Starting from a unit vector , Rayleigh quotient iteration sets , solvesand normalizes . The shift is updated from the new vector.
Let a real symmetric matrix have simple eigenpairs and letThen its Rayleigh quotient isAfter one shifted inverse solve and normalization, the relative coefficients areThus the eigenvector direction error is cubed at each sufficiently close iterate.
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Rayleigh Quotient Iteration is an iterative numerical method used for finding an eigenvalue and corresponding eigenvector of a matrix. It is particularly useful for finding the eigenvalue that is closest to a given initial estimate. This method can be seen as an extension of the standard power iteration and is more efficient, especially when searching for a dominant eigenvalue.