Auxiliary Polynomial Research Articles

FREE VIBRATIONS OF BEAMS WITH GENERAL BOUNDARY CONDITIONS

A simple and unified approach is presented for the vibration analysis of a generally supported beam. The flexural displacement of the beam is sought as the linear combination of a Fourier series and an auxiliary polynomial function. The polynomial function is introduced to take all the relevant discontinuities with the original displacement and its derivatives at the boundaries and the Fourier series now simply represents a residual or conditioned displacement that has at least three continuous derivatives. As a result, not only is it always possible to expand the displacement in a Fourier series for beams with any boundary conditions, but also the solution converges at a much faster speed. The reliability and robustness of the proposed technique are demonstrated through numerical examples.

Variations on Lanczos' tridiagonalization process

Lanczos' tridiagonalization processes transform a matrix into an equivalent tridiagonal one. In this paper, we propose several ways of implementing these procedures. They are based on different choices of the auxiliary polynomials which appear in the underlying theory of formal orthogonal polynomials and on a change in their normalization. We also give transpose-free variants of Lanczos processes in the spirit of the CGS and BiCGStab algorithms for solving a system of linear equations. Numerical experiments show that some of the variants proposed have a better numerical behavior than the original algorithms.

Diffuse reflection and transmission of light by a medium with an arbitrary scattering indicatrix

The diffuse reflection and transmission of light by a plane-parallel medium is considered, taking into account only the azimuth-independent part of the scattering indicatrix. The scattering indicatrix is expressed in terms of an expansion in an arbitrary number of Legendre polynomials. Two approaches to the problem are analyzed using Sobolev’s theory. Two systems of linear algebraic equations for the coefficients of the auxiliary polynomials qi(μ) and si(μ) are derived and compared. It is shown that one of these approaches is less preferable, since it is not valid for conservative scattering.

A simple proof of the Routh test

An elementary proof of the classic Routh method for counting the number of left half-plane and right half-plane zeros of a real coefficient polynomial P/sub n/(s) of degree n is given. Such a proof refers to the polynomials P/sub i/(s) of degree i/spl les/n formed from the entries of the rows of order i and i-1 of the relevant Routh array. In particular, it is based on the consideration of an auxiliary polynomial P/sub i/(s; q), linearly dependent on a real parameter q, which reduces to either polynomial P/sub i/(s) or to polynomial P/sub i-1/(s) for particular values of q. In this way, it is easy to show that i-1 zeroes of P/sub i/(s) lie in the same half-plane as the zeros of P/sub i/(s), and the remaining zero lies in the left or in the right half-plane according to the sign of the ratio of the leading coefficients of P/sub i/(s) and P/sub i-1/(s). By successively applying this property to all pairs of polynomials in the sequence, starting from P/sub o/(s) and P/sub 1/(s), the standard rule for determining the zero distribution of P/sub n/(s) is immediately derived.

New fast algorithms for polynomial interpolation and evaluation on the Chebyshev node set

For a polynomial p( x) of a degree n, we study its interpolation and evaluation on a set of Chebyshev nodes, x κ = cos( (2κ + 1)π (2n + 2) ), κ = 0, 1, …, n . This is easily reduced to applying discrete Fourier transforms (DFTs) to the auxiliary polynomial q( ω) = ω n p( x), where 2 x = αω + ( αω) −1, α = exp( π⇔−1 (2n) ) . We show the back and forth transition between p( x) and q( ω) based on the respective back and forth transformations of the variable: αω = (1 − z) (1 + z) , y = (x − 1) (x + 1) , y = z 2. All these transformations (like the DFTs) are performed by using O( n log n) arithmetic operations, which thus suffice in order to support both interpolation and evaluation of p( x) on the Chebychev set, as well as on some related sets of nodes. This improves, by factor log n, the known arithmetic time bound for Chebyshev interpolation and gives an alternative supporting algorithm for the record estimate of O( n log n) for Chebyshev evaluation, obtained by Gerasoulis in 1987 and based on a distinct algorithm.

