Full Reorthogonalization.

We could play it safe and do a full reorthogonalization either by modified or classical Gram-Schmidt, in the latter case computing the orthogonalization coefficients in a vector

$$h=V_j^{\ast} v_{j+1}$$

and then subtracting

$$v'_{j+1}=v_{j+1}-V_jh$$

to get the improved vector $v'_{j+1}$. If the norm is decreased by a nontrivial amount, say $\Vert v'_{j+1}\Vert<\frac{1}{\sqrt{2}}\Vert v_{j+1}\Vert$, this will have to be repeated, but this will hardly ever happen when treating properly computed Lanczos vectors.