We present a new algorithm for solving an eigenvalue problem for a realsymmetric arrowhead matrix. The algorithm computes all eigenvalues and allcomponents 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 theinverse of the shifted matrix. Each eigenvalue and the correspondingeigenvector can be computed separately, which makes the algorithm adaptable forparallel computing. Our results extend to Hermitian arrowhead matrices, realsymmetric diagonal-plus-rank-one matrices and singular value decomposition ofreal triangular arrowhead matrices.
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