Editing Rayleigh quotient iteration (section)

== 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 <math>\mu_0</math> as an initial eigenvalue guess for the Hermitian matrix <math>A</math>. An initial vector <math>b_0</math> must also be supplied as initial eigenvector guess.

Calculate the next approximation of the eigenvector <math>b_{i+1}</math> by
<math display="block">
b_{i+1} = \frac{(A-\mu_i I)^{-1}b_i}{\|(A-\mu_i I)^{-1}b_i\|},
</math>
where <math>I</math> is the identity matrix,
and set the next approximation of the eigenvalue to the Rayleigh quotient of the current iteration equal to<br>
<math display="block">
\mu_{i+1} = \frac{b^*_{i+1} A b_{i+1}}{b^*_{i+1} b_{i+1}}.
</math>

To compute more than one eigenvalue, the algorithm can be combined with a deflation technique.{{Citation needed|date=February 2025}}

Note that for very small problems it is beneficial to replace the [[matrix inverse]] with the [[adjugate matrix|adjugate]], which will yield the same iteration because it is 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.