Rayleigh quotient iteration

Rayleigh quotient iteration is an eigenvalue algorithm that extends inverse iteration by using the Rayleigh quotient to obtain increasingly accurate eigenvalue estimates. It is an iterative method: it produces a sequence of approximate solutions that converges to a true solution in the limit. Convergence is very rapid, and in practice only a few iterations are needed for a reasonable approximation; for Hermitian or symmetric matrices the algorithm converges cubically, given an initial vector that is sufficiently close to an eigenvector of the matrix being analyzed. The algorithm is much like inverse iteration, except that at the end of each iteration the estimated eigenvalue is replaced by the Rayleigh quotient. One starts from an initial eigenvalue guess and an initial eigenvector guess, computes the next eigenvector approximation by solving with the shifted matrix and normalizing, and then sets the next eigenvalue estimate to the Rayleigh quotient. More than one eigenvalue can be found by combining it with a deflation technique, and for very small problems the matrix inverse can be replaced by the adjugate.