Power iteration

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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 = λv. The algorithm is also known as the Von Mises iteration.^[1]

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 b₀, 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 $\left(b_{{k}}\right)$ converges to an eigenvector associated with the dominant eigenvalue

Note that the sequence $\left(b_{{k}}\right)$ 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 $\|r_{{k}}\|\rightarrow 0$ . The presence of the term $e^{{i\phi _{{k}}}}$ implies that $\left(b_{{k}}\right)$ does not converge unless $e^{{i\phi _{{k}}}}=1$ Under the two assumptions listed above, the sequence $\left(\mu _{{k}}\right)$ defined by: $\mu _{{k}}={\frac {b_{{k}}^{{*}}Ab_{{k}}}{b_{{k}}^{{*}}b_{{k}}}}$ 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; i<n; i++) {
         tmp[i] = 0;
         for (j=0; j<n; j++)
              tmp[i] += A[i][j] * b[j]; // dot product of ith col in A with b
    }
 
    // calculate the length of the resultant vector
    norm_sq=0;
    for (k=0; k<n; k++)
         norm_sq += tmp[k]*tmp[k]; 
    norm = sqrt(norm_sq);
 
    // normalize b to unit vector for next iteration
    b = tmp/norm;
}

The value of norm converges to the absolute value of the dominant eigenvalue, and the vector b to an associated eigenvector.

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

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

${\frac {b_{k}^{\top }Ab_{k}}{b_{k}^{\top }b_{k}}}={\frac {b_{{k+1}}^{\top }b_{k}}{b_{k}^{\top }b_{k}}}.$

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 $1\times 1$ matrix ${\begin{bmatrix}\lambda _{{1}}\end{bmatrix}}$ , 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 $c_{{1}}\neq 0$ .

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.
$\displaystyle {\begin{array}{lcl}b_{{k}}&=&{\frac {A^{{k}}b_{{0}}}{\|A^{{k}}b_{{0}}\|}}\\&=&{\frac {\left(VJV^{{-1}}\right)^{{k}}b_{{0}}}{\|\left(VJV^{{-1}}\right)^{{k}}b_{{0}}\|}}\\&=&{\frac {VJ^{{k}}V^{{-1}}b_{{0}}}{\|VJ^{{k}}V^{{-1}}b_{{0}}\|}}\\&=&{\frac {VJ^{{k}}V^{{-1}}\left(c_{{1}}v_{{1}}+c_{{2}}v_{{2}}+\cdots +c_{{n}}v_{{n}}\right)}{\|VJ^{{k}}V^{{-1}}\left(c_{{1}}v_{{1}}+c_{{2}}v_{{2}}+\cdots +c_{{n}}v_{{n}}\right)\|}}\\&=&{\frac {VJ^{{k}}\left(c_{{1}}e_{{1}}+c_{{2}}e_{{2}}+\cdots +c_{{n}}e_{{n}}\right)}{\|VJ^{{k}}\left(c_{{1}}e_{{1}}+c_{{2}}e_{{2}}+\cdots +c_{{n}}e_{{n}}\right)\|}}\\&=&\left({\frac {\lambda _{{1}}}{|\lambda _{{1}}|}}\right)^{{k}}{\frac {c_{{1}}}{|c_{{1}}|}}{\frac {v_{{1}}+{\frac {1}{c_{{1}}}}V\left({\frac {1}{\lambda _{1}}}J\right)^{{k}}\left(c_{{2}}e_{{2}}+\cdots +c_{{n}}e_{{n}}\right)}{\|v_{{1}}+{\frac {1}{c_{{1}}}}V\left({\frac {1}{\lambda _{1}}}J\right)^{{k}}\left(c_{{2}}e_{{2}}+\cdots +c_{{n}}e_{{n}}\right)\|}}\end{array}}$
The expression above simplifies as $k\rightarrow \infty$
$\left({\frac {1}{\lambda _{{1}}}}J\right)^{{k}}={\begin{bmatrix}[1]&&&&\\&\left({\frac {1}{\lambda _{{1}}}}J_{{2}}\right)^{{k}}&&&\\&&\ddots &\\&&&\left({\frac {1}{\lambda _{{1}}}}J_{{m}}\right)^{{k}}\\\end{bmatrix}}\rightarrow {\begin{bmatrix}1&&&&\\&0&&&\\&&\ddots &\\&&&0\\\end{bmatrix}}$ as $k\rightarrow \infty$ .
The limit follows from the fact that the eigenvalue of ${\frac {1}{\lambda _{{1}}}}J_{{i}}$ is less than 1 in magnitude, so $\left({\frac {1}{\lambda _{{1}}}}J_{{i}}\right)^{{k}}\rightarrow 0$ as $k\rightarrow \infty$
It follows that:
${\frac {1}{c_{{1}}}}V\left({\frac {1}{\lambda _{1}}}J\right)^{{k}}\left(c_{{2}}e_{{2}}+\cdots +c_{{n}}e_{{n}}\right)\rightarrow 0$ as $k\rightarrow \infty$
Using this fact, $b_{{k}}$ can be written in a form that emphasizes its relationship with $v_{{1}}$ when k is large:
${\begin{matrix}b_{{k}}&=&\left({\frac {\lambda _{{1}}}{|\lambda _{{1}}|}}\right)^{{k}}{\frac {c_{{1}}}{|c_{{1}}|}}{\frac {v_{{1}}+{\frac {1}{c_{{1}}}}V\left({\frac {1}{\lambda _{1}}}J\right)^{{k}}\left(c_{{2}}e_{{2}}+\cdots +c_{{n}}e_{{n}}\right)}{\|v_{{1}}+{\frac {1}{c_{{1}}}}V\left({\frac {1}{\lambda _{1}}}J\right)^{{k}}\left(c_{{2}}e_{{2}}+\cdots +c_{{n}}e_{{n}}\right)\|}}&=&e^{{i\phi _{{k}}}}{\frac {c_{{1}}}{|c_{{1}}|}}v_{{1}}+r_{{k}}\end{matrix}}$ where $e^{{i\phi _{{k}}}}=\left(\lambda _{{1}}/|\lambda _{{1}}|\right)^{{k}}$ and $\|r_{{k}}\|\rightarrow 0$ as $k\rightarrow \infty$
The sequence $\left(b_{{k}}\right)$ is bounded, so it contains a convergent subsequence. Note that the eigenvector corresponding to the dominant eigenvalue is only unique up to a scalar, so although the sequence $\left(b_{{k}}\right)$ may not converge, $b_{{k}}$ is nearly an eigenvector of A for large k.

