Derivative Of Polynomial Research Articles

Properties of expansions in Legendre polynomial derivatives that show up in analysis of nonlinear vibrations of a viscous liquid drop

Analytical expressions for the coefficients of expansions of Legendre polynomial products in the first and second derivatives of the polynomials with respect to polar angles, as well as for the coefficients of expansions of derivative products in Legendre polynomials and their first derivatives, are derived. An interrelation between these expansion coefficients and a relationship between these coefficients and the Clebsch-Gordan coefficients are found. When axisymmetric nonlinear vibrations of a viscous liquid drop are investigated analytically, the toroidal component of the velocity field can be ignored, which significantly cuts the body of computation.

Bounds for the derivatives of polynomials on centrally symmetric convex bodies

Exact multidimensional analogues of V.A. Markov's inequality are found for the derivatives of polynomials on centrally symmetric bodies.

Superconvergence of Discontinuous Finite Element Solutions for Transient Convection–diffusion Problems

We present a study of the local discontinuous Galerkin method for transient convection-diffusion problems in one dimension. We show that p-degree piece-wise polynomial discontinuous finite element solutions of convection-dominated problems are O(Δxp+2) superconvergent at Radau points. For diffusion-dominated problems, the solution's derivative is O(Δxp+2) superconvergent at the roots of the derivative of Radau polynomial of degree p+1. Using these results, we construct several asymptotically exact a posteriori finite element error estimates. Computational results reveal that the error estimates are asymptotically exact.

Derivative Free Inclusion Methods for Polynomial Zeros

Two iterative methods for the simultaneous inclusion of complex zeros of a polynomial are presented. Both methods are realized in circular interval arithmetic and do not use polynomial derivatives. The first method of the fourth order is composed as a combination of interval methods with the order of convergence two and three. The second method is constructed using double application of the inclusion method of Weierstrass’ type in serial mode. It is shown that its R-order of convergence is bounded below by the spectral radius of the corresponding matrix. Numerical examples illustrate the convergence rate of the presented methods

Roots of the Derivative of a Polynomial

Roots of the Derivative of a Polynomial

Bernstein's inequality for multivariate polynomials on the standard simplex

The classical Bernstein pointwise estimate of the (first) derivative of a univariate algebraic polynomial on an interval has natural extensions to the multivariate setting. However, in several variables the domain of boundedness, even if convex, has a considerable geometric variety. In 1990, Y. Sarantopoulos satisfactorily settled the case of a centrally symmetric convex body by a method we may call "the method of inscribed ellipses." On the other hand, for the general case of nonsymmetric convex bodies we are only within a constant factor of an exact inequality. The best known results suggest relevance of the generalized Minkowski functional, and a natural conjecture for the exact Bernstein factor was formulated with this geometric quantity. This work deals with the most natural and simple nonsymmetric case, that of a standard simplex in, and computes the exact yield of the method of inscribed ellipses. Although the known general estimates of the Bernstein factor are improved for the simplex here, we find that not even the exact yield of the inscribed ellipse method reaches the conjecture. However, we also show that for an arbitrary convex body the subset of ridge polynomials satisfies the conjecture.

Roots of the Derivative of a Polynomial

Roots of the Derivative of a Polynomial

Higher order pseudospectral differentiation matrices

A new explicit expression of the higher order pseudospectral differentiation matrices is presented by using an explicit formula for higher derivatives of Chebyshev polynomials. The roundoff errors incurred during computing differentiation matrices are investigated. The advantages of the suggested differentiation matrices emerged through comparisons with other ones.

