shifted inverse iteration

The inverse power method aimed at a specific target eigenvalue by first subtracting a guess for it off every diagonal entry.

Shifted inverse iteration runs the inverse power method on $(A-\sigma I)^{-1}$ for a chosen shift $\sigma$ , in practice by solving $(A-\sigma I)\mathbf w=\mathbf x_k$ . Since $(A-\sigma I)^{-1}$ 's eigenvalues are $1/(\lambda_i-\sigma)$ , its DOMINANT eigenvalue belongs to whichever $\lambda_i$ sits CLOSEST to $\sigma$ -- so choosing $\sigma$ near a specific target eigenvalue makes the method converge to exactly that one, and the closer the shift, the faster the convergence.

Examples

If $A$ 's eigenvalues are $6,3,1$ and $\sigma=3.1$ is chosen, shifted inverse iteration converges to $3$ -- the eigenvalue closest to the shift, not the largest or smallest.