A Newton-type method for two-dimensional eigenvalue problems

Two-dimensional eigenvalue problems (2DEVP) equations in complex Hermitian case contain complex quadratic forms which is non-holomorphic. In this case, standard Newton method fails to apply. An existing strategy to solve this problem is to transform them into real problems (TRN). However, this method doubles the size of equations and thus is time consuming. On the other hand, the non-isolation of the solution set in 2DEVP also complicates the analysis. In this article, we propose a Newton type method which solves the problem caused by non-holomophism and non-isolation. It has locally quadratic convergence rate and is about at least twice as much efficient as TRN. We hope our ideas can provide insights for solving other problems including non-holomorphism and non-isolation. We apply our algorithm to calculate the distance to instability. Numerical experiments show its advantages in efficiency while keeping good convergence compared with the current state of art algorithms.