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 situation the inverse iteration is the main and probably the only method to use. So typically the method is used in combination with some other methods which allows to find 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.
Dominant eigenvector. The dominant eigenvalue can be easily estimated for any matrix. For any induced norm it is true that for any eigenvalue . 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 may be millions matrices per second. In such applications typically the statistics of matrices is known in advance and one can take as approximate eigenvalue the average eigenvalue for some large matrix sample, or better one calculates the mean ratio of the eigenvalue to the trace or the norm of the matrix and eigenvalue is estimated as trace or norm multiplied on the average value the their ratio. Clearly such method can be used with much care and only in situations when the mistake in calculations is allowed. Actually such idea can be combined with other methods to avoid too big errors.
Read more about this topic: Inverse Iteration
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