Several tensor eigenpair definitions have been put forth in the past decade, but these can all be unified under generalized tensor eigenpair framework, introduced by Chang, Pearson, and Zhang [J. Math. Anal. Appl., 350 (2009), pp. 416--422]. Given mth-order, n-dimensional real-valued symmetric tensors @math and @math , the goal is to find @math and @math such that @math . Different choices for @math yield different versions of the tensor eigenvalue problem. We present our generalized eigenproblem adaptive power (GEAP) method for solving the problem, which is an extension of the shifted symmetric higher-order power method (SS-HOPM) for finding Z-eigenpairs. A major drawback of SS-HOPM is that its performance depended on choosing an appropriate shift, but our GEAP method also includes an adaptive method for choosing the shift automatically.
In general, Han's method represents an alternative approach to solving the generalized tensor eigenpair problem. In @cite_9 , Han's method was compared to SS-HOPM with a fixed shift (for Z-eigenpairs only) and was superior. However, GEAP is usually as fast as Han's method and perhaps a little more robust in terms of its convergence behavior. The speed being similar is thanks to the adaptive shift in GEAP. It may be that Han's method could avoid problems of converging to incorrect solutions with tighter tolerances, but then the speed would be slower.
@cite_9
2007036852
Let @math be a positive integer and @math be a positive even integer. Let @math be an @math order @math -dimensional real weakly symmetric tensor and @math be a real weakly symmetric positive definite tensor of the same size. @math is called a @math -eigenvalue of @math if @math for some @math . In this paper, we introduce two unconstrained optimization problems and obtain some variational characterizations for the minimum and maximum @math --eigenvalues of @math . Our results extend Auchmuty's unconstrained variational principles for eigenvalues of real symmetric matrices. This unconstrained optimization approach can be used to find a Z-, H-, or D-eigenvalue of an even order weakly symmetric tensor. We provide some numerical results to illustrate the effectiveness of this approach for finding a Z-eigenvalue and for determining the positive semidefiniteness of an even order symmetric tensor.