Algorithms for the Nonlinear Eigenvalue Problem

The following nonlinear eigenvalue problem is studied: Let T () be an n n matrix, whose elements are analytical functions of the complex number. We seek and vectors x and y, such that T () x = 0, and yH T () = 0. Several algorithms for the numerical solution of this problem are studied. These algorithms are extensions of algorithms for the linear eigenvalue problem such as inverse iteration and the QR algorithm, and algorithms that reduce the nonlinear problem into a sequence of linear problems. It is found that this latter method can be extended into a global strategy, finding a complete basis of eigenvectors in the cases where it is proved that such a basis exists. Numerical tests, performed in order to compare the different algorithms, are reported, and a few numerical examples illustrating their behavior are given.