Inverse Iteration

The inverse iteration is a numerical method for computing eigenvalues ​​and eigenvectors of matrices. It is a variant of the von Mises - iteration, with the aid, however, any eigenvalues ​​can be calculated. The process was introduced in 1944 by Helmut Wielandt in the stability analysis of structures that are small disturbances of known systems. In this case, a good approximation for the relevant properties are known values ​​, and rapid convergence is obtained.

Description

Is an eigenvalue of the square matrix, and x is the corresponding eigenvector, then the eigenvalue of the eigen vector x, where I is the identity matrix. Furthermore, it is an eigenvalue of the eigenvector x. Is now the eigenvalue of A that is closest, then the maximum absolute eigenvalue of. Applying now on the power method, so converges to the eigenvector corresponding to the eigenvalue of A, which is closest.

Instead mulitiplizieren as the power method in each step, the matrix to a vector, a linear system of equations will be solved, since it is not explicitly available. This matrix is conditioned poor, the closer is present, however, the error has a dominant component in the direction of this eigenvector, so that the method is of practical use.

Algorithm

Given a square matrix, a start vector and a shift is so regular. The starting vector can be chosen up to a Lebesgue -null set.

On the Rayleigh quotient obtained an approximation of the corresponding eigenvalue.

Extensions

If one chooses in each step a new shift we obtain the Rayleigh quotient iteration.