A global quadratic speed-up for computing the principal eigenvalue of Perron-like operators

2021/11/24 by Dong Li, Jianan Li, Li, Dong +1
Computer Science · Mathematics · #Benford’s Law and Fraud Detection #Mathematical Approximation and Integration #Matrix Theory and Algorithms #cs.NA #math.AP #math.NA

paper · pdf · doi:10.48550/arxiv.2111.12642

18 pages

arxiv created 2021/11/24 · arxiv updated 2021/11/25

Abstract

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.

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