Multidimensional Polynomial Research Articles

Identification and prediction of stochastic dynamical systems in a polynomial chaos basis

Non-parametric system identification techniques have been proposed for constructing predictive models of dynamical systems without detailed knowledge of the mechanisms of energy transfer and dissipation. In a class of such models, multi-dimensional Chebychev polynomials in the state variables are fitted to the observed dynamical state of the system. Due to the approximative nature of this non-parametric model as well as to various other sources of uncertainty such as measurement errors and non-anticipative excitations, the parameters of the model exhibit a scatter that is treated here in a probabilistic context. The statistics of these coefficients are related to the physical properties of the model being analyzed, and are used to endow the model predictions with a probabilistic structure. They are also used to obtain a parsimonious characterization of the predictive model while maintaining a desirable level of accuracy. The proposed methodology is quite simple and robust.

Polynomial approximations of multivariate smooth functions from quasi-random data

We improve a Monte Carlo algorithm which computes accurate approximations of smooth functions on multidimensional Tchebychef polynomials by using quasi-random sequences. We first show that the convergence of the previous algorithm is twice faster using these sequences. Then, we slightly modify this algorithm to make it work from a single set of random or quasi-random points. This especially leads to a Quasi-Monte Carlo method with an increased rate of convergence for numerical integration.

On multiindices and multivariables presentation of the Voigt functions

This paper aims at presenting multiindices and multivariables study of the unified (or generalized) Voigt functions which play an important rôle in the several diverse field of physics such as astrophysical spectroscopy and the theory of neutron reactions. Some expressions (representations) of these functions are given in terms of familiar special functions of multivariables. Further representations and series expansions involving multidimensional classical polynomials (Laguerre and Hermite) of mathematical physics are established.

Discrete truncated powers and lattice points in rational polytope

Discrete truncate power is very useful for studying the number of nonnegative integer solutions of linear Diophantine equations. In this paper, some detail information about discrete truncated power is presented. To study the number of integer solutions of linear Diophantine inequations, the generalized truncated power and generalized discrete truncated power are defined and discussed, respectively. We use generalized discrete truncated powers and multivariate splines to investigate the lattice points in rational polytopes. In particular, we present the degree and period of multidimensional Ehrhart polynomial.

Theory for structure and bulk modulus determination

A method for direct evaluation of both lattice parameters, atomic basis, bulk modulus ${B}_{0},$ and bulk-modulus pressure derivative ${B}_{0}^{\ensuremath{'}}$ of solid materials with complex crystal structures is presented. The explicit and exact results presented here permit a multidimensional polynomial fit of the total energy as a function of all relevant structure parameters to simultaneously determine the equilibrium configuration and the elastic properties. The method allows for inclusion of general (internal) structure parameters, e.g., bond lengths and angles within the unit cell, on an equal footing with the unit-cell lattice parameters. The method is illustrated by the calculation of ${B}_{0}$ and ${B}_{0}^{\ensuremath{'}}$ for a few selected materials with multiple structure parameters for which data are obtained by using first-principles density-functional theory.

Design of linear-phase variable 2-d digital filters using matrix-array decomposition

A novel approach for designing linear-phase variable two-dimensional (2-D) digital filters is described, which is based on the matrix-array decomposition (MAD) of real-valued multidimensional arrays. By using the MAD, the desired variable 2-D magnitude response can be decomposed into the real-valued frequency response specifications of the normal zero-phase constant 2-D filters and the approximation specifications of multidimensional polynomials. Consequently, the problem of approximating the desired variable 2-D magnitude response can be decomposed into sub-problems that involve the design of zero-phase constant 2-D filters and the approximation of multidimensional polynomials. Once the zero-phase constant 2-D filters and the multidimensional polynomials are obtained, interconnecting them yields a zero-phase variable 2-D digital filter. Finally, a linear-phase variable 2-D digital filter can be easily obtained by simply modifying the zero-phase constant 2-D filters (noncausal) to linear-phase ones (causal) through shifting the filter coefficients. Since the sub-problems are much easier to solve than the direct approximation of the given variable 2-D magnitude specification, this MAD-based design approach is extremely efficient and straightforward. In designing zero-phase constant 2-D filters and approximating multidimensional polynomials, we also propose a new objective criterion for selecting appropriate filter orders and polynomial degrees. Furthermore, we also show that the resulting linear-phase variable 2-D filters have highly parallel structures and modularity, which are suitable for high-speed multidimensional signal processing. Two numerical examples are given to demonstrate the effectiveness of the MAD-based design approach.

