Errors of solutions of linear algebraic systems

Error estimates are derived for the following methods: the sweepout method for tridiagonal systems, the method of square roots, the bordering method, and the method of reflection matrices. The book of S. K. Godunov is devoted to the last method; he altered the method so that it takes any matrix into a bidiagonal matrix; a considerable part of that book is devoted to the error of this alteration. In the present paper the method of reflection matrices is studied in the form in which it is expounded in the familiar book of D. K. Faddeev and V. N. Faddeeva. Recurrent formulas are obtained for the sweepout method which make it possible to successively estimate errors of the components of the solution vector. In the method of square roots the error of the solution vector is estimated by the quantity Here _i and are small quantities; the first characterizes the accuracy of small arithmetics effects, and the second the round-off error in the reverse step. Further, A is the matrix of the system, m, is its order, f is the vector of free terms, and C and are constants with 0 <1. We shall not present here the rather involved estimates for the bordering method. The error of the solution vector obtained by the method of reflection matrices is estimated by the quantity (p_A is the conditioning number of the matrix A) All estimates are obtained up to terms of higher order of smallness than and ₁. The estimates themselves are related to the classification of errors of computing processes proposed by the author in recent years.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 139, pp. 94–110, 1984.     

Structural properties and state of the surface layer of zirconium dioxide promoted by manganese cations

The effect of Mn cations on the structural properties of zirconium dioxide (phase composition and crystallite size) was studied. The cations were introduced by coprecipitation of hydroxide precursors followed by thermal processing at temperatures of 350 to 650°C. It was found by X-ray photoelectron spectroscopy that Mn ⁿ⁺ cations (4 ≥ n ≥ 2, n = 3 being the dominant state) were localized on the surface of MnO _x -ZrO₂ samples calcinated at 350 and 600°C.     

Conditions for the formation of copper nanoparticles by reduction of copper(II) ions with hydrazine hydrate solutions

Optimal conditions were found for the preparation of copper nanoparticles in aqueous solution via reduction of copper(II) ions with hydrazine hydrate. The effects of ligand environment of copper(II) in the initial solution (hydrate, ammonia, citrate, and glycine complexes), concentration, pH, surfactants, temperature, and mode of heating were examined. The obtained colloidal systems were studied by optical spectroscopy, X-ray photoelectron spectroscopy, X-ray powder diffraction, and atomic force microscopy. The examined colloids were found to contain generally spherical copper nanoparticles with a diameter of about 10 nm, which were coated with a copper(I) or copper(II) oxide and hydroxide film.     

Stability of similar nonlinear normal modes under random excitation

Nonlinear Dynamics - Using both analytical and numerical techniques, we investigate the 3-degree-of-freedom, autonomous, autoparametric, Hamiltonian spring–mass–pendulum system in...     

Variational-difference approximation

The article begins with a presentation of Courant's approach to the variational-difference method and of the definition of basis functions, as well as the method of construction of coordinate functions. Then the completeness of a system of coordinate functions is studied, and also the degree of approximation of functions of Sobolev classes and of some other (similar) classes, as well as basis functions of more complex structure. Some new problems related to approximate calculation of eigenvalues are also examined, as well as methods of construction of variational-difference equations and, in particular, difference schemes with a boundary layer; the stability of the variational-difference process is studied, as well as the condition number of the corresponding matrix. In the concluding part of the article it is shown that the obtained approximation formulas yield in the one-dimensional case the classical Euler-MacLaurin quadrature formula, and in the multidimensional case the obvious counterparts of this formula.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 48, pp. 32–188, 1974.     

Long-range chemical interactions in solid-state reactions: effect of an inert Ag interlayer on the formation of L10-FePd in epitaxial Pd(0 0 1)/Ag(0 0 1)/Fe(0 0 1) and Fe(0 0 1)/Ag(0 0 1)/Pd(0 0 1) trilayers

The effect of 0, 0.5, and 1?μm-thick Ag interlayers on the chemical interaction between Pd and Fe in epitaxial Pd(0?0?1)/Ag(0?0?1)/Fe(0?0?1)/MgO(0?0?1) and Fe(0?0?1)/Ag(0?0?1)/Pd(0?0?1)/MgO(0?0?1) trilayers has been studied using X-ray diffraction, ⁵⁷Fe Mössbauer spectroscopy, X-ray photoelectron spectroscopy, and magnetic structural measurements. No mixing of Pd and Fe occurs via the chemically inert Ag layer at annealing temperatures up to 400?°C. As the annealing temperature is increased above 400?°C, a solid-state synthesis of an ordered L1₀-FePd phase begins in the Pd(0?0?1)/Ag(0?0?1)/Fe(0?0?1) and Fe(0?0?1)/Ag(0?0?1)/Pd(0?0?1) film trilayers regardless of the thickness of the buffer Ag layer. In all samples, annealing above 500?°C leads to the formation of a disordered Fe_xPd_1?x(0?0?1) phase; however, in samples lacking the Ag layer, the synthesis of Fe_xPd_1?x is preceded by the formation of an ordered L1₂-FePd₃ phase. An analysis of the X-ray photoelectron spectroscopy results shows that Pd is the dominant moving species in the reaction between Pd and Fe. According to the preliminary results, the 2.2?μm-thick Ag film does not prevent the synthesis of the L1₀-FePd phase and only slightly increases the phase’s initiation temperature. Data showing the ultra-fast transport of Pd atoms via thick inert Ag layers are interpreted as direct evidence of the long-range character of the chemical interaction between Pd and Fe. Thus, in the reaction state, Pd and Fe interact chemically even though the distance between them is about 10⁴ times greater than an ordinary chemical bond length.