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{{merge|Ritz method|discuss=Talk:Ritz method#Merge proposal|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 [[
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 histroians.<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 [[
== Method ==
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* 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}}</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
* 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|url=https://arxiv.org/abs/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.
We now approximate <math>T</math> by <math>T_{\mathcal{L}}</math>, which is defined as the matrix with entries<ref name=daviesplum
<math display="block">(T_{\mathcal{L}})_{i,j} = (T \varphi_i, \varphi_j).</math>
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
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}}</ref>
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== Properties ==
=== Spectral pollution ===
It is possible for the
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}}</ref><ref>{{cite journal|last1=Bögli|first1=Sabine|last2=Marletta|first2=Marco|last3=Tretter|first3=Christiane|title=The essential numerical range for unbounded linear operators|journal=Journal of Functional Analysis|year=2020|url=https://arxiv.org/abs/1907.09599}}</ref>
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\end{bmatrix}.
</math>
Thus we already obtain the singular values 2 and 1 from <math>\Sigma</math> and from <math>\mathbf {U}</math> the corresponding two left singular vectors <math>u</math> as <math>[0, 1, 0, 0, 0]^*</math> and <math>[1, 0, 0, 0, 0]^*</math>, which span the column-space of the matrix <math>W</math>, explaining why the approximations are exact for the given <math>W</math>.
Finally, step 3 computes the matrix <math>V_h = \mathbf {V}_h W^*</math>
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Thus, for the given matrix <math>W</math> with its column-space that is spanned by two exact right singular vectors, we determine these right singular vectors, as well as the corresponding left singular vectors and the singular values, all exactly. For an arbitrary matrix <math>W</math>, we obtain approximate singular triplets which are optimal given <math>W</math> in the sense of optimality of the Rayleigh–Ritz method.
== Applications and examples ==
=== 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
=== In dynamical systems ===
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