Most recently, Dyksen has derived explicit closed-form expressions for the Hermite cubic approximations to both the eigenvalues and the eigenfunctions of the Laplace operator for both the Dirichlet and the Neumann problems. Moreover, for the Dirichlet case, he shows that optimal approximations are obtained using the Gauss points for collocation points.

Dyksen and Robert Lynch have recently developed a new decoupling technique for solving the linear systems arising from Hermite cubic collocation solutions to boundary value problems with both Dirichlet and Neumann boundary conditions. While the traditional approach yields a linear system of order 2 N × 2 N with bandwidth 2, this new technique decouples this system into two systems, one with a tridiagonal system of order N - 1 × N - 1 and the other with the identity matrix of order N × N.