Rayleigh quotient iteration

Rayleigh quotient iteration is an eigenvalue algorithm which extends the idea of the inverse iteration by using the Rayleigh quotient to obtain increasingly accurate eigenvalue estimates. Rayleigh quotient iteration is an eigenvalue algorithm which extends the idea of the inverse iteration by using the Rayleigh quotient to obtain increasingly accurate eigenvalue estimates. Rayleigh quotient iteration is an iterative method, that is, it delivers a sequence of approximate solutions that converges to a true solution in the limit. Very rapid convergence is guaranteed and no more than a few iterations are needed in practice to obtain a reasonable approximation. The Rayleigh quotient iteration algorithm converges cubically for Hermitian or symmetric matrices, given an initial vector that is sufficiently close to an eigenvector of the matrix that is being analyzed. 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 {displaystyle mu _{0}} as an initial eigenvalue guess for the Hermitian matrix A {displaystyle A} . An initial vector b 0 {displaystyle b_{0}} must also be supplied as initial eigenvector guess. Calculate the next approximation of the eigenvector b i + 1 {displaystyle b_{i+1}} by b i + 1 = ( A − μ i I ) − 1 b i | | ( A − μ i I ) − 1 b i | | , {displaystyle b_{i+1}={frac {(A-mu _{i}I)^{-1}b_{i}}{||(A-mu _{i}I)^{-1}b_{i}||}},} where I {displaystyle 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 + 1 = b i + 1 ∗ A b i + 1 b i + 1 ∗ b i + 1 . {displaystyle mu _{i+1}={frac {b_{i+1}^{*}Ab_{i+1}}{b_{i+1}^{*}b_{i+1}}}.}