Editing Inverse iteration (section)

In [[numerical analysis]], '''inverse iteration''' (also known as the ''inverse power method'') is an [[Iterative method|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.<ref name=Pohlhausen>Ernst Pohlhausen, ''Berechnung der Eigenschwingungen statisch-bestimmter Fachwerke'', ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik 1, 28-42 (1921).</ref>

The inverse power iteration algorithm starts with an approximation <math>\mu</math> for the [[eigenvalue]] corresponding to the desired [[eigenvector]] and a vector <math>b_0</math>, either a randomly selected vector or an approximation to the eigenvector. The method is described by the iteration

<math display="block"> b_{k+1} = \frac{(A - \mu I)^{-1}b_k}{C_k}, </math>

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

At every iteration, the vector <math>b_k</math> is multiplied by the matrix <math>(A - \mu I)^{-1}</math> and normalized.
It is exactly the same formula as in the [[power method]], except replacing the matrix <math>A</math> by <math>(A - \mu I)^{-1}. </math>
The closer the approximation <math>\mu</math> to the eigenvalue is chosen, the faster the algorithm converges; however, incorrect choice of  <math>\mu</math> can lead to slow convergence or to the convergence to an eigenvector other than the one desired. In practice, the method is used when a good approximation for the eigenvalue is known, and hence one needs only few (quite often just one) iterations.