The generalized minimum residual method (GMRES) is well known for solving large nonsymmetric systems of linear equations. It generally uses restarting, which slows the convergence. However, some information can be retained at the time of the restart and used in the next cycle. We present algorithms that use implicit restarting in order to retain this information. Approximate eigenvectors determ...

Iterative algorithms for large-scale eigenpair computation are mostly based subspace projections consisting of two main steps: a subspace update (SU) step that generates bases for approximate eigenspaces, followed by a Rayleigh-Ritz (RR) projection step that extracts approximate eigenpairs. A predominant methodology for the SU step makes use of Krylov subspaces that builds orthonormal bases pie...

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...

After giving a panoramic view of the " textbook " interpretation of the new quantum mechanics, as a sequel to the old quantum theory, the conceptual basis of quantum theory since the Copenhagen Interpretation is reviewed in the context of various proposals since Einstein and Niels Bohr, designed to throw light on possible new facets bearing on its foundations, the key issues to its inherent " i...

As Henri Poincaré once remarked, "solution of a mathematical problem" is a phrase of indefinite meaning. Pure mathematicians sometimes are satisfied with showing that the non-existence of a solution implies a logical contradiction, while engineers might consider a numerical result as the only reasonable goal. Such one sided views seem to reflect human limitations rather than objective values. I...

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 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 present a Ritz-Galerkin discretization on sparse grids using pre-wavelets, which allows to solve elliptic differential equations with variable coefficients for dimension d = 2, 3 and higher dimensions d > 3. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This alg...