Forward stable eigenvalue decomposition of rankone modifications of diagonal matrices
Abstract
We present a new algorithm for solving an eigenvalue problem for a real symmetric matrix which is a rankone modification of a diagonal matrix. The algorithm computes each eigenvalue and all components of the corresponding eigenvector with high relative accuracy in $O(n)$ operations. The algorithm is based on a shiftandinvert approach. Only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision. Each eigenvalue and the corresponding eigenvector can be computed separately, which makes the algorithm adaptable for parallel computing. Our results extend to the complex Hermitian case. The algorithm is similar to the algorithm for solving the eigenvalue problem for real symmetric arrowhead matrices from: N. Jakovčević~Stor, I. Slapničar and J. L. Barlow, {Accurate eigenvalue decomposition of real symmetric arrowhead matrices and applications}, Lin. Alg. Appl., 464 (2015).
 Publication:

arXiv eprints
 Pub Date:
 May 2014
 DOI:
 10.48550/arXiv.1405.7537
 arXiv:
 arXiv:1405.7537
 Bibcode:
 2014arXiv1405.7537J
 Keywords:

 Mathematics  Numerical Analysis;
 65F15;
 65G50;
 1504;
 15B99
 EPrint:
 arXiv admin note: text overlap with arXiv:1302.7203