On expansions in orthogonal polynomials

A recently introduced fast algorithm for the computation of the first N terms in an expansion of an analytic function into ultraspherical polynomials consists of three steps: Firstly, each expansion coefficient is represented as a linear combination of derivatives; secondly, it is represented, using the Cauchy integral formula, as a contour integral of the function multiplied by a kernel; finally, the integrand is transformed to accelerate the convergence of the Taylor expansion of the kernel, allowing for rapid computation using Fast Fourier Transform. In the current paper we demonstrate that the first two steps remain valid in the general setting of orthogonal polynomials on the real line with finite support, orthogonal polynomials on the unit circle and Laurent orthogonal polynomials on the unit circle.     

一个扩散问题的自然边界元法与有限元法组合

本文讨论由Helmholtz方程描述的扩散问题的自然边界元法与有限元法的组合．取一个圆作为公共边界，用Fourier展开建立边界积分方程，将无界区域上的问题化为有界区域上的非局部边值问题．在变分方程中公共边界上的未知量只包含函数本身而不包含其法向导数，从而减少了未知数的数目，并且边界元剐度矩阵只有极少量不同的元素，有利于数值计算．这种组台方法优越于建立在直接边界元法基础上的组合方法．文中证明了变分解的唯一性，数值解的收敛性和误差估计．最后讨论了数值技术并给出一个算倒．     

Complex velocity potential for an ellipse and a circle in translation and rotation

This paper investigates the analytical solutions of two-dimensional Neumann-type external boundary-value problem for an ellipse and a circle in complex analysis treatment. The transformation of harmonics between two planes is expanded in Laurent series. The circle theorem is extended so as to suit to the system of the ellipse and the circle. By means of Basset's process, the harmonic expressions of the system are found in the form of recurrence formulae which are suitable for numerical computations. The complex velocity potential of the system in translation and rotation is represented by the summation of the harmonic expressions.     

Numerical computation of the fourier transform using Laguerre functions and the Fast Fourier Transform

Summary In this paper we propose a numerical technique for the computation of Fourier transforms. It uses a bilateral expansion of the unknown transformed function with respect to Laguarre functions. The expansion coefficients are obtained via trigonometric interpolation and may be computed very efficiently by means of the Fast Fourier Transform. The convergence of the algorithm is analyzed and numerical results are presented which confirm that it works well.     

The use of phase portraits to visualize and investigate isolated singular points of complex functions

ABSTRACT

Undergraduate students usually study Laurent series in a standard course of Complex Analysis. One of the major applications of Laurent series is the classification of isolated singular points of complex functions. Although students are able to find series representations of functions, they may struggle to understand the meaning of the behaviour of the function near isolated singularities. In this paper, I briefly describe the method of domain colouring to create enhanced phase portraits to visualize and study isolated singularities of complex functions. Ultimately this method for plotting complex functions might help to enhance students' insight, in the spirit of learning by experimentation. By analysing the representations of singularities and the behaviour of the functions near their singularities, students can make conjectures and test them mathematically, which can help to create significant connections between visual representations, algebraic calculations and abstract mathematical concepts.     

Numerical solution of a class of nonlinear boundary value problems for analytic functions

Summary We analyse a numerical method for solving a nonlinear parameter-dependent boundary value problem for an analytic function on an annulus. The analytic function to be determined is expanded into its Laurent series. For the expansion coefficients we obtain an operator equation exhibiting bifurcation from a simple eigenvalue. We introduce a Galerkin approximation and analyse its convergence. A prominent problem falling into the class treated here is the computation of gravity waves of permanent type in a fluid. We present numerical examples for this case.

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Zusammenfassung Wir betrachten ein numerisches Verfahren zur Lösung eines nichtlinearen, parameterabhängigen Randwertproblems für analytische Funktionen auf einem Kreisring. Durch Laurent-Entwicklung erhalten wir für die Koeffizienten eine Operatorgleichung, die Verzweigung aufweist. Wir beweisen Konvergenz für das Galerkin-Verfahren. Als numerisches Beispiel untersuchen wir das Problem permanenter Wasserwellen unter dem Einfluß der Schwerkraft.

