Rayleigh–Ritz method

The Rayleigh–Ritz method is a direct numerical method of approximating <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a>, originated in the context of solving physical <a href="/facts/Boundary_value_problem/QEfmUqmP">boundary value problems</a> and named after <a href="/facts/Lord_Rayleigh/HcJyYdk5">Lord Rayleigh</a> and <a href="/facts/Walther_Ritz/jl9VvugR">Walther Ritz</a>.
In this method, an infinite-dimensional <a href="/facts/Linear_operator/Q1PBKNVV">linear operator</a> is approximated by a finite-dimensional <a href="/facts/Dilation_(operator_theory)/VaxIVN9T">compression</a>, on which we can use an <a href="/facts/Eigenvalue_algorithm/AYecadB0">eigenvalue algorithm</a>.
It is used in all applications that involve approximating <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues and eigenvectors</a>, often under different names. In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, where a system of particles is described using a <a href="/facts/Hamiltonian_(quantum_mechanics)/5XwK2HmD">Hamiltonian</a>, the Ritz method uses <a href="/facts/Ansatz/3epoB3Rg">trial wave functions</a> to approximate the ground state eigenfunction with the lowest energy. In the <a href="/facts/Finite_element_method/eUcfP1vn">finite element method</a> context, mathematically the same algorithm is commonly called the <a href="/facts/Ritz-Galerkin_method/oYm8Iylo">Ritz-Galerkin method</a>. The Rayleigh–Ritz method or Ritz method terminology is typical in mechanical and structural engineering to approximate the <a href="/facts/Normal_mode/XCsvQpuI">eigenmodes</a> and <a href="/facts/Resonance/q7VLALUz">resonant frequencies</a> of a structure.

Rayleigh–Ritz method open-in-new

Rayleigh–Ritz method