Rayleigh–Ritz method

The Rayleigh–Ritz method is a numerical method of finding approximations to eigenvalue equations that are difficult to solve analytically, particularly in the context of solving physical boundary value problems that can be expressed as matrix differential equations. It is used in mechanical engineering to approximate the eigenmodes of a physical system, such as finding the resonant frequencies of a structure to guide appropriate damping. The name is a common misnomer used to describe the method that is more appropriately termed the Ritz method or the Galerkin method. This method was invented by Walther Ritz in 1909, but it bears some similarity to the Rayleigh quotient and so the misnomer persists. The Rayleigh–Ritz method is a numerical method of finding approximations to eigenvalue equations that are difficult to solve analytically, particularly in the context of solving physical boundary value problems that can be expressed as matrix differential equations. It is used in mechanical engineering to approximate the eigenmodes of a physical system, such as finding the resonant frequencies of a structure to guide appropriate damping. The name is a common misnomer used to describe the method that is more appropriately termed the Ritz method or the Galerkin method. This method was invented by Walther Ritz in 1909, but it bears some similarity to the Rayleigh quotient and so the misnomer persists. The Rayleigh–Ritz method allows for the computation of Ritz pairs ( λ ~ i , x ~ i ) {displaystyle ({ ilde {lambda }}_{i},{ ilde { extbf {x}}}_{i})} which approximate the solutions to the eigenvalue problem where A ∈ C N × N {displaystyle Ain mathbb {C} ^{N imes N}} . The procedure is as follows: One can always compute the accuracy of such an approximation via ‖ A x ~ i − λ ~ i x ~ i ‖ {displaystyle |A{ ilde { extbf {x}}}_{i}-{ ilde {lambda }}_{i}{ ilde { extbf {x}}}_{i}|} If a Krylov subspace is used and A is a general matrix, then this is the Arnoldi algorithm. In this technique, we approximate the variational problem and end up with a finite dimensional problem. So let us start with the problem of seeking a function y ( x ) {displaystyle y(x)} that extremizes an integral I [ y ( x ) ] {displaystyle I} . Assume that we are able to approximate y(x) by a linear combination of certain linearly independent functions of the type: y ( x ) ≈ φ 0 ( x ) + c 1 φ 1 ( x ) + c 2 φ 2 ( x ) + ⋯ + c N φ N ( x ) {displaystyle y(x)approx varphi _{0}(x)+c_{1}varphi _{1}(x)+c_{2}varphi _{2}(x)+cdots +c_{N}varphi _{N}(x)} where c 1 , c 2 , ⋯ , c N {displaystyle c_{1},c_{2},cdots ,c_{N}} are constants to be determined by a variational method, such as one which will be described below.