Componentwise perturbation analysis for the generalized Schur decomposition

TL;DR

This paper develops linear and nonlinear componentwise perturbation bounds for the generalized Schur decomposition, providing new insights into the stability and sensitivity of the decomposition's factors and eigenvalues.

Contribution

It introduces basic perturbation vectors and linear bounds, deriving new perturbation bounds and condition numbers for the generalized Schur decomposition and eigenvalues.

Findings

Bounds are close to existing literature but do not include eigenvector info.

Numerical examples validate the bounds.

Nonlinear bounds are obtained via iterative methods.

Abstract

By defining two important terms called basic perturbation vectors and obtaining their linear bounds, we obtain the linear componentwise perturbation bounds for unitary factors and upper triangular factors of the generalized Schur decomposition. The perturbation bounds for the diagonal elements of the upper triangular factors and the generalized invariant subspace are also derived. From the former, we present an upper bound and a condition number of the generalized eigenvalue. Furthermore, with numerical iterative method, the nonlinear componentwise perturbation bounds of the generalized Schur decomposition are also provided. Numerical examples are given to test the obtained bounds. Among them, we compare our upper bound and condition number of the generalized eigenvalue with their counterparts given in the literature. Numerical results show that they are very close to each other but our results don't contain the information on the left and right generalized eigenvectors.