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## Roots of Quartic Polynomials

#### 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);

    }

#### 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 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.

#### 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.