Solving Eigenvalues Using an Analytic Approach for Conjunction Analysis
Sang Yang, Ryan J. Soldin, Roland Lang · 2012
This paper presents the use of an analytical method to compute eigenvalues for the application of conjunction analysis. Given two ellipsoids, the solution to the 4 th degree polynomial characteristic equation is shown using the classical numerical approach and the presented analytical method. It shows that if the characteristic equation has at least two negative real solutions then the ellipsoids are separated. In addition, if the ellipsoids share a common point, or touch each other, then the characteristic equation has a positive double real solution and the discriminant of the quadratic equations is zero. Moreover, two or more complex eigenvalues are obtained when it is a complete penetration of one ellipsoid by another. An advantage of using the analytical approach is not only that it yields direct solutions with numerical approximation and iteration but also it is computationally efficient.