A real square matrix is Perron-like if it has a real eigenvalue s, called the principal eigenvalue of the matrix, and Re µ < s for any other eigenvalue µ. Nonnegative matrices and symmetric ones are typical examples of this class of matrices. The main purpose of this paper is to develop a set of new schemes to compute the principal eigenvalues of Perron-like matrices and the associated generalized eigenspaces by using polynomial approximations of matrix exponentials. Numerical examples show that these schemes are effective in practice.

In this paper, we present some generalized versions of the Krein–Rutman theorem for sectorial operators. They are formulated in a fashion that can be easily applied to elliptic operators. Another feature of these generalized versions is that they contain some information on the generalized eigenspaces associated with nonprincipal eigenvalues, which are helpful in the study of the dynamics of evolution equations in ordered Banach spaces.

We present an iterative algorithm based on the concept of the forward and backward observers to reconstruct the initial state of the linear system with polynomial coefficients. Its discussions of the convergence in the continuous space, semi-discrete space and full-discrete space are considered. Finally numerical examples are given to illustrate the effectiveness of our algorithm.

In this article, we use a finite difference scheme to discretize the Cahn-Hilliard equation with the space step size h . We first prove that this semidiscrete system inherits two important properties, called the conservation of mass and the decrease of the total energy, from the original equation. Then, we show that the semidiscrete system has an attractor on a subspace of ℝ N + 1 . Finally, the convergence of attractors is established as the space step size h of the semidiscrete Cahn-Hilliard equation tends to 0.