Ritzdiagonalizáció

Ritzdiagonalizáció, also known as the Ritz method or Ritz–Galerkin method, is a powerful numerical technique used for approximating the solutions to certain types of differential equations, particularly eigenvalue problems. It is a variational method that seeks an approximate solution in a finite-dimensional subspace of the function space in which the true solution resides.

The core idea of the Ritz method is to transform an infinite-dimensional problem into a finite-dimensional one. This is achieved by choosing a set of basis functions, often called trial functions, that are known to be in the function space. The approximate solution is then expressed as a linear combination of these basis functions, with unknown coefficients.

The method then proceeds by minimizing a functional associated with the differential equation, or by projecting the residual onto the chosen subspace. For eigenvalue problems, the Ritz method aims to find the coefficients that best approximate the eigenvalues and corresponding eigenvectors. This minimization or projection process leads to a system of linear algebraic equations, often in the form of a matrix eigenvalue problem, which can then be solved using standard numerical linear algebra techniques.

The accuracy of the Ritz approximation depends on the choice of basis functions and the dimension of the subspace. A richer set of basis functions or a larger subspace generally leads to a more accurate approximation, but also to a larger system of equations to solve. The Ritz method is widely applied in fields such as structural mechanics, quantum mechanics, and computational physics and engineering.