Lagrange Formula Research Articles

On asymptotics of certain sums arising in coding theory

T. Klove (see ibid., vol.41, p.298-300, 1995) analyzed the average worst case probability of undetected error for linear [n, k; q] codes of length n and dimension k over an alphabet of size q. The following sum: S/sub n/=/spl Sigma//sub i=1//sup n/(/sub i//sup n/)(/sup i///sub n/)/sup i/((1-i)/n)/sup n-i/ arose, which also has applications in coding theory, average case analysis of algorithms, and combinatorics. Klove conjectured an asymptotic expansion of this sum, and we prove its enhanced version. Furthermore, we consider a more challenging sum arising in the upper bound of the average worst case probability of undetected error over systematic codes derived by Massey (1978). Namely S/sub n,k/=/spl Sigma//sub i=1//sup n/(/sub i//sup n-k/)(/sup i///sub n/)/sup i/((1-i)/n)/sup n-i/ for k/spl ges/0. We obtain an asymptotic expansion of S/sub n,k/, and this leads to a conclusion that Massey's bound on the average worst case probability over all systematic codes is better for every k than the corresponding Klove's bound over all codes [n, k; q]. The technique used belongs to the analytical analysis of algorithms and is based on some enumeration of trees, singularity analysis, Lagrange's inversion formula, and Ramanujan's identities. In fact, S/sub n/, turns out to be related to the so-called Ramanujan's Q-function which finds many applications (e.g. hashing with linear probing, the birthday paradox problem, random mappings, caching, memory conflicts, etc.).

Analyse non linéaire par éléments finis des pièces comprimées et fléchies à parois minces

A finite element analysis of the ultimate strength of steel thin-walled beam–columns is discussed in this paper. The numerical model used was developed by Akoussah, Beaulieu, and Dhatt and is based on a Lagrangian formula of thin-walled bars. In this model, the law of multiaxial elastic–plastic behaviour was adopted and incorporated on the section's surface by Radau or Gauss type points. Case studies were performed in which different parameters likely to influence the behaviour of beam–columns were used. Corresponding interaction curves were plotted. The results obtained were compared with those of other authors for H sections under biaxial bending, for square or rectangular tubular sections under biaxial bending, and for H sections bended in relation to each of the principal axes. The results emphasize the key role played by shear stress on the bearing capacity of open section compressed members under bending. Key words: shear, compression, finite elements, bending, interaction, elasticity, plasticity.

Parametric curve fitting: An alternative to Lagrange interpolation and splines

We shall discuss a method for obtaining parametric equations of a smooth curve that passes through a sequence of points, and compare it to cubic splines and to the method of interpolation by quadratic polynomials using the Lagrange formula. A mathematica program automating this method of curve fitting is also provided.

Curve fitting by Lagrange interpolation

It is demonstrated that interpolation by quadratic polynomials using the Lagrange formula may compete in accuracy with cubic spline interpolation, while being simpler to implement. It is also shown that Lagrange interpolation may easily be used to fit an almost arbitrary function to experimental data. FORTRAN routines for Lagrange interpolation as well as Lagrange fitting are supplied separately.

Centralized, decentralized, and independent control of a flexible manipulator on a flexible base

The dynamics and control of a flexible manipulator arm with payload mass on a flexible base in space are considered. The controllers are provided by one torquer at the center of the base and one torquer at the connection joint of the robot and the base. The nonlinear dynamics of the system is modeled by applying the finite element method and Lagrangian formula. Three control strategies are considered and compared, i.e. centralized control, decentralized control, and independent control. All these control designs are based on the linear quadratic regulator theory. A mathematical decomposition is used in the decentralization process so that the coupling between the subsystems is weak, while a physical decomposition is used in the independent control design process. For both the decentralized and the independent controls, the stability of the overall linear system is checked before a numerical simulation is initiated. Two numerical examples show that the responses of the independent control system are close to those of the centralized control system, while the responses of the decentralized control system are not.

