Chebyshev Coefficients Research Articles

On the Convergence Rates of Gauss and Clenshaw--Curtis Quadrature for Functions of Limited Regularity

We study the optimal general rate of convergence of the $n$-point quadrature rules of Gauss and Clenshaw--Curtis when applied to functions of limited regularity: if the Chebyshev coefficients decay at a rate $O(n^{-s-1})$ for some $s>0$, Clenshaw--Curtis and Gauss quadrature inherit exactly this rate. The proof (for Gauss, if $0<s<2$, there is numerical evidence only) is based on work of Curtis, Johnson, Riess, and Rabinowitz from the early 1970s and on a refined estimate for Gauss quadrature applied to Chebyshev polynomials due to Petras (1995). The convergence rate of both quadrature rules is up to one power of $n$ better than polynomial best approximation; hence, the classical proof strategy that bounds the error of a quadrature rule with positive weights by polynomial best approximation is doomed to fail in establishing the optimal rate.

Chebyshev polynomial solution of nonlinear integral equations

In the current work, the Chebyshev collocation method is adopted to find an approximate solution for nonlinear integral equations. Properties of the Chebyshev polynomials and operational matrix are used in the integral equation of a system consisting of nonlinear algebraic equations with the unknown Chebyshev coefficients. Numerical examples are presented to illustrate the method and results are discussed.

TTS in LAOS: validation of time-temperature superposition under large amplitude oscillatory shear

The validation of time-temperature superposition of non-linear parameters obtained from large amplitude oscillatory shear is investigated for a model viscoelastic fluid. Oscillatory time sweeps were performed on a 11 wt.% solution of high molecular weight polyisobutylene in pristane as a function of temperature and frequency and for a broad range of strain amplitudes varying from the linear to the highly non-linear regime. Lissajous curves show that this reference material displays strong non-linear behaviour when the strain amplitude is exceeding a critical value. Elastic and viscous Chebyshev coefficients and alternative non-linear parameters were obtained based on the framework of Ewoldt et al. (J Rheol 52(6):1427–1458, 2008) as a function of temperature, frequency and strain amplitude. For each strain amplitude, temperature shift factors a T (T) were calculated for the first order elastic and viscous Chebyshev coefficients simultaneously, so that master curves at a certain reference temperature T ref were obtained. It is shown that the expected independency of these shift factors on strain amplitude holds even in the non-linear regime. The shift factors a T (T) can be used to also superpose the higher order elastic and viscous Chebyshev coefficients and the alternative moduli and viscosities onto master curves. It was shown that the Rutgers-Delaware rule also holds for a viscoelastic solution at large strain amplitudes.

Chebyshev Spectral Methods and the Lane-Emden Problem

The three-dimensional spherical polytropic Lane-Emden problem is yrr + (2/r)yr + y m = 0, y(0) = 1, yr(0) = 0 where m∈ (0,5) is a constant parameter. The domain is r∈ (0,ξ) where ξ is the first root of y(r). We recast this as a non- linear eigenproblem, with three boundary conditions and ξ as the eigenvalue allow- ing imposition of the extra boundary condition, by making the change of coordinate x≡ r/ξ: yxx + (2/x)yx +ξ 2 y m = 0, y(0) = 1, yx(0) = 0, y(1) = 0. We find that a Newton-Kantorovich iteration always converges from an m-independent starting point y (0) (x) = cos((π/2)x),ξ (0) = 3. We apply a Chebyshev pseudospectral method to discretize x. The Lane-Emden equation has branch point singularities at the endpoint x = 1 whenever m is not an integer; we show that the Chebyshev coefficients are a n ∼ constant/n 2m+5 as n→∞. However, a Chebyshev truncation of N = 100 always gives at least ten decimal places of accuracy — much more accuracy when m is an inte- ger. The numerical algorithm is so simple that the complete code (in Maple) is given as a one page table. AMS subject classifications: 65L10, 65D05, 85A15

Spectral Method for Solving The General Form Linear Fredholm-Volterra Integro Differential Equations Based on Chebyshev Polynomials

The main aim of this paper reports a pseudo spectral method based on integrated Chebyshev polynomials for numerically solving higher order linear Fredholm Volterra integro differential equations (FVIDE) in the most general form. This method transforms FVIDE to matrix equations which correspond to a system of linear algebraic equations with unknown Chebyshev coefficients. The present numerical results are in satisfactory agreement with the exact solutions and show the advantages of this method to some other known methods.