Robust control for robots using auxiliary polynomials: Theory and experiments

An approach to robust control designs for trajectory tracking of robotic manipulators is presented. Two families of controllers are developed, one being continuous and the other being of switching type. The main feature of the control designs is the use of auxiliary polynomials in position and velocity tracking errors. The designs are simple in feedback gain selections and robust against non-zero initial tracking errors, disturbances and parameter uncertainties. The proposed controller structure consists of a PD feedback term and a nonlinear function in terms of the auxiliary polynomial in a simple form. The control parameters for the PD control and the nonlinear auxiliary polynomials are selected with minimum mutual dependence and based on only limited knowledge on bounds of robot dynamics. The bounds on the position and velocity tracking errors are explicitly specified by controller parameters. With very minimum knowledge of dynamic parameters, these controllers can still ensure that the closed-loop system will not become unstable and the tracking errors are bounded. The controllers are also simple to implement. Experiments are carried out using an industrial robot and results are presented.

Stability tests of N-dimensional discrete time systems using polynomial arrays

Tabular method has been applied to the difficult problem of stability test for multidimensional (N-D) systems. This paper further investigates a general test procedure using an (N-1)-level polynomial array structure. Each entry of the structure at the same level (say level p) is an p-D polynomial, or a set of numbers in an p-D space. A systematic procedure is proposed to compute those polynomial entries which involves only calculations of a set of second order determinants with numerical numbers. A simple formula is also given to form the auxiliary polynomial required in the test procedure. Another key issue is the countability of the zeros of an N-D polynomial. It is clarified that the properly defined zero clusters of an N-D polynomial with respect to some specified regions are countable. To illustrate the proposed method a step-by-step algorithm of the stability test for 3-D systems is described.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The computation of line spectral frequencies using Chebyshev polynomials

Line spectral frequencies provide an alternate parameterization of the analysis and synthesis filters used in linear predictive coding (LPC) of speech. In this paper, a new method of converting between the direct form predictor coefficients and line spectral frequencies is presented. The system polynomial for the analysis filter is converted to two even-order symmetric polynomial with interlacing roots on the unit circle. The line spectral frequencies are given by the positions of the roots of these two auxiliary polynomials. The response of each of these polynomials on the unit circle is expressed as a series expansion in Chebyshev polynomials. The line spectral frequencies are found using an iterative root finding algorithm which searches for real roots of a real function. The algorithm developed is simple in structure and is designed to constrain the maximum number of evaluations of the series expansions. The method is highly accurate and can be used in a form that avoids the storage of trigonometric tables or the computation of trigonometric functions. The reconversion of line spectral frequencies to predictor coefficients uses an efficient algorithm derived by expressing the root factors as an expansion in Chebyshev polynomials.

Eigenvalues of a symmetric tridiagonal matrix: A divide-and-conquer approach

The Symmetric Tridiagonal Eigenproblem has been the topic of some recent work. Many methods have been advanced for the computation of the eigenvalues of such a matrix. In this paper, we present a divide-and-conquer approach to the computation of the eigenvalues of a symmetric tridiagonal matrix via the evaluation of the characteristic polynomial. The problem of evaluation of the characteristic polynomial is partitioned into smaller parts which are solved and these solutions are then combined to form the solution to the original problem. We give the update equations for the characteristic polynomial and certain auxiliary polynomials used in the computation. Furthermore, this set of recursions can be implemented on a regulartree structure. If the concurrency exhibited by this algorithm is exploited, it can be shown that thetime for computation of all the eigenvalues becomesO(nlogn) instead ofO(n2) as is the case for the approach where the order is increased by only one at every step. We address the numerical problems associated with the use of the characteristic polynomial and present a numerically stable technique for the eigenvalue computation.

Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better

This paper will do the following: (1) Establish a (better than) Thue-Siegel-Roth-Schmidt theorem bounding the approximation of solutions of linear differential equations over valued differential fields; (2) establish an effective better than Thue-Siegel-Roth-Schmidt theorem bounding the approximation of irrational algebraic functions (of one variable over a constant field of characteristic zero) by rational functions; (3) extend Nevanlinna's Three Small Function Theorem to an n small function theorem (for each positve integer n), by removing Chuang's dependence of the bound upon the relative “number” of poles and zeros of an auxiliary function; (4) extend this n Small Function Theorem to the case in which the n small functions are algebroid (a case which has applications in functional equations); (5) solidly connect Thue-Siegel-Roth-Schmidt approximation theory for functions with many of the Nevanlinna theories. The method of proof is (ultimately) based upon using a Thue-Siegel-Roth-Schmidt type auxiliary polynomial to construct an auxiliary differential polynomial.

An analysis stability test for a certain class of distributed parameter systems with delays

The aim of this paper is to apply an algebraic test such as the Routh algorithm, Hurwitz determinant, or any similar method to a certain class of distributed parameter systems with multiple delays. As a first step, an auxiliary system is generated, and then two auxiliary polynomials are formed by using only the coefficients of the characteristic equation of the auxiliary system. The input-output stability of the original system is determined on the basis of the zero distribution of two auxiliary polynomials. An illustrative example is included.

A new general Routh-like algorithm to determine the number of RHP roots of a real or complex polynomial

A new Routh-like algorithm for determining the number of right-half plane (RHP) roots of a polynomial with real or complex coefficients is given. It includes the Routh algorithm for real polynomials as a special case. Moreover, the algorithm also applies directly to the singular case wherein the leading coefficient of a row, but not the entire row, vanishes, needing far fewer computations than the heuristic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">epsilon</tex> - method about which there was a vigorous discussion in these TRANSACTIONS a few years ago, and further not requiring investigation of an auxiliary polynomial. The algorithm is illustrated by a few examples. The proof of the algorithm is based on the Principle of the Argument, and thus also constitutes a simple proof of the Routh algorithm in the regular case.

Lagrangian Interpolation at the Chebyshev Points xn, cos ( /n), = 0(1)n; some Unnoted Advantages

Besides many applications of the Chebyshev points xnν ≡ cos(νπ/n),ν = 0(1)n, in approximation, interpolation by Chebyshev series, numerical integration and numerical differentiation, there are advantages in their use in the barycentric form of the Lagrange interpolation formula and in checking by divided differences. When n = 2m, we obtain X2m,ν with less than half the number of square roots that are required to find the other Chebyshev points X′2m,ν ≡ cos[(2ν – 1) π/2m+1], ν = 1(1)2m. Also, the barycentric interpolation formula may be applied to the solution of a near-minimax problem so as to avoid extensive calculation of auxiliary polynomials, and in a numerical differentiation procedure that conveniently by-passes direct differentiation of the interpolation polynomial.

Auxiliary polynomials for the direct determination of the Chebyshey interpolation polynomial

Auxiliary polynomials for the direct determination of the Chebyshey interpolation polynomial

Oscillations of a rigid heliostat mirror caused by fluctuating wind

In this paper, a unified solution method for free vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions is presented by using the first-order shear deformation theory and Rayleigh–Ritz procedure. The material properties of the structures are assumed to change continuously in the thickness direction according to the general four-parameter power-law distributions in terms of volume fractions of constituents. Each of displacements and rotations of those structures, regardless of boundary conditions, is expressed as a modified Fourier series, which is constructed as the linear superposition of a standard Fourier cosine series supplemented with auxiliary polynomial functions introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the edges and accelerate the convergence of series representations. The excellent accuracy and reliability of the current solutions are confirmed by comparing the present results with those available in the literatures, and numerous new results for functionally graded cylindrical, conical shells and annular plates with elastic boundary conditions are presented. The effects of boundary conditions and the material power-law distribution are also illustrated.