Some vector sequence transformations with applications to systems of equations

First, recursive algorithms for implementing some vector sequence transformations are given. In a particular case, these transformations are generalizations of Shanks transformation and the G-transformation. When the sequence of vectors under transformation is generated by linear fixed point iterations, Lanczos' method and the CGS are recovered respectively. In the case of a sequence generated by nonlinear fixed point iterations, a quadratically convergent method based on the -algorithm is recovered and a nonlinear analog of the CGS method is obtained.     

CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm

The Generalized Minimal Residual (GMRES) method and the Quasi-Minimal Residual (QMR) method are two Krylov methods for solving linear systems. The main difference between these methods is the generation of the basis vectors for the Krylov subspace. The GMRES method uses the Arnoldi process while QMR uses the Lanczos algorithm for constructing a basis of the Krylov subspace. In this paper we give a new method similar to QMR but based on the Hessenberg process instead of the Lanczos process. We call the new method the CMRH method. The CMRH method is less expensive and requires slightly less storage than GMRES. Numerical experiments suggest that it has behaviour similar to GMRES. This revised version was published online in June 2006 with corrections to the Cover Date.     

A new look at CMRH and its relation to GMRES

CMRH is a Krylov subspace method which uses the Hessenberg process to produce a basis of a Krylov method, and minimizes a quasiresidual. This method produces convergence curves which are very close to those of GMRES, but using fewer operations and storage. In this paper we present new analysis which explains why CMRH has this good convergence behavior. Numerical examples illustrate the new bounds.     

A review of formal orthogonality in Lanczos-based methods

Krylov subspace methods and their variants are presently the favorite iterative methods for solving a system of linear equations. Although it is a purely linear algebra problem, it can be tackled by the theory of formal orthogonal polynomials. This theory helps to understand the origin of the algorithms for the implementation of Krylov subspace methods and, moreover, the use of formal orthogonal polynomials brings a major simplification in the treatment of some numerical problems related to these algorithms. This paper reviews this approach in the case of Lanczos method and its variants, the novelty being the introduction of a preconditioner.     

Effect of chemical nature of atoms on the electronic,dielectric, and dynamical properties of ABX3 halide perovskite

First-principles calculations within density functional theory were performed on a series of halide perovskite compounds ABX₃(A: Cs or Rb; B:Pb or Sn). Their electronic structure, lattice dynamics, and dielectric properties were studied in relationship with the change in atom species at each one of the three inequivalent crystallographic sites, to explain the origin of these properties. Thus, the variation of the bandgap with the overlap between the B cation lone pair and the electronic states of halide atoms, as well as with the distortion of the BX₆ octahedra network is discussed. It is shown that the vibrational modes, phonon frequencies, atomic displacements, and the possible ferroelectric instability in these compounds are dependent on masses of atoms, volume of AX₁₂ polyhedron, as well as on streoactivity of Pb lone pair. Also, the Born effective charges, dielectric constant, spontaneous polarization, and infrared spectra are calculated. The relation between these dielectric properties and the ions dynamics is discussed.     

Avoiding breakdown in the CGS algorithm

The conjugate gradient squared algorithm can suffer of similar breakdowns as Lanczos type methods for the same reason that is the non-existence of some formal orthogonal polynomials. Thus curing such breakdowns is possible by jumping over these non-existing polynomials and using only those of them which exist. The technique used is similar to that employed for avoiding breakdowns in Lanczos type methods. The implementation of these new methods is discussed. Numerical examples are given.     

Multidimensional upwind convection schemes for flow in porous media on structured and unstructured quadrilateral grids

Standard reservoir simulation schemes employ first order upwind schemes for approximation of the convective fluxes when multiple phases or components are present. These convective flux schemes rely upon upwind information that is determined according to grid geometry. As a consequence directional diffusion is introduced into the solution that is grid dependent. The effect can be particularly important for cases where the flow is across grid coordinate lines and is known as cross-wind diffusion.Truly higher dimensional upwind schemes that minimize cross-wind diffusion are presented for convective flow approximation on quadrilateral unstructured grids. The schemes are locally conservative and yield improved results that are essentially free of spurious oscillations. The higher dimensional schemes are coupled with full tensor Darcy flux approximations.The benefits of the resulting schemes are demonstrated for classical test problems in reservoir simulation including cases with full tensor permeability fields. The test cases involve a range of structured and unstructured grids with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher dimensional formulations are shown to effectively reduce the numerical cross-wind diffusion effect, leading to improved resolution of concentration and saturation fronts.     

Silica-clay nanocomposites

A new class of porous nanocomposite materials have been prepared by reaction of alkoxysilanes with alkylammonium-exchanged phyllosilicates (clay minerals), using a sol-gel procedure which produces the complete delamination of these layered solids.     

Packing Dimensions of Basins Generated by Invariant Measures on a Sequence Space

Differential Equations - We consider the left shift mapping on the space $$X^{{\mathbb {N}}} $$ of infinite signals over a finite alphabet $$X $$ . For each signal, the iterated shifts generate a...