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Wikipedia / 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 problemss. 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.

Naming and attribution

The origin of this technique has been debated by historians. It has been called Ritz method after Walther Ritz, who published the numerical procedure 1908 and 1909. 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, Theory of Sound (1877), and elsewhere. According to Richard Courant, both Lord Rayleigh and Walther Ritz independently invented the idea of utilizing the equivalence between the boundary-value problems of partial differential equations on the one hand and problems of the calculus of variations on the other to numerically compute the solutions, by substituting for the variational problems simpler approximating extremum problems in which only a finite number of parameters need to be determined. Historian Jesper Lützen, who had examined the unpublished manuscripts of Joseph Liouville, discovered that Liouville had already known of this method as early as 1845.

Method

be a linear operator on a Hilbert space, with inner product. Now consider a finite set. 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.

, which is defined as the matrix with entries

is the compression.

For differential operators (such as Sturm-Liouville operators can be replaced by the weak formulation.

will be linear combinations.

Properties

Spectral pollution

It is possible for the Rayleigh–Ritz method to produce values which do not converge to actual values in the spectrum of the operator as the truncation gets large. These values are known as spectral pollution. In some cases (such as for the Schrödinger equation), there is no approximation which both includes all eigenvalues of the equation, and contains no pollution.

The spectrum of the compression (and thus pollution) is bounded by the numerical range of the operator; in many cases it is bounded by a subset of the numerical range known as the essential numerical range.

For matrix eigenvalue problems

In numerical linear algebra, the Rayleigh–Ritz method is commonly with orthonormal columns. The matrix version of the algorithm is the most simple:

  1. , called the Ritz pairs, to eigenvalues and eigenvectors.

, the Rayleigh–Ritz method determines the accuracy of such an approximation for every Ritz pair.

is a scalar that is equal to the Rayleigh quotient itself. Thus, the Rayleigh–Ritz method turns into computing of the Rayleigh quotient.

Another useful connection to the Rayleigh quotient from the corresponding theory for the Rayleigh quotient is a Hermitian matrix, its Rayleigh quotient.

Example

. A mathematical explanation for the exact approximation is based on the fact that the column space in this example.

For matrix singular value problems

Truncated singular value decomposition (SVD) in numerical linear algebra can also use the Rayleigh–Ritz method in given subspaces by turning the singular value problem into an eigenvalue problem.

Using the normal matrix

. Having found one set (left of right) of approximate singular vectors and singular values by applying naively the Rayleigh–Ritz method to the Hermitian normal matrix. However, the division is unstable or fails for small or zero singular values.

An alternative approach, e.g., defining the normal matrix with orthonormal. This interpretation allows simple simultaneous calculation of both left and right approximate singular vectors as follows.

  1. .

  2. Compute the thin, or economy-sized, SVD.

  3. singular vectors.

  4. representing an approximate Truncated singular value decomposition (SVD).

is an output of an eigenvalue solver, e.g., such as LOBPCG.

Example

and the corresponding thin SVD are the corresponding right singular vectors.

corresponding to the singular values 1 and 2.

and on step 2 its thin SVD.

in the sense of optimality of the Rayleigh–Ritz method.

Applications and examples

In quantum physics

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. In fact, for any system more complicated than a single hydrogen atom, there is no known exact solution for the spectrum of the Hamiltonian.

In this case, a trial wave function, is tested on the system. This trial function is selected to meet boundary conditions (and any other physical constraints). The exact function is not known; the trial function contains one or more adjustable parameters, which are varied to find a lowest energy configuration.

That is, the ground-state energy is less than this value. The trial wave-function will always give an expectation value larger than or equal to the ground-energy.

If the trial wave function is known to be orthogonal to the ground state, then it will provide a boundary for the energy of some excited state.

The Ritz ansatz function is a linear combination of N

The basis functions are usually not orthogonal, so that the overlap matrix S zero, the following equality is obtained for every k = 1, 2, ..., N which leads to a set of N secular equations

are unknown. With respect to c, this is a homogeneous set of linear equations, which has a solution when the determinant which in turn is true only for N. Furthermore, since the Hamiltonian is a hermitian operator, the H matrix is also hermitian, will be the best approximation to the ground state for the basis functions used. The remaining N-1 energies are estimates of excited state energies. An approximation for the wave function of state i from the corresponding secular equation.

In mechanical engineering

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. It is an extension of Rayleigh's method. It can also be used for finding buckling loads and post-buckling behaviour for columns.

. Next, find the total energy of the system, consisting of a kinetic energy term and a potential energy term. The kinetic energy term involves the square of the time derivative

By conservation of energy which is also known as the Rayleigh quotient will be close to the lowest possible actual eigenmode of the system. Thus, this finds the lowest eigenfrequency. If we find eigenmodes orthogonal to this approximated lowest eigenmode, we can approximately find the next few eigenfrequencies as well.

are the stiffness matrix and mass matrix of a discrete system respectively.

This gives a solution for the first N eigenfrequencies and eigenmodes of the system, with N being the number of approximating functions.

Simple case of double spring-mass system

The following discussion uses the simplest case, where the system has two lumped springs and two lumped masses, and only two mode shapes are assumed. Hence and .

A mode shape is assumed for the system, with two terms, one of which is weighted by a factorB, e.g. Y =[1,1]+B[1,−1]. Simple harmonic motion theory says that the velocity at the time when deflection is zero, is the angular frequency times the deflection (y) at time of maximum deflection. In this example the kinetic energy etc., and the potential energy (PE) for each spring etc.

The overall amplitude of the mode shape cancels out from each side, always. That is, the actual size of the assumed deflection does not matter, just the mode shape.

, in terms of B, which can be differentiated 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.

This method could be used iteratively, adding additional mode shapes to the previous best solution, or you can build up a long expression with many Bs and many mode shapes, and then differentiate them partially.

In dynamical systems

The Koopman operator allows a finite-dimensional nonlinear system to be encoded as an infinite-dimensional linear system. 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.

The relationship with the finite element method

is precisely the stiffness matrix is the mass matrix is a discretized eigenvector.

See also

Notes and references