Efficient initials for computing maximal eigenpair

TL;DR

This paper presents efficient initial estimates for inverse iteration algorithms to compute the maximal eigenpair of real matrices, improving stability and efficiency by leveraging analytic eigenvalue estimates and eigenvector approximations.

Contribution

It introduces novel initial estimates based on analytic bounds that enhance the stability and efficiency of inverse iteration for maximal eigenpair computation.

Findings

Initial estimates prevent algorithm collapse

Estimates are computationally efficient

Applicable to stochastic stability analysis

Abstract

This paper introduces some efficient initials for a well-known algorithm (an inverse iteration) for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algorithm but are also unexpectedly efficient. The initials presented here are based on our analytic estimates of the maximal eigenvalue and a mimic of its eigenvector for many years of accumulation in the study of stochastic stability speed. In parallel, the same problem for computing the next to the maximal eigenpair is also studied.