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  2. Rayleigh–Ritz method - Wikipedia

    en.wikipedia.org/wiki/RayleighRitz_method

    The RayleighRitz method is often used in mechanical engineering for finding the approximate real resonant frequencies of multi degree of freedom systems, such as spring mass systems or flywheels on a shaft with varying cross section. It is an extension of Rayleigh's method.

  3. LOBPCG - Wikipedia

    en.wikipedia.org/wiki/LOBPCG

    Reference [12] goes further applying the LOBPCG algorithm to each approximate eigenvector separately, i.e., running the unblocked version of the LOBPCG method for each desired eigenpair for a fixed number of iterations. The Rayleigh-Ritz procedures in these runs only need to solve a set of 3 × 3 projected eigenvalue problems.

  4. Calculus of variations - Wikipedia

    en.wikipedia.org/wiki/Calculus_of_Variations

    This variational characterization of eigenvalues leads to the RayleighRitz method: choose an approximating as a linear combination of basis functions (for example trigonometric functions) and carry out a finite-dimensional minimization among such linear combinations. This method is often surprisingly accurate.

  5. Dynamic substructuring - Wikipedia

    en.wikipedia.org/wiki/Dynamic_substructuring

    However, many of the analytical models of coupled continuous subdomains do not have closed-form solutions, which led to discretization and approximation techniques such as the Ritz method [1] (which is sometimes called the Rayleigh-Ritz method due to the similarity between Ritz's formulation and the Rayleigh ratio) the boundary element method ...

  6. Critical speed - Wikipedia

    en.wikipedia.org/wiki/Critical_speed

    There are two main methods used to calculate critical speed—the RayleighRitz method and Dunkerley's method. Both calculate an approximation of the first natural frequency of vibration, which is assumed to be nearly equal to the critical speed of rotation. The RayleighRitz method is discussed here.

  7. Arnoldi iteration - Wikipedia

    en.wikipedia.org/wiki/Arnoldi_iteration

    This is an example of the Rayleigh-Ritz method. It is often observed in practice that some of the Ritz eigenvalues converge to eigenvalues of A. Since H n is n-by-n, it has at most n eigenvalues, and not all eigenvalues of A can be approximated. Typically, the Ritz eigenvalues converge to the largest eigenvalues of A.

  8. Walther Ritz - Wikipedia

    en.wikipedia.org/wiki/Walther_Ritz

    In 1909 Ritz developed a direct method to find an approximate solution for boundary value problems. It converts the often insoluble differential equation into the solution of a matrix equation. It is a theoretical preparatory work for the finite element method (FEM). This method is also known as Ritz's variation principle and the Rayleigh-Ritz ...

  9. List of numerical analysis topics - Wikipedia

    en.wikipedia.org/wiki/List_of_numerical_analysis...

    Galerkin method — a finite element method in which the residual is orthogonal to the finite element space Discontinuous Galerkin method — a Galerkin method in which the approximate solution is not continuous; RayleighRitz method — a finite element method based on variational principles; Spectral element method — high-order finite ...