Computing accurate eigensystems of scaled diagonally dominant matrices
SIAM Journal on Numerical Analysis
Accurate singular values of bidiagonal matrices
SIAM Journal on Scientific and Statistical Computing
A Divide-and-Conquer Algorithm for the Symmetric TridiagonalEigenproblem
SIAM Journal on Matrix Analysis and Applications
Applied numerical linear algebra
Applied numerical linear algebra
ScaLAPACK user's guide
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
Faster Numerical Algorithms Via Exception Handling
IEEE Transactions on Computers
Accurate eigenvalues of a symmetric tri-diagonal matrix
Accurate eigenvalues of a symmetric tri-diagonal matrix
Perturbation Splitting for More Accurate Eigenvalues
SIAM Journal on Matrix Analysis and Applications
| Hi-index | 0.00 |
For the eigenvalues of a symmetric tridiagonal matrix $T$, the most accurate algorithms deliver approximations which are the exact eigenvalues of a matrix $\widetilde{T}$ whose entries differ from the corresponding entries of $T$ by small relative perturbations. However, for matrices with eigenvalues of different magnitudes, the number of correct digits in the computed approximations for eigenvalues of size smaller than $\Vert T\Vert_{2}$ depends on how well such eigenvalues are defined by the data. Some classes of matrices are known to define their eigenvalues to high relative accuracy but, in general, there is no simple way to estimate well the number of correct digits in the approximations. To remedy this, we propose a method that provides sharp bounds for the eigenvalues of $T$. We present some numerical examples to illustrate the usefulness of our method.