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
Famous quotes containing the word usage:
“I am using it [the word ‘perceive’] here in such a way that to say of an object that it is perceived does not entail saying that it exists in any sense at all. And this is a perfectly correct and familiar usage of the word.”
—A.J. (Alfred Jules)
“...Often the accurate answer to a usage question begins, “It depends.” And what it depends on most often is where you are, who you are, who your listeners or readers are, and what your purpose in speaking or writing is.”
—Kenneth G. Wilson (b. 1923)
“Girls who put out are tramps. Girls who don’t are ladies. This is, however, a rather archaic usage of the word. Should one of you boys happen upon a girl who doesn’t put out, do not jump to the conclusion that you have found a lady. What you have probably found is a lesbian.”
—Fran Lebowitz (b. 1951)