Differentiation matrices in polynomial bases

In the paper a comparison is made between two approximate methods of system analysis based on the use of the operational matrices of integration and differentiation. Both methods applied to model problems involving linear time-invariant differential equations yielded results of equal order of accuracy for the same number of orthogonal polynomial terms. However, the operational matrix of differentiation method is not limited to initial-value problems, but may easily be applied to boundary-value problems. In addition, due to the exact definition of this matrix, transformations from an orthogonal polynomial base to another may easily be performed with the use of the operational matrix of polynomial series transformation. The operational matrix of differentiation approach may be easily applied to the analysis of other types of systems such as time-varying and distributed parameter systems, and can be easily extended for other types of orthogonal differentiable base functions such as the Fourier trigonometric functions and the Bessel functions.

We propose an algorithm based on Newton's method and subdivision for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the intersections between a line and a surface, which has applications in graphics and computer-aided geometric design. The algorithm can operate on polynomials represented in any basis that satisfies a few conditions. The power basis, the Bernstein basis, and the first-kind Chebyshev basis are among those compatible with the algorithm. The main novelty of our algorithm is an analysis showing that its running is bounded only in terms of the condition number of the polynomial's zeros and a constant depending on the polynomial basis.

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Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies

Matrix methods for quadrature formulas on the unit circle. A survey

Cubature rules based on a bivariate degree-graded alternative orthogonal basis and their applications

Generalized Taylor series in hermitian Clifford analysis

Convergence properties of hypercomplex derivative bases of special monogenic polynomials are studied. These new results extend and improve a lot of known works from the complex case to Clifford setting. Copyright © 2009 John Wiley & Sons, Ltd.

We develop finite element data structures for T‐splines based on Bézier extraction generalizing our previous work for NURBS. As in traditional finite element analysis, the extracted Bézier elements are defined in terms of a fixed set of polynomial basis functions, the so‐called Bernstein basis. The Bézier elements may be processed in the same way as in a standard finite element computer program, utilizing exactly the same data processing arrays. In fact, only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same. A byproduct of the extraction process is the element extraction operator. This operator localizes the topological and global smoothness information to the element level, and represents a canonical treatment of T‐junctions, referred to as ‘hanging nodes’ in finite element analysis and a fundamental feature of T‐splines. A detailed example is presented to illustrate the ideas. Copyright © 2011 John Wiley & Sons, Ltd.

A general procedure to define non-smooth versions of classical approximants by means of fractal interpolation functions is proposed. A complete and explicit description in the frequency domain of the functions constructed is obtained through their exact Fourier transforms. In particular, the generalization of the polynomial interpolation is developed. The Lagrange basis of the space of polynomials of degree lower or equal than N is generalized to a basis of fractal polynomials. As a consequence of the process, the density of the polynomial fractal interpolation functions with non-null scale vector in the space of continuous functions in a compact interval is deduced. Furthermore, a method for the interpolation of real data is proposed, by the construction of a fractal function coming from any classical approximant. The convergence of the process when the partition is refined is proved, supposing the convergence of the smooth interpolant.

An orthogonal system of polynomials is constructed in the basis of Lagerra polynomials, and the problem of approximation of a function in the constructed orthogonal basis is considered. During the computational experiment, the influence of various errors on the approximation was investigated on model problems.

The structure of orthogonal polynomials on $\mathbb{R}^{2}$ with the weight function $| x_{1}^{2}-x_{2}^{2}|^{2k_{0}}| x_{1}x_{2}|^{2k_{1}}e^{-(x_{1}^{2}+x_{2}^{2})/2}$ is based on the Dunkl operators of type $B_{2}$. This refers to the full symmetry group of the square, generated by reflections in the lines $x_{1}=0$ and $x_{1}-x_{2}=0$. The weight function is integrable if $k_{0},k_{1},k_{0}+k_{1}>-\frac{1}{2}$. Dunkl operators can be defined for polynomials taking values in a module of the associated reflection group, that is, a vector space on which the group has an irreducible representation. The unique 2-dimensional representation of the group $B_{2}$ is used here. The specific operators for this group and an analysis of the inner products on the harmonic vector-valued polynomials are presented in this paper. An orthogonal basis for the harmonic polynomials is constructed, and is used to define an exponential-type kernel. In contrast to the ordinary scalar case the inner product structure is positive only when $(k_{0},k_{1})$ satisfy $-\frac{1}{2}<k_{0}\pm k_{1}<\frac{1}{2}$. For vector polynomials $(f_{i})_{i=1}^{2}$, $(g_{i})_{i=1}^{2}$ the inner product has the form $\iint_{\mathbb{R}^{2}}f(x) K(x) g(x)^{T}e^{-(x_{1}^{2}+x_{2}^{2})/2}dx_{1}dx_{2}$ where the matrix function $K(x)$ has to satisfy various transformation and boundary conditions. The matrix $K$ is expressed in terms of hypergeometric functions.

This paper is concerned with the extension of the notion of Hadamard product of bases of polynomials in one complex variable to the setting of Clifford analysis. It is proposed to study the effect of the algebraic property of a Hadamard product base of special monogenic polynomials on its convergence properties. Unlike the complex case some of these convergence properties may not be interchanged.

Polynomial chaos expansion (PCE) with Latin hypercube sampling (LHS) is employed for calculating the vibrational frequencies of an inviscid incompressible fluid partially filled in a rectangular tank with and without a baffle. Vibration frequencies of the coupled system are described through their projections on the PCE which uses orthogonal basis functions. PCE coefficients are evaluated using LHS. Convergence on the coefficient of variation is used to find the orthogonal polynomial basis function order which is employed in PCE. It is observed that the dispersion in the eigenvalues is more in the case of a rectangular tank with a baffle. The accuracy of the PCE method is verified with standard MCS results and is found to be more efficient.

In this paper, a new technique of decentralized control synthesis for linear time variant systems is proposed by using orthogonal functions. Making use of this mathematical tool allows primarily to transform the interconnected time-varying parameters differential state equations to a set of algebraic coupled ones. This is done by expanding the system input and output variables over a shifted Legendre polynomials basis and then computations are carried with the help of orthogonal basis operational matrices and Kronecker product properties. The decentralized state feedback control parameters for the interconnected linear time variant system is therefore determined by minimizing the gap between the closed loop system and a chosen linear reference model outputs coefficients in the orthogonal basis.