Alternatively, if A is diagonalizable, then the following proof yields the same result
Let λ₁, λ₂, …, λ_m be the <var>m</var> eigenvalues (counted with multiplicity) of <var>A</var> and let v₁, v₂, …, v_m be the corresponding eigenvectors. Suppose that $\lambda _{1}$ is the dominant eigenvalue, so that $|\lambda _{1}|>|\lambda _{j}|$ for $j>1$ .

The initial vector $b_{0}$ can be written:

$b_{0}=c_{{1}}v_{{1}}+c_{{2}}v_{{2}}+\cdots +c_{{m}}v_{{m}}.$

If $b_{0}$ is chosen randomly (with uniform probability), then c₁ ≠ 0 with probability 1. Now,

${\begin{array}{lcl}A^{{k}}b_{0}&=&c_{{1}}A^{{k}}v_{{1}}+c_{{2}}A^{{k}}v_{{2}}+\cdots +c_{{m}}A^{{k}}v_{{m}}\\&=&c_{{1}}\lambda _{{1}}^{{k}}v_{{1}}+c_{{2}}\lambda _{{2}}^{{k}}v_{{2}}+\cdots +c_{{m}}\lambda _{{m}}^{{k}}v_{{m}}\\&=&c_{{1}}\lambda _{{1}}^{{k}}\left(v_{{1}}+{\frac {c_{{2}}}{c_{{1}}}}\left({\frac {\lambda _{{2}}}{\lambda _{{1}}}}\right)^{{k}}v_{{2}}+\cdots +{\frac {c_{{m}}}{c_{{1}}}}\left({\frac {\lambda _{{m}}}{\lambda _{{1}}}}\right)^{{k}}v_{{m}}\right).\end{array}}$

The expression within parentheses converges to $v_{1}$ because $|\lambda _{j}/\lambda _{1}|<1$ for $j>1$ . On the other hand, we have

$b_{k}={\frac {A^{k}b_{0}}{\|A^{k}b_{0}\|}}.$

Therefore, $b_{k}$ converges to (a multiple of) the eigenvector $v_{1}$ . The convergence is geometric, with ratio

$\left|{\frac {\lambda _{2}}{\lambda _{1}}}\right|,$

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

Although the power iteration method approximates only one eigenvalue of a matrix, it remains useful for certain computational problems. For instance, Google uses it to calculate the PageRank of documents in their search engine.^[2] For matrices that are well-conditioned and as sparse as the Web matrix, the power iteration method can be more efficient than other methods of finding the dominant eigenvector.

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. Another variation of the power method that simultaneously gives n eigenvalues and eigenfunctions, as well as accelerated convergence as $\left|\lambda _{{n+1}}/\lambda _{1}\right|,$ is "Multiple extremal eigenpairs by the power method" in the Journal of Computational Physics Volume 227 Issue 19, October, 2008, Pages 8508-8522 (Also see pdf below for Los Alamos National Laboratory report LA-UR-07-4046)

References

↑ R. von Mises and H. Pollaczek-Geiringer, Praktische Verfahren der Gleichungsauflösung, ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik 9, 152-164 (1929).
↑ Ipsen, Ilse, and Rebecca M. Wills (5–8 May 2005). "Analysis and Computation of Google's PageRank". Fields Institute, Toronto, Canada. Unknown parameter |book= ignored (help); Check date values in: |date= (help)

External links

Power method, part of lecture notes on numerical linear algebra by E. Bruce Pitman, State University of New York.
Module for the Power Method
Los Alamos report LA-UR-07-4046 ""Multiple extremal eigenpairs by the power method"

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