Matrix analysis
ACM Transactions on Mathematical Software (TOMS)
Computing an Eigenvector with Inverse Iteration
SIAM Review
The symmetric eigenvalue problem
The symmetric eigenvalue problem
A new O (N(2)) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem
A new O (N(2)) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
The Anderson Model of Localization: A Challenge for Modern Eigenvalue Methods
SIAM Journal on Scientific Computing
Orthogonal Eigenvectors and Relative Gaps
SIAM Journal on Matrix Analysis and Applications
Glued Matrices and the MRRR Algorithm
SIAM Journal on Scientific Computing
On Large-Scale Diagonalization Techniques for the Anderson Model of Localization
SIAM Journal on Scientific Computing
The design and implementation of the MRRR algorithm
ACM Transactions on Mathematical Software (TOMS)
Algorithm 880: A testing infrastructure for symmetric tridiagonal eigensolvers
ACM Transactions on Mathematical Software (TOMS)
Performance and Accuracy of LAPACK's Symmetric Tridiagonal Eigensolvers
SIAM Journal on Scientific Computing
The Spectrum of a Glued Matrix
SIAM Journal on Matrix Analysis and Applications
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A normalized eigenvector, or, more interestingly, an invariant subspace, is localized if its significant entries are defined by just part(s) of the matrix and negligible elsewhere. This paper presents two new procedures to detect such localization in eigenvectors of a symmetric tridiagonal matrix. The procedures are intended for use before the actual eigenvector computation. If localization is found, one may reduce costs by computing the vectors just from the relevant matrix regions. Practical eigensolvers from numerical libraries such as LAPACK and ScaLAPACK already inspect a given tridiagonal $T$ for off-diagonal entries that are of small magnitude relative to the matrix norm. These so-called splitting points indicate that $T$ breaks into smaller blocks, each one defining a subset of eigenvalues and localized eigenvectors. However, localization can occur even when none of the off-diagonals is particularly small. Our study investigates this more complicated phenomenon in the context of invariant subspaces belonging to isolated eigenvalue clusters.