Content deleted Content added
authorlinks |
Link suggestions feature: 3 links added. |
||
| (2 intermediate revisions by 2 users not shown) | |||
Line 13:
== 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>
Line 38:
== 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:
Line 279:
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>
Line 324:
=== 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.<ref>{{cite arXiv|last1=Servadio|first1=Simone|last2=Arnas|first2=David|last3=Linares|first3=Richard|title=A Koopman Operator Tutorial with Orthogonal Polynomials|date=2021 |class=math.NA |eprint=2111.07485 }}</ref>
== The relationship with the finite element method ==
Line 338:
== Notes and references==
* {{cite journal|last=Ritz|first=Walther|author-link=Walther Ritz|title=Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik|journal=Journal für die Reine und Angewandte Mathematik|volume=135|pages=1–61|year=1909|doi=10.1515/crll.1909.135.1 |url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=261182|url-access=subscription}}
* {{cite journal|last=MacDonald|first=J. K.|title=Successive Approximations by the Rayleigh-Ritz Variation Method|journal=Phys. Rev.|volume=43|year=1933|issue=10 |pages=830–833 |doi=10.1103/PhysRev.43.830 |bibcode=1933PhRv...43..830M |url=http://journals.aps.org/pr/abstract/10.1103/PhysRev.43.830|url-access=subscription}}
{{Reflist}}
| |||