Rayleigh–Ritz method: Difference between revisions

Revision as of 13:26, 14 June 2024

Latest revision as of 11:06, 2 September 2025

(11 intermediate revisions by 8 users not shown)

Line 1:

{{Short description|Method for approximating eigenvalues}}

{{Additional citations|date=June 2024}}

The '''Rayleigh–Ritz method''' is a direct numerical method of approximating [[eigenvalues and eigenvectors|eigenvalues]], originated in the context of solving physical [[boundary value problem]]s and named after [[Lord Rayleigh]] and [[Walther Ritz]].

Line 5 ⟶ 6:

In this method, an infinite-dimensional [[linear operator]] is approximated by a finite-dimensional [[Dilation (operator theory)|compression]], on which we can use an [[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 (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 and structural engineering to approximate the [[Normal mode|eigenmodes]] and [[Resonance|resonant frequencies]] of a structure.

== Naming and attribution ==

The name of the method and its origin story have been debated by historians.<ref name="Leissa">{{cite journal|last1=Leissa|first1=A.W.|title=The historical bases of the Rayleigh and Ritz methods|journal=Journal of Sound and Vibration|volume=287|issue=4–5|year=2005|pages=961–978| doi=10.1016/j.jsv.2004.12.021| bibcode=2005JSV...287..961L| url=https://www.sciencedirect.com/science/article/abs/pii/S0022460X05000362 |url-access=subscription}}</ref><ref name="Ilanko">{{cite journal|last1=Ilanko|first1=Sinniah|title=Comments on the historical bases of the Rayleigh and Ritz methods|journal=Journal of Sound and Vibration|volume=319|issue=1–2|year=2009|pages=731–733 | doi=10.1016/j.jsv.2008.06.001|bibcode=2009JSV...319..731I }}</ref> It has been called [[Ritz method]] after [[Walther Ritz]], since the numerical procedure has been published by [[Walther Ritz]] in 1908-1909. According to A. W. Leissa,<ref name="Leissa" /> [[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 and in another publication. This statement, although later disputed, and the fact that the method in the trivial case of a single vector results in the [[Rayleigh quotient]] make the case for the name ''Rayleigh–Ritz'' method. According to S. Ilanko,<ref name="Ilanko"/> citing [[Richard Courant]], both [[Lord Rayleigh]] and [[Walther Ritz]] independently conceived the idea of utilizing the equivalence between [[boundary value problem]]s of [[partial differential equations]] on the one hand and problems of the [[calculus of variations]] on the other hand for numerical calculation of the solutions, by substituting for the variational problems simpler approximating extremum problems in which a finite number of parameters need to be determined; see the article [[Ritz method]] for details. Ironically for the debate, the modern justification of the algorithm drops the [[calculus of variations]] in favor of the simpler and more general approach of [[orthogonal projection]] as in [[Galerkin method]] named after [[Boris Galerkin]], thus leading also to the [[Ritz-Galerkin method]] naming.{{cn|reason=need historian reference here|date=June 2024}}

== 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|url=https://arxiv.org/abs/math/0302145|author-link1=E. Brian Davies|year=2003|arxiv=math/0302145 |bibcode=2003math......2145D }}</ref>

* A space of [[Spline (mathematics)|splines]] (as in the [[Galerkin method]]);<ref name=sulimayers>{{cite book|last1=Süli|first1=Endre|author-link1=Endre Süli|last2=Mayers|first2=David|title=An Introduction to Numerical Analysis|publisher=[[Cambridge University Press]]|isbn=0521007941|year=2003}}</ref>

* A set of functions which approximate the [[eigenfunctions]] of the operator.<ref name=levitinshargorodsky>{{cite journal|last1=Levitin|first1=Michael|last2=Shargorodsky|first2=Eugene|title=Spectral pollution and second order relative spectra for self-adjoint operators|journal=IMA Journal of Numerical Analysis|urlyear=https://arxiv2004|volume=24 |issue=3 |pages=393–416 |doi=10.org1093/absimanum/24.3.393 |arxiv=math/0212087|year=2004 }}</ref>

One could use the orthonormal basis generated from the eigenfunctions of the operator, which will produce [[diagonal matrix|diagonal]] approximating matrices, but in this case we would have already had to calculate the spectrum.

