If we can factor for a shift , we can generalize the inverse iteration to compute eigenvalues of the problem (5.1) near as shown in Algorithm 5.2.

Here are some comments on this algorithm.
In step (3) we multiply by , while in step (5) we solve
a system with the shifted matrix . In a practical case, we perform
an initial sparse Gaussian elimination and use
its and factors while operating.
Step (4) makes sure that the vector is of unit
norm. The quantity is a
Rayleigh quotient,

where is the current approximate eigenvalue. The inverse iteration converges under conditions similar to those for the standard HEP.

Susan Blackford 2000-11-20