New Schemes for Solving the Principal Eigenvalue Problems of Perron-like Matrices via Polynomial Approximations of Matrix Exponentials

A real square matrix is Perron-like if it has a real eigenvalue $s$, called the principal eigenvalue of the matrix, and $\mbox{Re}\,\mu<s$ for any other eigenvalue $\mu$. 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.