We are concerned with eigenvalue problems for definite and indefinite symmetric matrix pencils. First, Rayleigh-Ritz methods are formulated and, using Krylov subspaces, a convergence analysis is presented for definite pencils. Second, generalized symmetric Lanczos algorithms are introduced as a special Rayleigh-Ritz method. In particular, an a posteriori convergence criterion is demonstrated by...

New restarted Lanczos bidiagonalization methods for the computation of a few of the largest or smallest singular values of a large matrix are presented. Restarting is carried out by augmentation of Krylov subspaces that arise naturally in the standard Lanczos bidiagonalization method. The augmenting vectors are associated with certain Ritz or harmonic Ritz vectors. Computed examples show the ne...

After reviewing the harmonic Rayleigh–Ritz approach for the standard and generalized eigenvalue problem, we discuss different extraction processes for subspace methods for the polynomial eigenvalue problem. We generalize the harmonic and refined Rayleigh–Ritz approach, which are new approaches to extract promising approximate eigenpairs from a search space. We give theoretical as well as numeri...

In this paper, we use Hermite cubic finite elements to approximate the solutions of a nonlinear Euler– Bernoulli beam equation. The equation is derived from Hollomon’s generalized Hooke’s law for work hardening materials with the assumptions of the Euler–Bernoulli beam theory. The Ritz–Galerkin finite element procedure is used to form a finite dimensional nonlinear program problem, and a nonlin...

Inthis paper, -we considersthe linear quadratic optimal control 4, problem on infinite time interval for linear time-invariant systems defined on Hilbert spaces. The optimal control is given by a feedback form in terms of solution r1 to the associated algebraic Riccati equation (ARE). A Ritz type approximation is used to obtain a sequence rl of finite dimensional approximations of the solution ...

In this paper we present the convergence analysis of iterative schemes for solving linear systems resulting from diacretizing multidimensional linear second order elliptic partial differential equations (PDEs) defined in a hyper-parallelepiped n and subject to Dirichlet boundary conditions on some faces of n and Neumann on the others, using a new class of line cubic spline collocation (LeSC) me...

this paper may be regarded as a new numerical method for the analysis of triangular thin plates using the natural area coordinates. previous studies on the solution of triangular plates with different boundary conditions are mostly based on the rayleigh-ritz principle which is performed in the cartesian coordinates. consequently, manipulation of the geometry and numerical calculation of the int...