Rayleigh–Ritz method: Difference between revisions

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== Method ==

Let <math>T</math> be a [[linear operator]] on a [[Hilbert space]] <math>\mathcal{H}</math>, with [[inner product]] <math>(\cdot, \cdot)</math>. Now consider a [[finite set]] of functions <math>\mathcal{L} = \{\varphi_1, ...,\varphi_n\}</math>. Depending on the application these functions may be:

* A subset of the [[orthonormal basis]] of the original operator;<ref name=daviesplum>{{cite journal|last1=Davies|first1=E. B.|last2=Plum|first2=M.|title=Spectral Pollution|journal=IMA Journal of Numerical Analysis|author-link1=E. Brian Davies|year=2003|arxiv=math/0302145 |bibcode=2003math......2145D }}</ref>

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== For matrix eigenvalue problems ==

In [[numerical linear algebra]], the '''Rayleigh–Ritz method''' is commonly<ref name="TrefethenIII1997">{{cite book| last1=Trefethen| first1=Lloyd N. | last2= Bau, III|first2=David|title=Numerical Linear Algebra|url=https://books.google.com/books?id=JaPtxOytY7kC| year=1997| publisher=SIAM| isbn=978-0-89871-957-4|page=254}}</ref> applied to approximate an eigenvalue problem

<math display="block"> A \mathbf{x} = \lambda \mathbf{x}</math>

for the matrix <math> A \in \mathbb{C}^{N \times N}</math> of size <math>N</math> using a projected matrix of a smaller size <math>m < N</math>, generated from a given matrix <math> V \in \mathbb{C}^{N \times m} </math> with [[orthonormal]] columns. The matrix version of the algorithm is the most simple:

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If the subspace with the orthonormal basis given by the columns of the matrix <math> V \in \mathbb{C}^{N \times m} </math> contains <math> k \leq m </math> vectors that are close to eigenvectors of the matrix <math>A</math>, the '''Rayleigh–Ritz method''' above finds <math>k</math> Ritz vectors that well approximate these eigenvectors. The easily computable quantity <math> \| A \tilde{\mathbf{x}}_i - \tilde{\lambda}_i \tilde{\mathbf{x}}_i\|</math> determines the accuracy of such an approximation for every Ritz pair.

In the easiest case <math>m = 1</math>, the <math> N \times m </math> matrix <math>V</math> turns into a unit column-vector <math>v</math>, the <math> m \times m </math> matrix <math> V^* A V </math> is a scalar that is equal to the [[Rayleigh quotient]] <math>\rho(v) = v^*Av/v^*v</math>, the only <math>i = 1</math> solution to the eigenvalue problem is <math>y_i = 1</math> and <math>\mu_i = \rho(v)</math>, and the only one Ritz vector is <math>v</math> itself. Thus, the Rayleigh–Ritz method turns into computing of the [[Rayleigh quotient]] if <math>m = 1</math>.

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{\mathbf{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>.

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Consider the case whereby we want to find the resonant frequency of oscillation of a system. First, write the oscillation in the form,

<math display="block">y(x,t) = Y(x) \cos\omega t</math>

with an unknown mode shape <math>Y(x)</math>. 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]] of <math>y(x,t)</math> and thus gains a factor of <math>\omega ^2</math>. Thus, we can calculate the total energy of the system and express it in the following form:

<math display="block">E = T + V \equiv A[Y(x)] \omega^2\sin^2 \omega t + B[Y(x)] \cos^2 \omega t</math>