We estimate by applying ARPACK to a Galerkin-spectral discretization of . We chose the spectral method over both finite differences and elements for the reason that the improvement in accuracy outweighed the cost of working with dense linear systems. The Galerkin basis of choice was
We denote by a piecewise constant approximation to at n points and denote by the nonnormal, complex, dense, matrix of order 2m that results from the Galerkin discretization of . Rather than computing the entire spectrum of only to keep the element with the largest real part we invoke, with care, the shift and invert strategy offered by ARPACK and effectively compute the half dozen or so right-most eigenvalues. For m on the order of 64 this method was faster, by a factor of 10, than the LAPACK approach.
With regard to the minimization of we recall the excellent performance of PDS on the related though simpler problems (shear buildings and strings) treated in . The fact that does not possess a bounded derivative in a neighborhood of its minimizer all but forces us to exploit derivative-free search techniques.