Solving a nonlinear spectral problem for a matrix

This paper examines the solving of the eigenvalue problem for a matrix $M(\lambda)$ with a nonlinear occurrence of the spectral parameter. Two methods are suggested for replacing the equation $\det M(\lambda)=0$ by a scalar equation $f(\lambda)=0$. Here the function $f(\lambda)$ is not written formally, but a rule for computing $f(\lambda)$ at a fixed point of the domain in which the desired roots lie is indicated. Mьller's method is used to solve the equation $f(\lambda)=0$. The eigenvalue found is refined by Newton's method based on the normalized expansion of matrix $M(\lambda)$ and the linearly independent vectors corresponding to it are computed. An ALGOL program and test examples are presented.