Parallel Multi-step Spectral Indicator Method for Nonlinear Eigenvalue Problems

Consider nonlinear eigenvalue problems to compute all eigenvalues in a given region on the complex plane. We propose a parallel multi-step spectral indicator method with the contour integrals being the main ingredient. Step 1 (screening) divides the given region into subregion and compute an indicator for each subregion. Then it decides candidate regions that contain eigenvalues. Step 2 (computation) computes eigenvalues in each candidate region. Step 3 (validation) double checks each eigenvalue by substituting it back to the system and computing the smallest eigenvalue. Each step is carried out in parallel. Two examples are presented for demonstration.