Two-index multidimensional Gegenbauer polynomials and their integral representations

The authors make use of a recently introduced integral representation of multivariable Chebyshev polynomials in order to derive a number of new results concerning, among other things, addition and duplication theorems for some families of two-index multidimensional Gegenbauer (or ultraspherical) polynomials.

A genetic algorithm approach to the problem of factorization of general multidimensional polynomials

In this paper, a solution to the problem of the multidimensional (m-D) polynomial factorization is attempted by using genetic algorithms (GAs). The proposed method is based on an appropriate minimization of the norm of the difference between the original polynomial and its desirable factorized form. Using GAs, we can obtain better results than with other methods of minimization (numerical techniques, neural networks, etc.). The present methodology, which can also be used for every type of m-D factorization, is illustrated by means of a numerical example.

Un algorithme probabiliste de calcul d'approximations polynômiales sur un hypercube

We describe a Monte Carlo method which enables an iterative computation of the L 2 approximation of a function on any orthonormal basis. We use it for the approximation of smooth functions on an hypercube with the help of multidimensional orthogonal polynomial basis containing only few terms. The algorithm is both a tool for approximation and numerical integration. To cite this article: S. Maire, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

New algorithms for multidimensional discrete Hartley transform

New fast algorithms for multidimensional discrete Hartley transform (MD-DHT) are presented. Based on the index mapping and multidimensional polynomial transform (PT), the MD-DHT is converted into a series of one-dimensional type-II discrete W transform and PTs. The proposed fast algorithms achieve considerable savings on the number of operations. For example, the number of multiplications for computing an r-dimensional DHT is only 1/ r times that needed by the row-column method, and the number of additions is also reduced considerably. Detailed analysis of the computational complexity is also presented.

Multidimensional Polynomial Transform Algorithm For Multidimensional Discrete Hartley Transform

As the generalisation of the multidimensional discrete Hartley transform the multidimensional discrete W transform (MDDWT) is a tool for multidimensional signal processing problems. The multidimensional polynomial transform is used to convert the MDDWT into a series of one-dimensional W transform (DWT). Thus, new polynomial transform algorithms for the MDDWT is obtained. The algorithm needs no complex number operations and is simple in structure. The number of multiplications for computing a r-d DWT is only 1/r times that of the common used row-column method. The number of additions is also reduced considerably.

Fast algorithm for multi-dimensional discrete Hartley transform with size ql1× ql2×⋯× qlr

Fast algorithms are presented for multi-dimensional discrete Hartley transform (MD-DHT) with size q l 1 × q l 2 ×⋯× q l r , where q is an odd prime number, and r>1 is the number of dimensions. By using the multi-dimensional polynomial transform, the MD-DHT can be converted into a series of reduced one-dimensional DHT. Compared to other fast algorithms, the proposed one substantially reduces the overall computational complexity and has a simple computational structure.

The generalized unified computation of multidimensional discrete orthogonal transforms

By introducing a form of reorder for multidimensional data, we propose a unified fast algorithm that jointly employs one-dimensional W transform and multidimensional discrete polynomial transform to compute eleven types of multidimensional discrete orthogonal transforms, which contain three types ofm-dimensional discrete cosine transforms (m-D DCTs), four types ofm-dimensional discrete W transforms (m-D DWTs) (m-dimensional Hartley transform as a special case), and four types of generalized discrete Fourier transforms (m-D GDFTs). For real input, the number of multiplications for all eleven types of them-D discrete orthogonal transforms needed by the proposed algorithm are only 1/m times that of the commonly used corresponding row-column methods, and for complex input, it is further reduced to 1/(2m) times. The number of additions required is also reduced considerably. Furthermore, the proposed algorithm has a simple computational structure and is also easy to be implemented on computer, and the numerical experiments show that the computational efficiency is consistent with the theoretic analysis.