     

Determination of reflector surfaces from near-field scattering data II. Numerical solution

Let be a nonisotropic point source of light, and the power intensity of this source in direction . Suppose that the light rays emitted by the source through an aperture fall on a perfectly reflecting surface and reflect off it so that the reflected rays illuminate a closed domain on some plane with intensity . The inverse problem consists of constructing the reflector surface from given position of the source , the input aperture , function , “target” set , and output intensity . For example, the input intensity may have a “bell”-like shape and we may wish to redistribute the energy uniformly over a prespecified region. The analytical formulation of the described above problem leads to a non-linear partial differential equation of Monge-Ampère type. In our previous paper we proved existence of weak solutions to this inverse problem and in this paper we describe and illustrate with examples an algorithm for its numerical solution. The proposed numerical method can be easily modified for the case when is a closed domain on an arbitrary surface. Received November 26, 1996     

On the generating function for consecutively weighted permutations

We show that the analytic continuation of the exponential generating function associated to consecutive weighted pattern enumeration of permutations only has poles and no essential singularities. The proof uses the connection between permutation enumeration and functional analysis, and as well as the Laurent expansion of the associated resolvent. As a consequence, we give a partial answer to a question of Elizalde and Noy: when is the multiplicative inverse of the exponential generating function for the number permutations avoiding a single pattern an entire function? Our work implies that it is enough to verify that this function has no zeros to conclude that the inverse function is entire.     

Spectral factorization of Laurent polynomials

We analyse the performance of five numerical methods for factoring a Laurent polynomial, which is positive on the unit circle, as the modulus squared of a real algebraic polynomial. It is found that there is a wide disparity between the methods, and all but one of the methods are significantly influenced by the variation in magnitude of the coefficients of the Laurent polynomial, by the closeness of the zeros of this polynomial to the unit circle, and by the spacing of these zeros. This revised version was published online in August 2006 with corrections to the Cover Date.     

Singularities of solutions to linear,second order analytic elliptic equations in two independent variables I. The completely regular boundary

Two methods are described for the a priori location of singularities of solutions to exterior boundary value problems. One uses an expansion for the solution in a circle centered on a regular exterior point P. A singularity lies on the circle of convergence. The envelope of these circles, generated as P makes a circuit about the closed boundary, circumscribes the singularities. The radius of convergence depends on singularities of the solution u(s) and its normal derivative v(s) on the boundary. The second method employs complex characteristics to relate singularities of the boundary data to real singularities of the solution. Integral equations connecting (y), v(s) and the analytic boundary condition are used to continue the data into the complex s-plane and to locate their singularities. Explicit solution of the integral equations is unnecessary; some nonlinear boundary conditions can be handled.     

The cracked-beam problem solved by the boundary approximation method

The cracked-beam problem, as a variant of Motz’s problem, is discussed, and its very accurate solution in double precision is explicitly provided by the boundary approximation method (BAM) (i.e., the Trefftz method). Half of its expansion coefficients are zero, which is supported by an a posteriori analysis. Finding a good model of singularity problems is important for studying numerical methods. As a singularity model, the cracked-beam problem given in this paper seems to be superior to Motz’s problem in Li et al. [Z.C. Li, R. Mathon, P. Serman, Boundary methods for solving elliptic problem with singularities and interfaces, SIAM J. Numer. Anal. 24 (1987) 487–498].     

Slender body expansions in potential theory along a finite straight line

Consider a distribution of singularities in a potential field along a finite straight line such that the potential satisfies the Laplace equation. An example is a distribution of sources representing a ship or missile moving with forward velocity in a potential inviscid flow field. Such bodies are often truncated or bluff at the ends, and so the strength of the resulting distributions may not gradually tend to zero close to these ends and may instead be non-zero finite. A near-field expansion is obtained which accounts for this using the slender body theory integral splitting method. All terms in the expansion are obtained, and the coefficient of each term in the infinite sequence is given in terms of differentials of the distribution strength. Hence an exact separation of variables solution (separating the axial distance from the cross-sectional distances) is obtained for the potential. This is different from previous representations in that it represents a distribution over a finite length, and the resulting expansion is a simple single summation expression that is straightforward to apply. The resulting numerical scheme is discussed, in particular the evaluation close to the ends and also a comparison between the presented slender body theory and existing numerical methods.     

Numerical solution of the Painlevé IV equation

A numerical method for solving the Cauchy problem for the fourth Painlevé equation is proposed. The difficulty of the problem is that the unknown function can have movable singular points of the pole type; moreover, the equation may have singularities at the points where the solution vanishes. The positions of poles and zeros of the solution are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities in the corresponding point and its neighborhood. Numerical results confirming the efficiency of this method are presented.     

