Rayleigh-Ritz method

The Rayleigh-Ritz method is a direct numerical method for approximating eigenvalues, and it began as a way of solving physical boundary-value problems. It replaces an infinite-dimensional linear operator by a finite-dimensional compression, built from a chosen set of basis or trial functions, so that an ordinary numerical eigenvalue algorithm can then be applied to the resulting matrix. In numerical linear algebra the matrix version solves a small eigenvalue problem inside a subspace and returns Ritz values and Ritz vectors as approximations to the eigenvalues and eigenvectors of the original matrix; truncated singular value decompositions can be approximated the same way. The same idea appears under other names in quantum mechanics, where trial wave functions approximate a ground state, in the finite-element method, where it matches the Ritz-Galerkin method, and in structural engineering, where it estimates resonant frequencies. It is named for Lord Rayleigh and Walther Ritz, and historians have debated how credit should be shared. A known limitation is spectral pollution, in which computed values fail to converge to the true spectrum as the truncation grows.