Computing accurate eigensystems of scaled diagonally dominant matrices
SIAM Journal on Numerical Analysis
A Divide-and-Conquer Algorithm for the Symmetric TridiagonalEigenproblem
SIAM Journal on Matrix Analysis and Applications
Applied numerical linear algebra
Applied numerical linear algebra
SIAM Journal on Matrix Analysis and Applications
The symmetric eigenvalue problem
The symmetric eigenvalue problem
Quadratic Residual Bounds for the Hermitian Eigenvalue Problem
SIAM Journal on Matrix Analysis and Applications
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
The Multishift QR Algorithm. Part II: Aggressive Early Deflation
SIAM Journal on Matrix Analysis and Applications
The Effect of Aggressive Early Deflation on the Convergence of the QR Algorithm
SIAM Journal on Matrix Analysis and Applications
Structured Hölder Condition Numbers for Multiple Eigenvalues
SIAM Journal on Matrix Analysis and Applications
Refined Perturbation Bounds for Eigenvalues of Hermitian and Non-Hermitian Matrices
SIAM Journal on Matrix Analysis and Applications
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We derive new perturbation bounds for eigenvalues of Hermitian matrices with block tridiagonal structure. The main message of this paper is that an eigenvalue is insensitive to blockwise perturbation, if it is well-separated from the spectrum of the diagonal blocks nearby the perturbed blocks. Our bound is particularly effective when the matrix is block-diagonally dominant and graded. Our approach is to obtain eigenvalue bounds via bounding eigenvector components, which is based on the observation that an eigenvalue is insensitive to componentwise perturbation if the corresponding eigenvector components are small. We use the same idea to explain two well-known phenomena, one concerning aggressive early deflation used in the symmetric tridiagonal QR algorithm and the other concerning the extremal eigenvalues of Wilkinson matrices.