The solution of high-order nonlinear ordinary differential equations by Chebyshev Series

By the use of the Chebyshev series, a direct computational method for solving the higher order nonlinear differential equations has been developed in this paper. This method transforms the nonlinear differential equation into the matrix equation, which corresponds to a system of nonlinear algebraic equations with unknown Chebyshev coefficients, via Chebyshev collocation points. The solution of this system yields the Chebyshev coefficients of the solution function. An algorithm for this nonlinear system is also proposed in this paper. The method is valid for both initial-value and boundary-value problems. Several examples are presented to illustrate the accuracy and effectiveness of the method.

Análise dos métodos de diferenças finitas e expansão rápida na migração reversa no tempo

Neste trabalho mostramos atraves da solucao analitica da equacao da onda no tempo e do metodo de expansao rapida (REM) que e possivel obter a solucao da equacao da onda que utiliza esquemas de diferencas-finitas de qualquer ordem no tempo. Alem disso, demonstramos que a grande vantagem do REM e que o metodo permite usar qualquer intervalo de amostragem temporal para realizar a extrapolacao do campo de ondas, enquanto que o metodo de diferencas finitas impoe limites ao intervalo usado, devido a condicao de estabilidade e a dispersao numerica. Fizemos tambem uma analise das aproximacoes em series de Taylor e em polinomios de Chebyshev para a funcao cosseno que aparece na solucao analitica da equacao da onda no tempo. E finalizamos este trabalho mostrando o desempenho dos metodos numericos na migracao reversa no tempo pos e pre-empilhamento e demonstramos que o REM, combinado com o metodo espectral para calcular as derivadas espaciais, pode ser usado para obter resultados numericamente estaveis e com menor custo computacional do que um metodo de extrapolacao do campo de ondas no tempo com um esquema de diferencas-finitas, dentro do mesmo nivel de precisao.

Application of orthogonal expansions for approximate integration of ordinary differential equations

An approximate method to solve the Cauchy problem for normal and canonical systems of second-order ordinary differential equations is proposed. The method is based on the representation of a solution and its derivative at each integration step in the form of partial sums of series in shifted Chebyshev polynomials of the first kind. A Markov quadrature formula is used to derive the equations for the approximate values of Chebyshev coefficients in the right-hand sides of systems. Some sufficient convergence conditions are obtained for the iterative method solving these equations. Several error estimates for the approximate Chebyshev coefficients and for the solution are given with respect to the integration step size.

A Karatsuba-Based Algorithm for Polynomial Multiplication in Chebyshev Form

In this paper, we present a new method for multiplying polynomials in Chebyshev form. Our approach has two steps. First, the well-known Karatsuba's algorithm is applied to polynomials constructed by using Chebyshev coefficients. Then, from the obtained result, extra arithmetic operations are used to write the final result in Chebyshev form. The proposed algorithm has a quadratic computational complexity. We also compare our method to other approaches.

