Inverse Iteration

In numerical analysis, inverse iteration 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 and is also known as the inverse 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 number which is an approximation for the eigenvalue corresponding to the searched eigenvector, and vector b₀, which is an approximation to the eigenvector or a random vector. The method is described by the iteration

where C_k are some constants usually chosen as 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 are discussed below.

So, at every iteration, the vector b_k is multiplied by the inverse of the matrix and normalized. It is exactly the same formula as in the power method modula change of matrix A, by The better approximation to the eigenvalue is chosen, the faster convergence one gets, however incorrect choice of can lead to slow convergence or to the convergence of a different eigenvector. Usually in practice the method is used when good approximation for the eigenvalue is known, and hence one needs only few (quite often just one) iteration.