Algorithms for solving Hermite interpolation problems using the Fast Fourier Transform

We present a method for computing the Hermite interpolation polynomial based on equally spaced nodes on the unit circle with an arbitrary number of derivatives in the case of algebraic and Laurent polynomials. It is an adaptation of the method of the Fast Fourier Transform (FFT) for this type of problems with the following characteristics: easy computation, small number of operations and easy implementation.In the second part of the paper we adapt the algorithm for computing the Hermite interpolation polynomial based on the nodes of the Tchebycheff polynomials and we also study Hermite trigonometric interpolation problems.     

On a Uniform Treatment of Darboux’s Method

Let F(z) be an analytic function in |z| < 1. If F(z) has only a finite number of algebraic singularities on the unit circle |z| = 1, then Darbouxs method can be used to give an asymptotic expansion for the coefficient of zⁿ in the Maclaurin expansion of F(z). However, the validity of this expansion ceases to hold, when the singularities are allowed to approach each other. A special case of this confluence was studied by Fields in 1968. His results have been considered by others to be too complicated, and desires have been expressed to investigate whether any simplification is feasible. In this paper, we shall show that simplification is indeed possible. In the case of two coalescing algebraic singularities, our expansion involves only two Bessel functions of the first kind.     

A singular function boundary integral method for elliptic problems with singularities

In this work we present a singular function boundary integral method for elliptic problems with boundary singularities. In this method, the approximation is constructed from the truncated asymptotic expansion for the solution near the singular point and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier functions. The resulting discrete problem is posed and solved on the boundary of the domain, away from the point of singularity. We are able to show that the method approximates the generalized stress intensity factors, i.e. the coe cients in the asymptotic expansion, at an exponential rate. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of the Painlevé V equation

A numerical method for solving the Cauchy problem for the fifth Painlevé equation is proposed. The difficulty of the problem is that the unknown function can have movable singular points of the pole type; moreover, the equation has singularities at the points where the solution vanishes or takes the value 1. The positions of all of these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Numerical results illustrating the potentials of this method are presented.     

RCMARS: Robustification of CMARS with different scenarios under polyhedral uncertainty set

Our recently developed CMARS is powerful in handling complex and heterogeneous data. We include into CMARS the existence of uncertainty about the scenarios. Indeed, data include noise in both output and input variables. Therefore, solutions of the optimization problem may reveal a remarkable sensitivity to perturbations in the parameters of the problem. The data uncertainty results in uncertain constraints and objective function. To overcome this difficulty, we refine our CMARS algorithm by a robust optimization technique proposed to cope with data uncertainty. In our previous study, we present the new robust CMARS (RCMARS) in theory and method and illustrate it with a numerical example. In this study, we present RCMARS results with different uncertainty scenarios for our numerical example.     

Nonlinear inversion of a band-limited Fourier transform

We consider the problem of reconstructing a compactly supported function with singularities either from values of its Fourier transform available only in a bounded interval or from a limited number of its Fourier coefficients. Our results are based on several observations and algorithms in [G. Beylkin, L. Monzón, On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal. 19 (1) (2005) 17–48]. We avoid both the Gibbs phenomenon and the use of windows or filtering by constructing approximations to the available Fourier data via a short sum of decaying exponentials. Using these exponentials, we extrapolate the Fourier data to the whole real line and, on taking the inverse Fourier transform, obtain an efficient rational representation in the spatial domain. An important feature of this rational representation is that the positions of its poles indicate location of singularities of the function. We consider these representations in the absence of noise and discuss the impact of adding white noise to the Fourier data. We also compare our results with those obtained by other techniques. As an example of application, we consider our approach in the context of the kernel polynomial method for estimating density of states (eigenvalues) of Hermitian operators. We briefly consider the related problem of approximation by rational functions and provide numerical examples using our approach.     

Contact of shells with a sharp linear stamp of finite length

The action of a plane, absolutely rigid stamp on a transversely isotropic shell is investigated. The use of the equations of shells with finite shear stiffness enables the correct formulation of the problem of the action on a shell by a stamp of fixed length. The problem is reduced to an integral equation. Applying the Fourier transform, the kernel of the integral equation is represented in the form of an expansion with respect to Chebyshev polynomials. By the representation of the solution of the integral equation in the form of a product, of a series of Chebyshev polynomials and a function that takes into account the singularities of the solution at the boundary of the contact zone, the considered problem is reduced to the solving of an infinite system of linear algebraic equations, whose coefficients have been determined by the methods of numerical integration. As an example a problem for a cylindrical shell has been solved.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 59–63, 1989.