A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem

An algorithm is presented for computing the eigendecomposition of a symmetric rank-one modification of a symmetric matrix whose eigendecomposition is known. Previous algorithms for this problem suffer a potential loss of orthogonality among the computed eigenvectors, unless extended precision arithmetic is used. This algorithm is based on a novel, stable method for computing the eigenvectors. It does not require extended precision and is as efficient as previous approaches.