  We consider a new algorithm in light of the min-max Collatz-Wielandt
formalism to compute the principal eigenvalue and the eigenvector
(eigen-function) for a class of positive Perron-Frobenius-like operators. Such
operators are natural generalizations of the usual nonnegative primitive
matrices. These have nontrivial applications in PDE problems such as computing
the principal eigenvalue of Dirichlet Laplacian operators on general domains.
We rigorously prove that for general initial data the corresponding numerical
iterates converge globally to the unique principal eigenvalue with quadratic
convergence. We show that the quadratic convergence is sharp with compatible
upper and lower bounds. We demonstrate the effectiveness of the scheme via
several illustrative numerical examples.