Electromagnetic induction modelling based on satellite magnetic vector data

SUMMARY The first results on the modelling of magnetic signals induced by transient equatorial ring currents in the magnetosphere with a timescale of the order of days and recorded at satellite altitudes are presented. The input of modelling consists of the X -component of the time-series recorded by the CHAMP vector magnetometer along individual night-time, mid-latitude satellite tracks. We have not considered the magnetic signals measured along day-time tracks and above polar regions because they are disturbed by signals with sources different from magnetospheric ring currents. The modelling procedure is divided into two parts. First, the X -components of satellite magnetic signals are processed by a two-step least-squares analysis. As a result, the X -data are represented in terms of series of Legendre polynomial derivatives. Four examples of the two-step analysis of the signals recorded by the CHAMP vector magnetometer are presented. Second, forward modelling of the electromagnetic induction response of a 2-D heterogeneous conducting sphere to a transient external current excitation is carried out in the time domain using a recently developed spectral finite-element technique (Martinec et al. 2003), which has been modified to include satellite magnetic data. The modified approach has been verified against a semi-analytical solution for a 2-D geometry: a spherical inclusion eccentrically nested in a homogeneous sphere. The output of the forward modelling of electromagnetic induction, that is, the predicted Z-component at satellite altitude, can then be compared with the satellite observations to obtain data to be minimized when adjusting electrical conductivity models in future inversion modelling. To outline such a procedure, we use a 5-layer spherically symmetric model of electrical conductivity and compare the predicted and observed Z-data for the magnetic storm recorded by the CHAMP satellite between 2001 September 25 and 2001 October 7. The agreement between observations and predictions is quantitatively satisfactory. The main advantage of the present method is its ability to use the satellite magnetic data directly, without the need to continue them from the satellite altitude to the ground or to decompose them into the inducing and induced parts by spherical harmonic analysis.

Pointwise convergence of derivatives of Lagrange interpolation polynomials for exponential weights

For a general class of exponential weights on the line and on (−1,1), we study pointwise convergence of the derivatives of Lagrange interpolation. Our weights include even weights of smooth polynomial decay near ±∞ (Freud weights), even weights of faster than smooth polynomial decay near ±∞ (Erdös weights) and even weights which vanish strongly near ±1, for example Pollaczek type weights.

Pointwise convergence of derivatives of Lagrange interpolation polynomials for exponential weights

For a general class of exponential weights on the line and on (−1,1), we study pointwise convergence of the derivatives of Lagrange interpolation. Our weights include even weights of smooth polynomial decay near ±∞ (Freud weights), even weights of faster than smooth polynomial decay near ±∞ (Erdös weights) and even weights which vanish strongly near ±1, for example Pollaczek type weights.

Jacobi Pseudospectral Method for Solving Optimal Control Problems

The generalization of pseudospectral method for collocation based on roots of derivatives of general Jacobi polynomials was described. The Legendre and Chebyshev nodes were also obtained as particular cases of more general formulation. The state equations were enforced by differentiation of approximating polynomial at Gauss-Lobatto points. The appropriate selection of Jacobi parameters also determined real-time solution to nonlinear optimal control problems.

Numerical factorization of multivariate complex polynomials

One can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components. The importance and nature of this problem however justify a special treatment. We exploit the reduction to the univariate root finding problem as a way to sample the polynomial more efficiently, certify the decomposition with linear traces, and apply interpolation techniques to construct the irreducible factors. With a random combination of differentials we lower multiplicities and reduce to the regular case. Estimates on the location of the zeroes of the derivative of polynomials provide bounds on the required precision. We apply our software to study the singularities of Stewart–Gough platforms.

On the connection coefficients and recurrence relations arising from expansions in series of hermite polynomials

A formula expressing the Hermite coefficients of a general-order derivative of an infinitely differentiable function in terms of its original coefficients is proved, and a formula expressing explicitly the derivatives of Hermite polynomials of any degree and for any order as a linear combination of suitable Hermite polynomials is deduced. A formula for the Hermite coefficients of the moments of one single Hermite polynomial of certain degree is given. Formulae for the Hermite coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Hermite coefficients are also obtained. Two numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in Hermite coefficients, are discussed. A simple approach in order to build and solve recursively for the connection coefficients between Jacobi–Hermite and Laguerre–Hermite polynomials is described. Explicit formula for these coefficients between Jacobi and Hermite polynomials is given, of which the ultraspherical polynomials and the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. Analytical formula for the connection coefficients between Laguerre and Hermite polynomials is also obtained.