Approximate Stability Analysis and Computation of Solutions of Nonlinear Delay Differential Algebraic Equations with Time Periodic Coefficients

Approximate stability analysis of nonlinear delay differential algebraic equations (DDAEs) with periodic coefficients is proposed with a geometric interpretation of evolution of the linearized system. Firstly, a numerical algorithm based on direct integration by expansion in terms of Chebyshev polynomials is derived for linear analysis. The proposed algorithm is shown to have deeper connections with and be computationally less cumbersome than the solution of the underlying semi-explicit system via a similarity transformation. The stability of time periodic DDAE systems is characterized by the spectral radius of a “monodromy matrix”, which is a finite-dimensional approximation of a compact infinite-dimensional operator. The monodromy matrix is essentially a map of the Chebyshev coefficients (or collocation vector) of the state from the delay interval to the next adjacent interval of time. The computations are entirely performed with the original system to avoid cumbersome transformations associated with the semi-explicit form of the system. Next, two computational algorithms, the first based on perturbation series and the second based on Chebyshev spectral collocation, are detailed to obtain solutions of nonlinear DDAEs with periodic coefficients for consistent initial functions.

On the convergence of the method for indefinite integration of oscillatory and singular functions

We consider the problem of convergence and error estimation of the method for computing indefinite integrals proposed in Keller [8]. To this end, we have analysed the properties of the difference operator related to the difference equation for the Chebyshev coefficients of a function that satisfies a given linear differential equation with polynomial coefficients. Properties of this operator were never investigated before. The obtained results lead us to the conclusion that the studied method is always convergent. We also give a rigorous proof of the error estimates.

Effective solution of a linear system with Chebyshev coefficients

This paper presents an efficient algorithm for a special triangular linear system with Chebyshev coefficients. We present two methods of derivations, the first is based on formulae where the nth power of x is solved as the sum of Chebyshev polynomials and modified for a linear system. The second deduction is more complex and is based on the Gauss–Banachiewicz decomposition for orthogonal polynomials and the theory of hypergeometric functions which are well known in the context of orthogonal polynomials. The proposed procedure involves 𝒪(n m) operations only, where n is matrix size of the triangular linear system L and m is number of the nonzero elements of vector b. Memory requirements are 𝒪(m), and no recursion formula is needed. The linear system is closely related to the optimal pulse-wide modulation problem.

Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms

When a function is expanded as $$f(x) \approx \sum \limits _{n=0}^{N-1}\, a_{n}\phi_{n}(x)$$ for some set of basis functions $${\phi_{j}(x)}$$ , its spectral coefficients a n generally have an asymptotic approximation, as n→∞, in the form of an inverse power series plus terms that decrease exponentially with n. If f(x) is analytic on the expansion interval, then all the coefficients of the inverse power series are zero and the problem becomes one of “beyond-all-orders” or “exponential” asymptotics. The method of steepest descent for integrals and other complex-path integration techniques can successfully connect the exponentially small behavior of the spectral coefficients to the singularities of f(x) off the expansion interval. Many examples are given in both one and two dimensions.

Chebyshev polynomial approximation for high‐order partial differential equations with complicated conditions

AbstractIn this article, a new method is presented for the solution of high‐order linear partial differential equations (PDEs) with variable coefficients under the most general conditions. The method is based on the approximation by the truncated double Chebyshev series. PDE and conditions are transformed into the matrix equations, which corresponds to a system of linear algebraic equations with the unknown Chebyshev coefficients, via Chebyshev collocation points. Combining these matrix equations and then solving the system yields the Chebyshev coefficients of the solution function. Some numerical results are included to demonstrate the validity and applicability of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Computational techniques for a quantum control problem with H1-cost

The accurate numerical computation of optimal controls of infinite-dimensional quantum control problems is a very difficult task that requires us to take into account the features of the original infinite-dimensional problem. An important issue is the choice of the functional space where the minimization process is defined. A systematic comparison of L2- versus H1-based minimization shows that the choice of the appropriate functional space matters and has many consequences in the implementation of some optimization techniques. A matrix-free cascadic BFGS algorithm is introduced in the L2 and H1 settings, and it is demonstrated that the choice of H1 over L2 results in a substantial performance and robustness increase. A comparison between optimal control resulting from function space minimization and the control obtained by minimization over Chebyshev and proper orthogonal decomposition basis function coefficients is presented.