Multi-dimensional Hermite polynomials in quantum optics

We study a class of optical circuits with vacuum input states consisting of Gaussian sources without coherent displacements such as down-converters and squeezers, together with detectors and passive interferometry (beam-splitters, polarisation rotations, phase-shifters etc.). We show that the outgoing state leaving the optical circuit can be expressed in terms of so-called multi-dimensional Hermite polynomials and give their recursion and orthogonality relations. We show how quantum teleportation of photon polarisation can be modelled using this description.

Combined polynomial transform and radix-q algorithm for MD discrete W transform

The type-II r-dimensional discrete W transform (rD-DWT-II) with size q/sup l1//spl times/q/sup l2//spl times//spl middot//spl middot//spl middot//spl times//sup lr/ q where q is an odd prime number, is converted into a series of one-dimensional (1-D) reduced DWT-IIs by using the multidimensional polynomial transform and an index permutation. Then, a radix-q algorithm and a cyclic convolution algorithm are presented for the computation of the 1-D reduced DWT-IIs. The new fast algorithm substantially reduces the overall computational complexity compared with the row-column method. Especially, the number of multiplications required by the proposed algorithm for computing an rD-DWT-II is only 1/r times that needed by the commonly used row-column method.

An improved method for designing variable recursive digital filters with guaranteed stability

Digital filters with variable frequency responses are called variable digital filters. Generally speaking, variable recursive digital filters require lower orders and less computational complexity to satisfy the same desired variable magnitude responses than nonrecursive filters, but the stability of variable recursive filters is difficult to guarantee since their coefficients are also varied in some manner. This paper presents an improved version of an existing algorithm for designing variable recursive one-dimensional (1-D) digital filters with guaranteed stability, which is an improved version of the existing one. The basic idea is to find both the numerator and the denominator coefficients of the transfer function of a variable recursive filter as multi-dimensional (M-D) polynomials of the spectral parameters that define variable magnitude characteristics. To guarantee the stability, some stability constraints have to be imposed on the denominator coefficients. Therefore, we first substitute the denominator coefficients by another set of variables whose values can be arbitrary but without affecting the stability. Then both the numerator and the new denominator coefficients are determined as M-D polynomials of the spectral parameters. Compared to the existing design method, the new one is simpler, but does not degrade the final design accuracy. Two design examples are given to compare the new design method with the existing one.

Limit Cycles Appearing After Perturbation of Certain Multidimensional Vector Fields

Let an unperturbed multidimensional polynomial vector field have an invariant plane L and let the system restricted to this plane be Hamiltonian with a quadratic Hamilton function. Now take a polynomial perturbation of this system. The new system has an invariant surface close to L and the system restricted to it has a certain number of limit cycles. We strive to estimate this number. The linearization of this problem leads to estimation of the number of zeros of certain integral, which is a generalization of the abelian integral. We estimate this number of zeros by C 1+C 2 n, where n is the degree of the perturbation.

Orthogonal and pseudo-orthogonal multi-dimensional Appell polynomials

More than one hundred years ago, Appell introduced a polynomials class characterized by the exponential form of its generator function. This class is well-known on R and the aim of this paper is to offer a multi-dimensional analysis in two directions: we indicate a classical characterization of the Appell polynomials on R d via their generator function and we investigate their orthogonality. Two types of orthogonality are characterized. We relate this study to the theory of Lie algebras and martingales.

On the Robust Stability of 2D Schur Polynomials

In this note, the problem of the robust stability for a two-dimensional (two-variable) Schur polynomial which is the characteristic polynomial of a discrete-time linear time-invariant system is investigated. A new approach based on the Rouche theorem is adopted. The extension to the robust stability for multidimensional (multivariable) polynomials is also provided. Interesting sufficient conditions for such robust stability are derived. A two-dimensional example is included to support the theoretical result.

Multidimensional polynomial transform algorithmfor multidimensional DFT

A method is presented for converting the m-dimensional discrete Fourier transform (MD-DFT) into a number of one-dimensional DFTs (1D-DFTs) by rearranging the order of the input sequence. The result of this conversion is that the number of multiplications for computing an m-dimensional DFT is only 1/m times that of the usually used row-column DFT algorithm. To reduce the number of additions, a multidimensional polynomial transform (MD-PT) is then used and a considerable saving in the number of addition operations is also achieved.