This project involves the development of advanced computational methods for solving genuine nonlinear eigenvalue problems. In this project, skillful combination of Kublanovskaya's nonlinear QR algorithm with modern rank-revealing and structure-preserving techniques for small and medium size dense problems enhances the capabilities of new methods. A novel trimmed linearization via Pade rational approximation extends the enhancements for solving large but sparse problems. The investigators seek to develop a systematic and unified treatment of the relevant mathematical theory, and produce numerical methods and software tools for the genuine nonlinear eigenvalue problems. In addition to advancing research in nonlinear eigenvalue problems, the project provides training for graduate students in computational mathematics and interdisciplinary research tools.
Eigenvalue problems are ubiquitous in computational science and engineering, where they arise in the study of dynamics of structures, simulation of nanostructured photovoltaic conversion materials to advance energy science, and many other scenarios. Eigenvalues explain a wide range of physical phenomena such as vibrations and frequencies, (in)stabilities of dynamical systems, and energy excitation states of electrons and molecules. Many eigenvalue problems occur naturally in nonlinear form. In this project, the investigators study the underlying nonlinear problems without relying on linearization approximations. The promise of substantially improved methods for computing solutions of nonlinear eigenvalue problems, brings immediate benefits to a wide range of practical applications.