Rayleigh quotient iteration

Algorithm

The algorithm is very similar to inverse iteration, but replaces the estimated eigenvalue at the end of each iteration with the Rayleigh quotient. Begin by choosing some value μ 0 as an initial eigenvalue guess for the Hermitian matrix A . An initial vector b 0 must also be supplied as initial eigenvector guess.

Calculate the next approximation of the eigenvector b i + 1 by

b i + 1 = ( A − μ i I ) − 1 b i | | ( A − μ i I ) − 1 b i | | ,
where I is the identity matrix, and set the next approximation of the eigenvalue to the Rayleigh quotient of the current iteration equal to
μ i = b i ∗ A b i b i ∗ b i .

To compute more than one eigenvalue, the algorithm can be combined with a deflation technique.

Note that for very small problems it is beneficial to replace the matrix inverse with the adjugate, which will yield the same iteration because it's equal to the inverse up to an irrelevant scale (the inverse of the determinant, specifically). The adjugate is easier to compute explicitly than the inverse (though the inverse is easier to apply to a vector for problems that aren't small), and is more numerically sound because it remains well defined as the eigenvalue converges.

Example

Consider the matrix

for which the exact eigenvalues are λ 1 = 3 + 5 , λ 2 = 3 − 5 and λ 3 = − 2 , with corresponding eigenvectors

(where φ = 1 + 5 2 is the golden ratio).

The largest eigenvalue is λ 1 ≈ 5.2361 and corresponds to any eigenvector proportional to v 1 ≈ [ 1 0.6180 1 ] .

We begin with an initial eigenvalue guess of

Then, the first iteration yields

the second iteration,

and the third,

from which the cubic convergence is evident.

Octave Implementation

The following is a simple implementation of the algorithm in Octave.