The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz.
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, the Ritz method uses trial wave functions to approximate the ground state eigenfunction with the lowest energy. In the finite element method context, mathematically the same algorithm is commonly called the Ritz-Galerkin method. The Rayleigh–Ritz method or Ritz method terminology is typical in mechanical and structural engineering to approximate the eigenmodes and resonant frequencies of a structure.
Naming and attribution
The name Rayleigh–Ritz is being debated vs. the Ritz method after Walther Ritz, since the numerical procedure has been published by Walther Ritz in 1908-1909. According to A. W. Leissa, Lord Rayleigh wrote a paper congratulating Ritz on his work in 1911, but stating that he himself had used Ritz's method in many places in his book and in another publication. This statement, although later disputed, and the fact that the method in the trivial case of a single vector results in the Rayleigh quotient make the arguable misnomer persist. According to S. Ilanko, citing Richard Courant, both Lord Rayleigh and Walther Ritz independently conceived the idea of utilizing the equivalence between boundary value problems of partial differential equations on the one hand and problems of the calculus of variations on the other hand for numerical calculation of the solutions, by substituting for the variational problems simpler approximating extremum problems in which a finite number of parameters need to be determined; see the article Ritz method for details. Ironically for the debate, the modern justification of the algorithm drops the calculus of variations in favor of the simpler and more general approach of orthogonal projection as in Galerkin method named after Boris Galerkin, thus leading also to the Ritz-Galerkin method naming.
For matrix eigenvalue problems
In numerical linear algebra, the Rayleigh–Ritz method is commonly applied to approximate an eigenvalue problem
for the matrix of size using a projected matrix of a smaller size , generated from a given matrix with orthonormal columns. The matrix version of the algorithm is the most simple:
Compute the matrix , where denotes the complex-conjugate transpose of
Solve the eigenvalue problem
Compute the Ritz vectors and the Ritz value
Output approximations , called the Ritz pairs, to eigenvalues and eigenvectors of the original matrix .
If the subspace with the orthonormal basis given by the columns of the matrix contains vectors that are close to eigenvectors of the matrix , the Rayleigh–Ritz method above finds Ritz vectors that well approximate these eigenvectors. The easily computable quantity determines the accuracy of such an approximation for every Ritz pair.
In the easiest case , the matrix turns into a unit column-vector , the matrix is a scalar that is equal to the Rayleigh quotient , the only solution to the eigenvalue problem is and , and the only one Ritz vector is itself. Thus, the Rayleigh–Ritz method turns into computing of the Rayleigh quotient if .
Another useful connection to the Rayleigh quotient is that for every Ritz pair , allowing to derive some properties of Ritz values from the corresponding theory for the Rayleigh quotient. For example, if is a Hermitian matrix, its Rayleigh quotient (and thus its every Ritz value) is real and takes values within the closed interval of the smallest and largest eigenvalues of .
Example
The matrix
has eigenvalues and the corresponding eigenvectors
Let us take
then
with eigenvalues and the corresponding eigenvectors
so that the Ritz values are and the Ritz vectors are
We observe that each one of the Ritz vectors is exactly one of the eigenvectors of for the given as well as the Ritz values give exactly two of the three eigenvalues of . A mathematical explanation for the exact approximation is based on the fact that the column space of the matrix happens to be exactly the same as the subspace spanned by the two eigenvectors and in this example.