Rayleigh–Ritz method: Difference between revisions

Revision as of 17:01, 23 March 2021

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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 (quantum mechanics)|Hamiltonian]], the [[Ritz method]] uses [[ansatz|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 engineering to approximate the [[Normal mode|eigenmodes]] of a physical system, such as finding the [[Resonance|resonant frequencies]] of a structure to guide appropriate [[Damping ratio|damping]].

== Algorithm description forFor matrices ==

In numerical linear algebra, the '''Rayleigh–Ritz method''' is commonly<ref name="TrefethenIII1997" /> applied to approximate an eigenvalue problem

:<math> A \textbf{x} = \lambda \textbf{x}</math>

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Another useful connection to the [[Rayleigh quotient]] is that <math>\mu_i=\rho(v_i)</math> for every Ritz pair <math>(\tilde{\lambda}_i,\tilde{\textbf{x}}_i)</math>, allowing to derive some properties of Ritz values <math>\mu_i</math> from the corresponding theory for the [[Rayleigh quotient]]. For example, if <math>A</math> 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 <math>A</math>.

=== Examples ===

The matrix

:<math>A = \begin{bmatrix}

2 & 0 & 0\\

0 & 2 & 1\\

0 & 1 & 2

\end{bmatrix}</math>

has eigenvalues <math>1, 2, 3</math> and the corresponding eigenvectors

:<math>\mathbf x_{\lambda=1} = \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix}, \quad \mathbf x_{\lambda=2} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad \mathbf x_{\lambda=3} = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}.</math>

Let us take

:<math>V = \begin{bmatrix}

0 & 0\\

1 & 0\\

0 & 1

\end{bmatrix},</math>

then

:<math>V^* A V = \begin{bmatrix}

2 & 1\\

1 & 2

\end{bmatrix}</math>

with eigenvalues <math>1, 3</math> and the corresponding eigenvectors

:<math>\mathbf y_{\mu=1} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \quad \mathbf y_{\mu=3} = \begin{bmatrix} 1 \\ 1 \end{bmatrix},</math>

so that the Ritz values are <math>1, 3</math> and the Ritz vectors are

:<math>\mathbf \tilde{x}_{\tilde{\lambda}=1} = \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix}, \quad \mathbf \tilde{x}_{\tilde{\lambda}=3} = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}.</math>

We observe that each one of the Ritz vectors is exactly one of the eigenvectors of <math>A</math> for the given <math>V</math> as well as the Ritz values give exactly two of the three eigenvalues of <math>A</math>. A mathematical explanation for the exact approximation is based on the fact that the [[column space]] of the matrix <math>V</math> happens to be exactly the same as the subspace spanned by the two eigenvectors <math>\mathbf x_{\lambda=1}</math> and <math>\mathbf x_{\lambda=3}</math> in this example.

== Derivation from calculus of variations ==