Rapid Resolution of Truncated Intermediate Problems

Abstract

The determinant method of truncation is a useful computational adjunct in seeking tight lower bounds to eigenvalues of self-adjoint operators commonly found in science and engineering. The primary computational effort and bottleneck in this method involves finding and labeling selected roots of a determinant of a nonlinear matrix-valued function — the Weinstein-Aronszajn perturbation determinant. The best previous algorithms for resolving this problem directly were simply variants of the secant method and had at best a local superlinear rate of convergence. Worse, these methods frequently failed either due to encounters with singularities that lie interspersed among the desired roots or due to a mislabeling of the index of a root — thus leading to erroneous eigenvalue bounds.