Inverse power iteration

In numerical analysis, inverse iteration (also known as the inverse power method) is an iterative eigenvalue algorithm. It allows one to find an approximate eigenvector when an approximation to a corresponding eigenvalue is already known. The method is conceptually similar to the power method. It appears to have originally been developed to compute resonance frequencies in the field of structural mechanics.

The inverse power iteration algorithm starts with an approximation ${\displaystyle \mu }$ for the eigenvalue corresponding to the desired eigenvector and a vector b₀, either a randomly selected vector or an approximation to the eigenvector. The method is described by the iteration

where C_k are some constants usually chosen as ${\displaystyle C_{k}=\|(A-\mu I)^{-1}b_{k}\|.}$ Since eigenvectors are defined up to multiplication by constant, the choice of C_k can be arbitrary in theory; practical aspects of the choice of ${\displaystyle C_{k}}$ are discussed below.

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