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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

\begin{displaymath}h=V_j^{\ast} v_{j+1}\end{displaymath}

and then subtracting


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.

Susan Blackford 2000-11-20