Numerical Methods For Polynomial Eigenvalue Problems

Solving nonlinear eigenvalue problems is a hot spot of computational mathematics and science and engineering field. polynomial eigenvalue problems (PEP for short), especially quadratic eigenvalue problem(QEP) and cubic eigenvalue problem(CEP), is a typical nonlinear eigenvalue problems. Most of them are arising in the dynamic analysis of structural mechanical, and acoustic systems, in electrical circuit simulation, in fluid mechanics, and, more recently, in modeling microelectronic mechanical systems and so on.For solving the large scale cubic eigenvalue problem(CEP) L (λ) x = (λ~ 3 A +λ~ 2B +λC + D ) x= 0, First of all, the thesis introduced iterated shift-and-invert Arnoldi algorithm. This algorithm, which is a direct projection method combined with the shift-and-invert technique in Arnoldi Process, has the virtues of higher computational efficiency and rapid convergence. The thesis also studied Jacobi-Davidson method for polynomial eigenvalue problems. Took Quadratic Eigenvalue Problem (QEP for short) for example, the thesis studied linearlization Jacobi-Davidson method and direct Jacobi-Davidson method, and did a comparative analysis. Finally, wrote programs to achieve the above algorithms, then did numerical experimentations and compared the results. Numerical examples given in this thesis confirmed new method’s effectiveness and advantage compared to the existing clipping algorithm.