Assessing the Perron-Frobenius Root of Symmetric Positive Semidefinite Matrices by the Adaptive Steepest Descent Method

We discuss the maximum eigenvalue problem which is fundamental in many cutting-edge research fields. We provide the necessary theoretical background required for applying the fully adaptive steepest descent method (or ASDM) to estimate the Perron-Frobenius root of symmetric positive semidefinite matrices. We reduce the problem of assessing the Perron-Frobenius root of a certain matrix to the problem of unconstrained optimization of the quadratic function associated with this matrix. We experimentally investigated the ability of ASDM to approximate the Perron-Frobenius root and carry out a comparative analysis of the obtained computational results with some others presented earlier in the literature. This study also provides some insight into the choice of parameters, which are computationally important, for ASDM. The study revealed that ASDM is suitable for estimating the Perron-Frobenius root of matrices regardless of whether or not their elements are positive and regardless of the dimension of these matrices.