Rayleigh Quotient Iteration

Example

Take the matrix

$A = \left[\begin{matrix} 1 & 2 & 3\\ 1 & 2 & 1\\ 3 & 2 & 1\\ \end{matrix}\right]$

with the largest eigenvalue and corresponding eigenvector of,

$\lambda_1 \approx 5.2361, v_1 \approx \left[ \begin{matrix} 0.64794 \\ 0.40045 \\ 0.64794 \\ \end{matrix}\right].$

We begin with an initial eigenvalue guess of and eigenvector of . Then the first iteration yields,

$b_1 \approx \left[\begin{matrix} -0.57927 \\ -0.57348 \\ -0.57927 \\ \end{matrix}\right], \mu_1 \approx 5.3355$

the second iteration,

$b_2 \approx \left[\begin{matrix} 0.64676 \\ 0.40422 \\ 0.64676 \\ \end{matrix}\right], \mu_2 \approx 5.2418$

and the third,

$b_3 \approx \left[\begin{matrix} -0.64793 \\ -0.40045 \\ -0.64793 \\ \end{matrix}\right], \mu_3 \approx 5.2361$

from which the cubic convergence is evident.

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Rayleigh Quotient Iteration - Example

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