  This work demonstrates an innovative numerical method for counting and
locating eigenvalues with the Evans function. Utilizing the geometric phase in
the Hopf bundle, the technique calculates the winding of the Evans function
about a contour in the spectral plane, describing the eigenvalues enclosed by
the contour for the Hocking-Stewartson pulse of the complex Ginzburg-Landau
equation. Locating eigenvalues with the geometric phase in the Hopf bundle was
proposed by Way, and proven by Grudzien, Bridges & Jones. Way demonstrated his
proposed method for the Hocking-Stewartson pulse, and this manuscript
redevelops this example as in the proof of the method, modifying his numerical
shooting argument, and introduces new numerical results concerning the phase
transition.
