For the nonlinear matrix eigenvalue problem: Find \(\lambda\in {\mathbb{R}}\) and \(u\in {\mathbb{R}}^ N\setminus \{0\}\) with \(T(\lambda)u=0\) (T(\(\lambda)\) a real symmetric \(N\times N\) matrix), new algorithms are introduced. One of it is an extension of the well-known inverse iteration technique for linear matrix eigenvalue problems \((T(\lambda)=\lambda B-A)\), which converges (locally) at least superquadratic in the case of simple eigenvalues. Also the bisection method for eigenvalues is extended, so global convergence is ensured.
For boundary eigenvalue problems from technical dynamics a new method of discretization is developed. Following the finite element idea it turns out, that the dynamic stiffness element matrices \(Z_ e(\lambda)\) depend highly nonlinear from the eigenparameter \(\lambda\). \(Z_ e(\lambda)\) is replaced by a polynomial for instance of degree three by a least squares method. The resulting nonlinear matrix eigenvalue problem with \(T(\lambda)=\lambda^ 3D+\lambda^ 2C+\lambda B-A\) yields much better approximation qualities than the normally used linear problem.