Recursive Integral Method with Cayley Transformation
For researchers solving non-Hermitian eigenvalue problems, this work offers a more efficient variant of a recently proposed eigensolver, but the improvement is incremental.
The authors improve the Recursive Integral Method (RIM) for non-Hermitian eigenvalue problems by incorporating Cayley transformation and Arnoldi's method, significantly reducing computational cost. Numerical examples show the new method is effective and efficient compared to MATLAB's eigs.
Recently, a non-classical eigenvalue solver, called RIM, was proposed to compute (all) eigenvalues in a region on the complex plane. Without solving any eigenvalue problem, it tests if a region contains eigenvalues using an approximate spectral projection. Regions that contain eigenvalues are subdivided and tested recursively until eigenvalues are isolated with a specified precision. This makes RIM an eigensolver distinct from all existing methods. Furthermore, it requires no a priori spectral information. In this paper, we propose an improved version of {\bf RIM} for non-Hermitian eigenvalue problems. Using Cayley transformation and Arnoldi's method, the computation cost is reduced significantly. Effectiveness and efficiency of the new method are demonstrated by numerical examples and compared with 'eigs' in Matlab.