  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.
