The following problem: $$\bf A(\lambda)X=0$$ is called a nonlinear matrix eigenvalue problem where $\bf A(\lambda)$ is a square matrix with $\lambda$-dependent entries (nonlinear functions such as sine and exponential). The eigenvalues $\lambda$ and eigenvectors $\bf X$ are unknown. I am looking for a basic reference that explains efficient numerical methods to solve these problems.
Emphasis: The goal is to find as many eigenvalues as is needed one by one in ascending order.