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.
In this method, an infinite-dimensional linear operator is approximated by a finite-dimensional compression, on which we can use an eigenvalue algorithm.
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.
According to S. Ilanko,[2] 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.
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.
Depending on the application these functions may be: One could use the orthonormal basis generated from the eigenfunctions of the operator, which will produce diagonal approximating matrices, but in this case we would have already had to calculate the spectrum.
will be linear combinations of orthonormal basis functions, and as a result they will be approximations of the eigenvectors of
[10][11] In numerical linear algebra, the Rayleigh–Ritz method is commonly[12] applied to approximate an eigenvalue problem
A mathematical explanation for the exact approximation is based on the fact that the column space of the matrix
This interpretation allows simple simultaneous calculation of both left and right approximate singular vectors as follows.
is an output of an eigenvalue solver, e.g., such as LOBPCG, approximating numerically selected eigenvectors of the normal matrix
where the columns of the first multiplier from the complete set of the left singular vectors of the matrix
, the diagonal entries of the middle term are the singular values, and the columns of the last multiplier transposed (although the transposition does not change it)
In quantum physics, where the spectrum of the Hamiltonian is the set of discrete energy levels allowed by a quantum mechanical system, the Rayleigh–Ritz method is used to approximate the energy states and wavefunctions of a complicated atomic or nuclear system.
[7] In fact, for any system more complicated than a single hydrogen atom, there is no known exact solution for the spectrum of the Hamiltonian.
This trial function is selected to meet boundary conditions (and any other physical constraints).
The trial wave-function will always give an expectation value larger than or equal to the ground-energy.
The basis functions are usually not orthogonal, so that the overlap matrix S has nonzero nondiagonal elements.
With respect to c, this is a homogeneous set of linear equations, which has a solution when the determinant of the coefficients to these unknowns is zero:
The Rayleigh–Ritz method is often used in mechanical engineering for finding the approximate real resonant frequencies of multi degree of freedom systems, such as spring mass systems or flywheels on a shaft with varying cross section.
The kinetic energy term involves the square of the time derivative of
This gives a solution for the first N eigenfrequencies and eigenmodes of the system, with N being the number of approximating functions.
Simple harmonic motion theory says that the velocity at the time when deflection is zero, is the angular frequency
That is, the actual size of the assumed deflection does not matter, just the mode shape.
is hoped to be the predicted fundamental frequency of the system because the mode shape is assumed, but we have found the lowest value of that upper bound, given our assumptions, because B is used to find the optimal 'mix' of the two assumed mode shape functions.
There are many tricks with this method, the most important is to try and choose realistic assumed mode shapes.
For example, in the case of beam deflection problems it is wise to use a deformed shape that is analytically similar to the expected solution.
A quartic may fit most of the easy problems of simply linked beams even if the order of the deformed solution may be lower.
The springs and masses do not have to be discrete, they can be continuous (or a mixture), and this method can be easily used in a spreadsheet to find the natural frequencies of quite complex distributed systems, if you can describe the distributed KE and PE terms easily, or else break the continuous elements up into discrete parts.
In general, both of these problems are difficult to solve, but for the latter we can use the Ritz-Galerkin method to approximate a solution.