Rayleigh–Ritz method

The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, which originated in the context of solving physical boundary-value problems. It is named after Lord Rayleigh and Walther Ritz. In this method, an infinite-dimensional linear operator is approximated by a finite-dimensional compression, enabling the use of a numerical eigenvalue algorithm.

It is used in all applications that involve approximating eigenvalues and eigenvectors, often under different names. In quantum mechanics, where a system of particles is described using a Hamiltonian, it uses trial wave functions to approximate the ground-state eigenfunction. In the context of the finite-element method, it is mathematically the same as the Ritz-Galerkin method. In mechanical and structural engineering, it is used to approximate the eigenmodes and resonant frequencies of a structure. A related adaption of Rayleigh-Ritz known as Hamiltonian truncation can be used in quantum field theory.