Editing Inverse iteration (section)

== Usage ==

The main application of the method is the situation when an approximation to an eigenvalue is found and one needs to find the corresponding approximate eigenvector. In such a situation the inverse iteration is the main and probably the only method to use.

=== Methods to find approximate eigenvalues ===

Typically, the method is used in combination with some other method which finds approximate eigenvalues: the standard example is the [[bisection eigenvalue algorithm]], another example is the [[Rayleigh quotient iteration]], which is actually the same inverse iteration with the choice of the approximate eigenvalue as the [[Rayleigh quotient]] corresponding to the vector obtained on the previous step of the iteration.

There are some situations where the method can be used by itself, however they are quite marginal.

=== Norm of matrix as approximation to the ''dominant'' eigenvalue ===

The dominant eigenvalue can be easily estimated for any matrix. 
For any [[Matrix norm#Induced norm|induced norm]] it is true that <math>\left \| A \right \| \ge |\lambda| , </math> for any eigenvalue <math>\lambda</math>. 
So taking the norm of the matrix as an approximate eigenvalue one can see that the method will converge to the dominant eigenvector.

=== Estimates based on statistics ===

In some real-time applications one needs to find  eigenvectors  for matrices with  a speed of millions of matrices per second. In such applications, typically the statistics of matrices is known in advance and one can take as an approximate eigenvalue the average eigenvalue for some large matrix sample.
Better, one may calculate the mean ratio of the eigenvalues to the trace or the norm of the  matrix and estimate the average eigenvalue as the trace or norm multiplied by the average value of that ratio.  Clearly such a method can be used only with discretion and only when high precision is not critical. 
This approach of estimating an average eigenvalue can be combined with other methods to avoid excessively large error.