#include <acts/blob/sPHENIX/Core/include/Acts/Utilities/detail/RealQuadraticEquation.hpp>

Detailed Description

Mathematic struct for solving real quadratic equations

Mathematical motivation:
The equation is given by:
$\alpha x^{2} + \beta x + \gamma = 0$ and has therefore the analytical solution:
$x_{1, 2} = - \frac{\beta \pm \sqrt{\beta^{2}-4\alpha\gamma}}{2\alpha}$

case $\beta > 0$ :
$x_{1} = - \frac{\beta + \sqrt{\beta^{2}-4\alpha\gamma}}{2\alpha} := \frac{q}{\alpha}$ ,
so that $q= -\frac{1}{2}(\beta+sqrt{\beta^{2}-4\alpha\gamma})$ . $x_{2}$ can now be written as: $x_{2} = \frac{\gamma}{q} = -\frac{2\gamma}{\beta+sqrt{\beta^{2}-4\alpha\gamma}}$ , since:
$-\frac{2\gamma}{\beta+sqrt{\beta^{2}-4\alpha\gamma}} = -\frac{2\gamma}{\beta}\frac{1}{1+\sqrt{1-4\alpha\gamma/\beta^{2}}}$ , and
$x_{2}\frac{1}{1-\sqrt{1-4\alpha\gamma/\beta^{2}}} = -\frac{2\gamma}{\beta}\frac{1}{1-1+4\alpha\gamma/\beta^{2}}=-\frac{\beta}{2\alpha}.$
Hence, $-\frac{\beta(1-\sqrt{1-4\alpha\gamma/\beta^{2}}}{2\alpha} = - \frac{\beta - \sqrt{\beta^{2}-4\alpha\gamma}}{2\alpha}$ .
case $\beta > 0$ is similar.

The documentation for this struct was generated from the following file:

acts/blob/sPHENIX/Core/include/Acts/Utilities/detail/RealQuadraticEquation.hpp