Reliable determination of interpolating polynomials

This paper addresses a fundamental problem in mathematics and numerical analysis, that of determining a polynomial interpolant to specified data. The data is taken as consisting of a set of points (abscissae), at each of which is specified a function value. Additionally, at each point, any number of leading derivative values of the function may be given. Mathematically, this problem is solved. The classical Lagrangian interpolation formula applies in the derivative-free case, and the Newton form of the interpolating polynomial in general.Numerically, few reliable algorithms are available; most published algorithms concentrate on speed of computation. This paper describes an algorithm that delivers the required polynomial in Chebyshev form. It is based on the repeated use of the Newton representation, with a data ordering strategy and iterative refinement to improve accuracy, using a carefully devised merit function to measure success. The algorithm attempts to provide a polynomial that is stable in the sense of backward error analysis, i.e. that is exact for slightly perturbed data. Implementations of the algorithm have been in use since the early 1980s in the NAG Library and NPL's Data Approximation Subroutine Library (DASL). In addition to providing polynomial interpolants in their own right, these implementations are used as computational modules in the NAG and DASL routines for constrained least-squares polynomial data fitting. This paper constitutes the first detailed presentation of the algorithm.

Construction of modified third-order upwind schemes for stretched meshes

Our concept for numerical schemes is based on two relations: 1) a polynomial and the derivatives, and 2) a polynomial and the integrations. The Taylor series expansion of a function q about a certain point and the Lagrangian interpolation formula are utilized to construct the upwind scheme on a physical space. Three results in two dimensions are shown: a motion of viscous vortex with a uniform background flow, a driven cavity flow, and a flow past a circular cylinder. The method presented produces smooth and realistic results for computations with stretched meshes

Applied mathematics and mechanics

Based on the Coriolis acceleration and the Lagrangian strain formula, a gen- eralized equation for the transverse vibration system of convection belts is derived using Newton's second law. The method of multiple scales is directly applied to the govern- ing equations, and an approximate solution of the primary parameter resonance of the system is obtained. The detuning parameter, cross-section area, elastic and viscoelastic parameters, and axial moving speed have a significant influences on the amplitudes of steady-state response and their existence boundaries. Some new dynamical phenomena are revealed.

Third-order nonoscillatory schemes for the Euler equations

Two time-level third-order finite-difference shock-capturin g schemes based on applying the characteristic flux difference splitting to a modified flux that may have high-order accuracy and either monotonicity preserving or essentially nonoscillatory (ENO) property have been developed for the Euler equations of gas dynamics. Two ways to achieve high-order accuracy are described. One is based on upstream interpolation using Lagrange's formula with van Leer's smoothness monitors. The other is based on ENO interpolation using reconstruction via primitive function approaches. For multidimensional problems, dimensional splitting is adopted for explicit schemes, and standard alternating direction implicit approximate factorization procedures are used for implicit schemes. Numerical examples to illustrate the performance of the proposed schemes are given. ERY recently, a new class of uniformly high-order accurate, essentially nonoscillatory (ENO) schemes has been developed by Harten and Osher,1 Marten,2 and Harten et al.3'4 They presented a hierarchy of uniformly high-order accurate schemes that generalize Godunov's scheme5 and its secondorder accurate MUSCL extension6'7 and total variation diminishing (TVD) schemes8'9 to an arbitrary order of accuracy. In contrast to the earlier second-order TVD schemes that drop to first-order accuracy at local extrema and maintain second-order accuracy in smooth regions, the new ENO schemes are uniformly high-order accurate throughout, even at critical points. Theoretical results for the scalar coefficient case and numerical results for the scalar conservation law and for the one-dimensional Euler equations of gas dynamics have been reported with highly accurate results. Preliminary results for two-dimensional problems were reported in Ref. 2. In this paper, following van Leer,10'11 Harten,2 and Harten et al.3'4 we describe a class of third-order, essentially nonoscillatory shock-capturing schemes for the Euler equations of gas dynamics. These schemes are obtained by applying the characteristic flux-difference splitting to an appropriately modified flux vector that may have high-order accuracy and nonoscillatory property. Third-order schemes are constructed using upstream interpolation and ENO interpolation. Both explicit and implicit schemes are derived. Implicit schemes for two-dimensional Euler equations in general coordinates are also given. We apply the resulting schemes to simulate one-dimensional and two-dimensional unsteady shock tube flows and steady two-dimensional flows involving strong shocks to illustrate the performance of the schemes. In Sec. II, characteristic properties of the Euler equations related to the numerical advections are briefly summarized. Upstream interpolation using Lagrange's formula to generate a class of two time-level, 2p + 1 space point, (2p - l)thorder accurate schemes is described in Sec. III. In particular, a third-order scheme with smoothness monitors due to van Leer10 is described. In Sec. IV, the essentially nonoscillatory interpolation of Harten2 and Harten et al.3'4 using reconstruction via primitive function approach is employed to yield a third-order nonoscillatory scheme.

