An algorithm is proposed for calculating the eigenvectors of a diagonally dominant matrix all of whose elements are known to high relative accuracy. Eigenvectors corresponding to pathologically close eigenvalues are treated by computing the invariant subspace that they span. If the off-diagonal elements of the matrix are sufficiently small, the method is superior to standard techniques, and indeed it may produce a complete set of eigenvectors with an amount of work proportional to the square of the order of the matrix. An analysis is given of the effects of perturbations in the matrix on the eigenvectors.