Now we can rewrite Eq. (6) as $$ \mathbf{v} - k\mathbf{e} = 2\mathbf{u}( \mathbf{u}^T\mathbf{v}), $$ and taking the scalar product of this equation with itself and obtain $$ \begin{equation} 2( \mathbf{u}^T\mathbf{v})^2=(v^2\pm a_{21}v), \tag{7} \end{equation} $$ which finally determines $$ \mathbf{u}=\frac{\mathbf{v}-k\mathbf{e}}{2( \mathbf{u}^T\mathbf{v})}. $$ In solving Eq. (7) great care has to be exercised so as to choose those values which make the right-hand largest in order to avoid loss of numerical precision. The above steps are then repeated for every transformations till we have a tridiagonal matrix suitable for obtaining the eigenvalues.