with , where . Here, it is understood that is a specified relative accuracy tolerance between and .

If is ``wanted,'' it is desirable to lock .
However, in order to accomplish this, it will be necessary to
arrange a transformation of the current Lanczos factorization
to one with a small subdiagonal to isolate .
This may be accomplished by constructing a orthogonal
matrix using Algorithm 4.9:

with .

The end result of these transformations is
Av_1 &=& v_1 + r , where
v_1^* r = 0,

A V_2 &=& V_2 T_2 + r e_k-1^* ,
where
.

This means that subsequent implicit restart takes place as if

with all the subsequent orthogonal transformations associated with implicit restart applied to and never disturbing the relation . In subsequent Lanczos steps, participates in the orthogonalization so that the selective orthogonalization recommended by Parlett and Scott [363,353] is accomplished automatically.