Line 26 ⟶ 27:

and solve the eigenvalue problem <math>T_{\mathcal{L}}u = \lambda u</math>. It can be shown that the matrix <math>T_{\mathcal{L}}</math> is the [[Dilation (operator theory)|compression]] of <math>T</math> to <math>\mathcal{L}</math>.<ref name=daviesplum />

For [[differential operators]] (such as [[Sturm-Liouville problem|Sturm-Liouville operators]]), the inner product <math>(\cdot, \cdot)</math> can be replaced by the [[weak formulation]] <math>\mathcal{A}(\cdot, \cdot)</math>.<ref name=sulimayers /><ref name=pryce>{{cite book|last1=Pryce|first1=John D.|title=Numerical Solution of Sturm-Liouville Problems|ISBNisbn=0198534159|publisher=Oxford University Press|year=1994}}</ref>

If a subset of the orthonormal basis was used to find the matrix, the eigenvectors of <math>T_{\mathcal{L}}</math> will be [[linear combinations]] of orthonormal basis functions, and as a result they will be approximations of the eigenvectors of <math>T</math>.<ref name=arfkenweber>{{cite book|last1=Arfken|first1 = George B.|author-link1=George B. Arfken|last2 = Weber| first2 = Hans J.|year = 2005|title= Mathematical Methods For Physicists|url= https://books.google.com/books?id=tNtijk2iBSMC&pg=PA83|edition= 6th|publisher=Academic Press| isbn=978-0-08-047069-6 }}</ref>

== 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.<ref name=daviesplum /><ref name=levitinshargorodsky /><ref>{{cite magazine|url=https://ima.org.uk/16912/unscrambling-the-infinite-can-we-compute-spectra/|last1=Colbrook|first1=Matthew|title=Unscrambling the Infinite: Can we Compute Spectra?|magazine=Mathematics Today|publisher=Institute of Mathematics and its Applications}}</ref> 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.<ref>{{cite journal|last1=Colbrook|first1=Matthew|last2=Roman|first2=Bogdan|last3=Hansen|first3=Anders|title=How to Compute Spectra with Error Control|url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.250201|journal=Physical Review Letters|year=2019|volume=122 |issue=25 |page=250201 |doi=10.1103/PhysRevLett.122.250201 |pmid=31347861 |bibcode=2019PhRvL.122y0201C }}</ref>

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]].<ref>{{cite journal|last1=Pokrzywa|first1=Andrzej|title=Method of orthogonal projections and approximation of the spectrum of a bounded operator|year=1979|journal=Studia Mathematica|volume=65 |pages=21–29 |doi=10.4064/sm-65-1-21-29 }}</ref><ref>{{cite journal|last1=Bögli|first1=Sabine|author1-link=Sabine Bögli|last2=Marletta|first2=Marco|last3=Tretter|first3=Christiane|author3-link=Christiane Tretter|title=The essential numerical range for unbounded linear operators|journal=Journal of Functional Analysis|year=2020|urlvolume=https://arxiv279 |doi=10.org/abs1016/j.jfa.2020.108509 |arxiv=1907.09599 }}</ref>

== 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 48 ⟶ 49:

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>.

Line 246 ⟶ 247:

=== In quantum physics ===

In quantum physics, where the spectrum of the [[Hamiltonian (quantum mechanics)|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.<ref name=arfkenweber /> In fact, for any system more complicated than a single hydrogen atom, there is no known exact solution for the spectrum of the Hamiltonian.<ref name=pryce />

In this case, ana [[ansatz|trial wave function]], <math>\Psi</math>, 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.

It can be shown that the ground state energy, <math>E_0</math>, satisfies an inequality:

Line 278 ⟶ 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 323 ⟶ 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 webarXiv|last1=Servadio|first1=Simone|last2=Arnas|first2=David|last3=Linares|first3=Richard|title=A Koopman Operator Tutorial with Orthogonal Polynomials|publisherdate=arXiv2021 |urlclass=https://arxivmath.org/abs/NA |eprint=2111.07485 }}</ref>

== The relationship with the finite element method ==

Line 337 ⟶ 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=1-611–61|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}}

Line 344 ⟶ 345:

*[https://web.archive.org/web/20081010161336/http://www.math.nps.navy.mil/~bneta/4311.pdf Course on Calculus of Variations, has a section on Rayleigh–Ritz method].

* [https://encyclopediaofmath.org/wiki/Ritz_method Ritz method] in the ''[[Encyclopedia of Mathematics]]''

*{{cite journal | title=From Euler, Ritz, and Galerkin to Modern Computing | last1=Gander | first1=Martin J.| last2=Wanner | first2=Gerhard | journal=SIAM Review | year=2012 | volume=54 | issue=4 | pages=627–666 | doi=10.1137/100804036| url=https://archive-ouverte.unige.ch/unige:171273 | citeseerx=10.1.1.297.5697 }}

{{DEFAULTSORT:Rayleigh-Ritz Method}}