Open AccessFast verification for the Perron pair of an irreducible nonnegative matrix
Author(s) -
Shinya Miyajima
Publication year - 2021
Publication title -
the electronic journal of linear algebra
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.447
H-Index - 31
eISSN - 1537-9582
pISSN - 1081-3810
DOI - 10.13001/ela.2021.5181
Subject(s) - mathematics , matrix (chemical analysis) , nonnegative matrix , root (linguistics) , collatz conjecture , algorithm , symmetric matrix , combinatorics , eigenvalues and eigenvectors , linguistics , materials science , physics , philosophy , conjecture , quantum mechanics , composite material
Fast algorithms are proposed for calculating error bounds for a numerically computed Perron root and vector of an irreducible nonnegative matrix. Emphasis is put on the computational efficiency of these algorithms. Error bounds for the root and vector are based on the Collatz--Wielandt theorem, and estimating a solution of a linear system whose coefficient matrix is an $M$-matrix, respectively. We introduce a technique for obtaining better error bounds. Numerical results show properties of the algorithms.