Multivariate polynomial interpolation under projectivities part I: lagrange and newton interpolation formulas

In this note interpolation by real polynomials of several real variables is treated. Existence and unicity of the interpolant for knot systems being the perspective images of certain regular knot systems is discussed. Moreover, for such systems a Newton interpolation formula is derived allowing a recursive computation of the interpolant via multivariate divided differences. A numerical example is given.

Optimal large angle maneuvers of a flexible spacecraft

The optimal control of three-dimensional large-angle rapid maneuvers and vibrations of a Shuttle-mast-reflector system is considered. The nonlinear equations of motion are formulated by using Lagrange's formula, with the mast modeled as a continuous beam subject to three-dimensional deformations. The nonlinear terms in the equations come from the coupling between the angular velocities, the modal coordinates, and the modal rates. Pontryagin's Maximum Principle is applied to the slewing problem, to derive the necessary conditions for the optimal controls, which are bounded by given saturation levels. The resulting two-point boundary-value problem is then solved by using the quasilin-earization algorithm and the method of particular solutions. The numerical results for the flexible nonlinear system, the flexible linearized system, and the rigidized nonlinear system are presented to compare the differences in their time responses.

The β-Extension of the Multivariable Lagrange Inversion Formula

We show that Gessel's combinatorial proof of the multivariable Lagrange inversion formula can be given a ,β-extension, which generalizes Foata and Zeilberger's, β-extension of MacMahon's master theorem. Moreover, we show that there is no need to use Jacobi's identity in the derivation of the Lagrange formula. Finally, combining Gessel's method and ours, we obtain a new proof of Jacobi's identity.

When does the surplus reach a given target?

In the classical model of risk theory a martingale method is used to compute the distributions of the first and last passage times (and of their difference) of the surplus process at a given level. As a byproduct a certain probabilistic identity related to Lagrange's formula is derived; furthermore Consul's generalized Poisson distribution is explained.

Application of FEM to anodic smoothing

This paper deals with the development and potential application of a finite element method to anodic smoothing in electrochemical machining. The tests of convergence using a direct method of calculating the potential gradients are compared with those using Lagrangian differentiation formula. Experimental verification of the FEM model and comparison with results from a model based on perturbation technique are reported. The predictions of FEM model are found to be superior.

Approximation with exact Lagrange-type interpolation

Under stated favorable conditions, the Lagrange formula for polynomial interpolation is computationally exact at the nodes. This is applied to approximation with Lagrange-type interpolation.

Operator methods and Lagrange inversion: a unified approach to Lagrange formulas

On presente une methode generale de demonstration des formules d'inversion de Lagrange et de leurs variantes. On considere le cas de plusieurs variables

Operator Methods and Lagrange Inversion: A Unified Approach to Lagrange Formulas

Operator Methods and Lagrange Inversion: A Unified Approach to Lagrange Formulas

Quadrature Formulae for Cauchy Principal Value Integrals of Oscillatory Kind

The problem considered is that of evaluating numerically an integral of the form $f_{ - 1}^1\;{e^{i\omega x}}f(x) dx$, where f has one simple pole in the interval $[ - 1,1]$. Modified forms of the Lagrangian interpolation formula, taking account of the simple pole are obtained, and form the bases for the numerical quadrature rules obtained. Further modification to deal with the case when an abscissa in the interpolation formula is coincident with the pole is also considered. An error bound is provided and some numerical examples are given to illustrate the formulae developed.

Quadrature formulae for Cauchy principal value integrals of oscillatory kind

The problem considered is that of evaluating numerically an integral of the form f − 1 1 e i ω x f ( x ) d x f_{ - 1}^1\;{e^{i\omega x}}f(x)\,dx , where f has one simple pole in the interval [ − 1 , 1 ] [ - 1,1] . Modified forms of the Lagrangian interpolation formula, taking account of the simple pole are obtained, and form the bases for the numerical quadrature rules obtained. Further modification to deal with the case when an abscissa in the interpolation formula is coincident with the pole is also considered. An error bound is provided and some numerical examples are given to illustrate the formulae developed.

Osculatory Interpolation in $\mathbb{R}^n $

In this work we study the Hermite osculatory interpolation problems in several variables. We give general Lagrange formulae for the solution of these problems and exhibit examples where these formulae apply in the context of $P_m $-correct Hermite problems. We finally give an inductive characterization of a family of $P_m $-correct Hermite problems, which we call totally reducible.