We analyze the Ritz–Galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates. For a simple eigenvalue, we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants. For a multiple eigenvalue we prove, in addition, apparently the first truly a priori error estimates th...

Abstract. We present a detailed convergence analysis of preconditioned MINRES for approximately solving the linear systems that arise when Rayleigh Quotient Iteration is used to compute the lowest eigenpair of a symmetric positive definite matrix. We provide insight into the “slow start” of MINRES iteration in both a qualitative and quantitative way, and show that the convergence of MINRES main...

Many problems in science and engineering field require the solution of shifted linear systems with multiple right hand sides and multiple shifts. To solve such systems efficiently, the implicitly restarted global GMRES algorithm is extended in this paper. However, the shift invariant property could no longer hold over the augmented global Krylov subspace due to adding the harmonic Ritz matrices...

Experimentally determined natural frequencies and modes shapes are presented for an elastically point-supported isotropic plate with attached masses under impulsive loading. These results are compared to frequencies and to modes shapes determined from the Rayleigh–Ritz method and a finite element analysis using COMSOL. Accelerometers mounted at three locations on the plate, provide input for ME...

The Rational Krylov algorithm computes eigenvalues and eigenvectors of a regular not necessarily symmetric matrix pencil. It is a generalization of the shifted and inverted Arnoldi algorithm, where several factorizations with di erent shifts are used in one run. It computes an orthogonal basis and a small Hessenberg pencil. The eigensolution of the Hessenberg pencil approximates the solution of...

In this paper we explore two sets of polynomials, the orthogonal polynomi-als and the residual polynomials, associated with a preconditioned conjugate gradient iteration for the solution of the linear system Ax = b. In the context of preconditioning by the matrix C, we show that the roots of the orthogonal polynomials, also known as generalized Ritz values, are the eigenvalues of an orthogonal ...

The main objective of this research work was to investigate three-dimensional free vibration of thick annular plates which are composed of carbon nanotube (CNT) reinforced composites materials using the Chebyshev–Ritz method. In order to obtain precise results, a new form of the rule of mixtures including an exponential shape function, length efficiency parameter, orientation efficiency factor,...

As a model of the second order elliptic equation with non-trivial boundary conditions, we consider the Laplace equation with mixed Dirichlet and Neumann boundary conditions on convex polygonal domains. Our goal is to establish that finite element discrete harmonic functions with mixed Dirichlet and Neumann boundary conditions satisfy a weak (Agmon-Miranda) discrete maximum principle, and then p...

We investigate the behavior of the Lanczos process when it is used to find all the eigenvalues of large sparse symmetric matrices. We study the convergence of classical Lanczos (i.e., without reorthogonalization) to the point where there is a cluster of Ritz values around each eigenvalue of the input matrix A. At that point, convergence to all the eigenvalues can be ascertained if A has no mult...

We study the Lanczos method for computing extreme eigenvalues of a symmetric or Hermitian matrix. It is not guaranteed that the extreme Ritz values are close to the extreme eigenvalues—even when the norms of the corresponding residual vectors are small. Assuming that the starting vector has been chosen randomly, we compute probabilistic bounds for the extreme eigenvalues from data available dur...