Accurate eigenvalue decomposition of arrowhead matrices and applications
This work provides a more accurate and efficient method for eigenvalue decomposition of arrowhead matrices, which are important in applications like signal processing and control theory.
The paper presents an O(n²) algorithm for computing eigenvalues and eigenvectors of real symmetric arrowhead matrices with high relative accuracy, extending to related matrix types. The algorithm enables separate computation of each eigenpair, facilitating parallel computing.
We present a new algorithm for solving an eigenvalue problem for a real symmetric arrowhead matrix. The algorithm computes all eigenvalues and all components of the corresponding eigenvectors with high relative accuracy in $O(n^{2})$ operations. The algorithm is based on a shift-and-invert approach. Double precision is eventually needed to compute only one element of the inverse of the shifted matrix. Each eigenvalue and the corresponding eigenvector can be computed separately, which makes the algorithm adaptable for parallel computing. Our results extend to Hermitian arrowhead matrices, real symmetric diagonal-plus-rank-one matrices and singular value decomposition of real triangular arrowhead matrices.