46,051 research outputs found
Solving eigenvalue problems on curved surfaces using the Closest Point Method
Eigenvalue problems are fundamental to mathematics and science. We present a
simple algorithm for determining eigenvalues and eigenfunctions of the
Laplace--Beltrami operator on rather general curved surfaces. Our algorithm,
which is based on the Closest Point Method, relies on an embedding of the
surface in a higher-dimensional space, where standard Cartesian finite
difference and interpolation schemes can be easily applied. We show that there
is a one-to-one correspondence between a problem defined in the embedding space
and the original surface problem. For open surfaces, we present a simple way to
impose Dirichlet and Neumann boundary conditions while maintaining second-order
accuracy. Convergence studies and a series of examples demonstrate the
effectiveness and generality of our approach
Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method
Given the matrix polynomial , we
consider the associated polynomial eigenvalue problem. This problem, viewed in
terms of computing the roots of the scalar polynomial , is treated
in polynomial form rather than in matrix form by means of the Ehrlich-Aberth
iteration. The main computational issues are discussed, namely, the choice of
the starting approximations needed to start the Ehrlich-Aberth iteration, the
computation of the Newton correction, the halting criterion, and the treatment
of eigenvalues at infinity. We arrive at an effective implementation which
provides more accurate approximations to the eigenvalues with respect to the
methods based on the QZ algorithm. The case of polynomials having special
structures, like palindromic, Hamiltonian, symplectic, etc., where the
eigenvalues have special symmetries in the complex plane, is considered. A
general way to adapt the Ehrlich-Aberth iteration to structured matrix
polynomial is introduced. Numerical experiments which confirm the effectiveness
of this approach are reported.Comment: Submitted to Linear Algebra App
- …