Computation of a Few Small Eigenvalues of a Large Matrix with Application to Liquid Crystal Modeling

Equilibrium configurations of liquid crystals in a finite containment are minimizers of the thermodynamic free energy of the system. It is important to be able to track an equilibrium configuration as the temperature of the liquid crystals is decreased. The path of the minimal energy configuration at a bifurcation point can be computed from the null space of a sparse symmetric matrix, which typically is very large, e.g., of order 3 × 105. We describe an implicitly restarted block Lanczos method designed for the computation of a few extreme multiple or close eigenvalues and associated eigenvectors of a large sparse symmetric matrix and apply this method to determine the desired null space. Our method generalizes the implicitly restarted Lanczos method introduced by Sorensen. The method requires that certain acceleration parameters, referred to as shifts, be chosen. The storage requirement depends on the choice of shifts. We propose a new strategy for choosing shifts. Numerical examples illustrate that the implicitly restarted block Lanczos method with shifts chosen in this manner gives rapid convergence, reliably detects extreme multiple or close eigenvalues, and requires little computer storage in addition to the storage used for the desired eigenvectors. These features make the method well suited for the application of tracking an equilibrium configuration of liquid crystals.