Forward stable eigenvalue decomposition of rank-one modifications of diagonal matrices
This work provides a fast and accurate method for a specific structured eigenvalue problem, which is useful in applications like signal processing and numerical linear algebra.
The paper presents an O(n) algorithm for computing all eigenvalues and eigenvectors of a rank-one modification of a diagonal matrix with high relative accuracy, extending to complex Hermitian matrices.
We present a new algorithm for solving an eigenvalue problem for a real symmetric matrix which is a rank-one 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 shift-and-invert 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).