Accurate singular values of bidiagonal matrices
SIAM Journal on Scientific and Statistical Computing
Computing an Eigenvector with Inverse Iteration
SIAM Review
On Computing an Eigenvector of a Tridiagonal Matrix. Part I: Basic Results
SIAM Journal on Matrix Analysis and Applications
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
For tridiagonals T replace T with LDLt
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. III: linear algebra
A framework for symmetric band reduction
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
Computing Approximate Eigenpairs of Symmetric Block Tridiagonal Matrices
SIAM Journal on Scientific Computing
Block tridiagonalization of "effectively" sparse symmetric matrices
ACM Transactions on Mathematical Software (TOMS)
The design and implementation of the MRRR algorithm
ACM Transactions on Mathematical Software (TOMS)
Hi-index | 7.29 |
New methods for computing eigenvectors of symmetric block tridiagonal matrices based on twisted block factorizations are explored. The relation of the block where two twisted factorizations meet to an eigenvector of the block tridiagonal matrix is reviewed. Based on this, several new algorithmic strategies for computing the eigenvector efficiently are motivated and designed. The underlying idea is to determine a good starting vector for an inverse iteration process from the twisted block factorizations such that a good eigenvector approximation can be computed with a single step of inverse iteration. An implementation of the new algorithms is presented and experimental data for runtime behaviour and numerical accuracy based on a wide range of test cases are summarized. Compared with competing state-of-the-art tridiagonalization-based methods, the algorithms proposed here show strong reductions in runtime, especially for very large matrices and/or small bandwidths. The residuals of the computed eigenvectors are in general comparable with state-of-the-art methods. In some cases, especially for strongly clustered eigenvalues, a loss in orthogonality of some eigenvectors is observed. This is not surprising, and future work will focus on investigating ways for improving these cases.