Power Iteration - The Method

The Method

The power iteration algorithm starts with a vector b0, which may be an approximation to the dominant eigenvector or a random vector. The method is described by the iteration

So, at every iteration, the vector bk is multiplied by the matrix A and normalized.

Under the assumptions:

  • A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues
  • The starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.

then:

  • A subsequence of converges to an eigenvector associated with the dominant eigenvalue

Note that the sequence does not necessarily converge. It can be shown that:
where: is an eigenvector associated with the dominant eigenvalue, and . The presence of the term implies that does not converge unless Under the two assumptions listed above, the sequence defined by: converges to the dominant eigenvalue.

This can be run as a simulation program with the following simple algorithm:

for each(''simulation'') { // calculate the matrix-by-vector product Ab for(i=0; iThe value of norm converges to the dominant eigenvalue, and the vector b to an associated eigenvector.

Note: The above code assumes real A,b. To handle complex; A becomes conj(A), and tmp*tmp becomes conj(tmp)*tmp

This algorithm is the one used to calculate such things as the Google PageRank.

The method can also be used to calculate the spectral radius of a matrix by computing the Rayleigh quotient

Read more about this topic:  Power Iteration

Famous quotes containing the word method:

    “I have a new method of poetry. All you got to do is look over your notebooks ... or lay down on a couch, and think of anything that comes into your head, especially the miseries.... Then arrange in lines of two, three or four words each, don’t bother about sentences, in sections of two, three or four lines each.”
    —Allen Ginsberg (b. 1926)

    “Women are denied masturbation even more severely than men and that’s another method of control—they’re not taught to please themselves.... Most women—it takes them a while to warm up to the “situation” but once they get into it, I’m sure they’re going to get just as hooked as—well, everyone I know is!”
    —Lydia Lunch (b. 1959)