Simplified Polynomial Derivative Technique for the Reduction of Large-Scale Interval Systems

A new method for generating low order models for large-scale interval systems using simplified polynomial derivative technique is proposed. New algorithms are developed for obtaining the numerator and denominator polynomials of the reduced order model which also retain the steady state response characteristics of the original high order interval system in its low order model, unlike existing methods [1,2]. The method is computationally simple avoiding the necessity of formation of long Routh-type differentiation arrays and reciprocal transformations and always retains stability of the original system in the low-order models, unlike the existing methods [3-5]. Typical numerical examples illustrating the proposed method are provided.

The fourth Painlevé equation and associated special polynomials

In this article rational solutions and associated polynomials for the fourth Painlevé equation are studied. These rational solutions of the fourth Painlevé equation are expressible as the logarithmic derivative of special polynomials, the Okamoto polynomials. The structure of the roots of these Okamoto polynomials is studied and it is shown that these have a highly regular structure. The properties of the Okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials. Further representations are given of the associated rational solutions in the form of determinants through Schur functions.

Algorithms for the integration and derivation of Chebyshev series

General formulas for the mth integral and derivative of a Chebyshev polynomial of the first or second kind are presented. The result is expressed as a finite series of the same kind of Chebyshev polynomials. These formulas permit to accelerate the determination of such integrals or derivatives. Besides, it is presented formulas for the mth integral and derivative of finite Chebyshev series and a numerical algorithm for the direct evaluation of the mth derivative of such a series.

Harmonic mean, random polynomials and stochastic matrices

Motivated by a problem in learning theory, we are led to study the dominant eigenvalue of a class of random matrices. This turns out to be related to the roots of the derivative of random polynomials (generated by picking their roots uniformly at random in the interval [0,1], although our results extend to other distributions). We produce sharp results on the statistical properties of the smallest critical point. This, in turn, requires the study of the statistical behavior of the harmonic mean of i.i.d. random variables, and we produce a number of limiting distributions and laws of large numbers.

An efficient method for capturing free boundaries in multi‐fluid simulations

AbstractAn easy‐to‐use front capturing method is devised by directly solving the transport equation for a volume of fluid (VOF) function. The key to this method is a semi‐Lagrangian conservative scheme, namely CIP_CSL3, recently proposed by the author. In the CIP_CSL3 scheme, the first‐order derivative of the interpolation polynomial at each cell centre is used to control the shape of the reconstructed profile. We show in the present paper that the first‐order derivative, which plays a crucial role in reconstructing the interpolation profile, can also be used to eliminate numerical diffusion. The resulting algorithm can be directly used to compute the VOF‐like function and retain the compact thickness of the moving interface in multi‐fluid simulations. No surface reconstruction based on the value of VOF function is required in the method, which makes it quite economical and easy to use. The presented method has been tested with various interfacial flows including pure rotation, vortex shearing, multi‐vortex deformation and the moving boundaries in real fluid as well. The method gives promising results to all computed problems. Copyright © 2003 John Wiley & Sons, Ltd.

Saddle points in the chaotic analytic function and Ginibre characteristic polynomial

Comparison is made between the distribution of saddle points in the chaotic analytic function and in the characteristic polynomials of the Ginibre ensemble. Realising the logarithmic derivative of these infinite polynomials as the electric field of a distribution of coulombic charges at the zeros, a simple mean-field electrostatic argument shows that the density of saddles minus zeros falls off as $\pi^{-1}|z|^{-4}$ from the origin. This behaviour is expected to be general for finite or infinite polynomials with zeros uniformly randomly distributed in the complex plane, and which repel quadratically.