Power iteration

In mathematics, the power iteration is an eigenvalue algorithm: given a matrix "A", the algorithm will produce a number λ (the eigenvalue) and a nonzero vector "v" (the eigenvector), such that "Av" = λ"v".

The power iteration is a very simple algorithm. It does not compute a matrix decomposition, and hence it can be used when "A" is a very large sparse matrix. However, it will find only one eigenvalue (the one with the greatest absolute value) and it may converge only slowly.

The method

The power iteration algorithm starts with a vector "b"₀, which may be an approximation to the dominant eigenvector or a random vector. The method is described by the iteration:$$b_{k+1} = frac{Ab_k}{|Ab_k. So, at every iteration, the vector "b"_"k" 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 $$b_{0} has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.then:
*A subsequence of $$left( b_{k} ight) converges to an eigenvector associated with the dominant eigenvalueNote that the sequence $$left( b_{k} ight) does not necessarily converge. It can be shown that:
$$b_{k} = e^{i phi k} v_{1} + r_{k}where: $$v_{1} is an eigenvector associated with the dominant eigenvalue, and $$ | r_{k} | ightarrow 0. The presence of the term $$e^{i phi k}implies that $$left( b_{k} ight) does not converge unless $$e^{i phi k} = 1Under the two assumptions listed above, the sequence $$left( mu_{k} ight) defined by:$$mu_{k} = frac{b_{k}^{*}Ab_{k{b_{k}^{*}b_{kconverges to the dominant eigenvalue.

The method can also be used to calculate the spectral radius of a matrix by computing the Rayleigh quotient:$$frac{b_k^ op A b_k}{b_k^ op b_k} = frac{b_{k+1}^ op b_k}{b_k^ op b_k}.

Analysis

Let $$A be decomposed into its Jordan canonical form: $$A=VJV^{-1},where the first column of $$V is an eigenvector of $$A corresponding to the dominant eigenvalue $$lambda_{1}.Since the dominant eigenvalue of $$A is unique,the first Jordan block of $$J is the $$1 imes 1 matrix$$egin{bmatrix} lambda_{1} end{bmatrix} , where$$lambda_{1} is the largest eigenvalue of "A" in magnitude. The starting vector $$b_{0} can be written as a linear combination of the columns of V:$$b_{0} = c_{1}v_{1} + c_{2}v_{2} + cdots + c_{n}v_{n}.By assumption, $$b_{0} has a nonzero component in the direction of the dominant eigenvalue, so $$c_{1} e 0.

The computationally useful recurrence relation for $$b_{k+1}can be rewritten as: $$b_{k+1}=frac{Ab_{k{|Ab_{k}=frac{A^{k+1}b_{0{|A^{k+1}b_{0},where the expression: $$frac{A^{k+1}b_{0{|A^{k+1}b_{0} is more amenable to the following analysis.
$$egin{matrix}b_{k} &=& frac{A^{k}b_{0{| A^{k} b_{0} \ &=& frac{left( VJV^{-1} ight)^{k} b_{0{|left( VJV^{-1} ight)^{k}b_{0} \ &=& frac{ VJ^{k}V^{-1} b_{0{| V J^{k} V^{-1} b_{0} \ &=& frac{ VJ^{k}V^{-1} left( c_{1}v_{1} + c_{2}v_{2} + cdots + c_{n}v_{n} ight)} {| V J^{k} V^{-1} left( c_{1}v_{1} + c_{2}v_{2} + cdots + c_{n}v_{n} ight) \ &=& frac{ VJ^{k}left( c_{1}e_{1} + c_{2}e_{2} + cdots + c_{n}e_{n} ight)} {| V J^{k} left( c_{1}e_{1} + c_{2}e_{2} + cdots + c_{n}e_{n} ight) \ &=& left( frac{lambda_{1 ight)^{k} frac{c_{1. Therefore, $$b_k converges to (a multiple of) the eigenvector $$v_1. The convergence is geometric, with ratio:$$left| frac{lambda_2}{lambda_1} ight|, where $$lambda_2 denotes the second dominant eigenvalue. Thus, the method converges slowly if there is an eigenvalue close in magnitude to the dominant eigenvalue.

Applications

Power iteration is not used very much because it can find only the dominant eigenvalue. Nevertheless, the algorithm is very useful in some specific situations. For instance, Google uses it to calculate the page rank of documents in their search engine. [Ipsen, Ilse, and Rebecca M. Wills, " [http://www4.ncsu.edu/~ipsen/ps/slides_imacs.pdf Analysis and Computation of Google's PageRank] ", 7th IMACS International Symposium on Iterative Methods in Scientific Computing, Fields Institute, Toronto, Canada, 5–8 May 2005.]

Some of the more advanced eigenvalue algorithms can be understood as variations of the power iteration. For instance, the inverse iteration method applies power iteration to the matrix $$A^{-1}. Other algorithms look at the whole subspace generated by the vectors $$b_k. This subspace is known as the Krylov subspace. It can be computed by Arnoldi iteration or Lanczos iteration.

See also

* Rayleigh quotient iteration
* Inverse iteration

References

External links

* [http://www.math.buffalo.edu/~pitman/courses/mth437/na2/node17.html Power method] , part of lecture notes on numerical linear algebra by E. Bruce Pitman, State University of New York.
* [http://www.math.gatech.edu/~carlen/2605S04/Power.pdf Power method] on www.math.gatech.edu
* [http://math.fullerton.edu/mathews/n2003/PowerMethodMod.html Module for the Power Method]