include <boost/math/tools/quartic_roots.hpp>
namespace boost::math::tools {
// Solves ax⁴ + bx³ + cx² + dx + e = 0.
std::array<Real,3> quartic_roots(Real a, Real b, Real c, Real d, Real e);
}
```
[heading Background]
The `quartic_roots` function extracts all real roots of a quartic polynomial ax⁴+ bx³ + cx² + dx + e.
The result is a `std::array<Real, 4>`, which has length four, irrespective of the number of real roots the polynomial possesses.
(This is to prevent the performance overhead of allocating a vector, which often exceeds the time to extract the roots.)
The roots are returned in nondecreasing order. If a root is complex, then it is placed at the back of the array and set to a nan.
The algorithm uses the classical method of Ferrari, and follows [@https://github.com/erich666/GraphicsGems/blob/master/gems/Roots3And4.c Graphics Gems V],
with an additional Halley iterate for root polishing.
A typical use of a quartic real-root solver is to raytrace a torus.
[heading Performance and Accuracy]
On a consumer laptop, we observe extraction of the roots taking ~90ns.
The file `reporting/performance/quartic_roots_performance.cpp` allows determination of the speed on your system.
[endsect]
[/section:quartic_roots]
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