Boost C++ Libraries of the most highly regarded and expertly designed C++ library projects in the world. Herb Sutter and Andrei Alexandrescu, C++ Coding Standards

This is the documentation for an old version of Boost. Click here to view this page for the latest version.

Roots of Quartic Polynomials


#include <boost/math/roots/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);



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.