Symmetric rank-1 update of a size-n symmetric eigenvalue problem

The matrix A to diagonalize is given in the form

A = D +uu^T,

where D is a diagonal matrix and u is a column-vector.

Given the diagonal elements of the matrix D and the elements of the vector u find the eigenvalues of the matrix A using only O(n²) operations (see section "Eigenvalues of updated matrix" in the book).