Rayleigh–Ritz method: Difference between revisions

Revision as of 18:40, 19 June 2025

Latest revision as of 16:39, 17 August 2025

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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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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>