A method for indefinite integration of oscillatory and singular functions

We propose a general method for computing indefinite integrals of the form $$I(y) = {\int_0^y {g(t)k} }(t)dt\;\;\;\;(0 \leqslant y \leqslant y_{{{\text{max}}}} ),$$ where g is a smooth function, and k is a function that contains a singular factor or is rapidly oscillatory. The only assumption on k is that it satisfies a linear differential equation with polynomial coefficients. The approximate value of the integral is given in terms of Chebyshev coefficients of functions that form a solution of a certain system of differential equations. As an illustration, we present effective algorithms for computing indefinite integrals of the functions g(t)|t–d|αeiωt, g(t)log|t–d| eiωt, g(t) tαJν(ct).

Application of Chebyshev II–Bernstein basis transformations to degree reduction of Bézier curves

A polynomial curve on [ 0 , 1 ] can be expressed in terms of Bernstein polynomials and Chebyshev polynomials of the second kind. We derive the transformation matrices that map the Bernstein and Chebyshev coefficients into each other, and examine the stability of this linear map. In the p = 1 and ∞ norms, the condition number of the Chebyshev–Bernstein transformation matrix grows at a significantly slower rate with n than in the power–Bernstein case, and the rate is very close (somewhat faster) to the Legendre–Bernstein case. Using the transformation matrices, we present a method for the best multi-degree reduction with respect to the t − t 2 -weighted square norm for the unconstrained case, which is further developed to provide a good approximation to the best multi-degree reduction with constraints of endpoints continuity of orders r , s ( r , s ≥ 0 ). This method has a quadratic complexity, and may be ill-conditioned when it is applied to the curves of high degree. We estimate the posterior L 1 -error bounds for degree reduction.

PRODUCTION OF ANTICENTAURO EVENTS IN ULTRA-RELATIVISTIC HEAVY ION COLLISIONS

We propose a novel method for studying the production of anticentauro events in high energy heavy ion collisions utilizing Chebyshev expansion coefficients. These coefficients have proved to be very efficient in investigating the pattern of fluctuations in neutral pion fraction. For the anticentauro like events, the magnitude of first few coefficients is strongly enhanced (≈ 3 times) as compared to those of normal HIJING events. Various characteristics of Chebyshev coefficients are studied in detail and the probability of formation of exotic events is calculated from the simulated events.

Computing the zeros of a Fourier series or a Chebyshev series or general orthogonal polynomial series with parity symmetries

In recent years, good algorithms have been developed for finding the zeros of trigonometric polynomials and of ordinary polynomials when written in the form of a truncated Chebyshev polynomial or Legendre polynomial series. In each case, the roots can be found from the eigenvalues of a generalized Frobenius companion matrix whose elements are trivial functions of the Fourier coefficients or Chebyshev coefficients. However, the QR method for computing the companion matrix eigenvalues has a cost that grows proportionally to N 3 where N is the polynomial degree. (By exploiting the special structure of the companion matrices, the cost can be reduced to O ( N 2 ) , but only for large N .) Here, we show that if the polynomial has definite parity, such as a trigonometric polynomial composed only of cosines or a polynomial that is a sum only of Chebyshev polynomials of odd degree, one can exploit these symmetries to halve the size of the problem. This reduces costs in the companion matrix method by a factor ranging between four and eight. For trigonometric polynomials, we give transformations that dramatically reduce costs even if the roots are found by an algorithm other than the companion matrix procedure. We further give reductions for trigonometric polynomials with double parity symmetries which save a factor of sixteen to a factor of sixty-four in the companion matrix algorithm. Special functions such as spherical harmonics, Mathieu functions, prolate spheroidal wavefunctions and Hough functions, all represented by truncated Fourier series with double parity, are a rich source of applications.

A simple method for prediction of first-order modal field and cladding decay parameter in graded index fiber

Based on a series expansion method involving the Chebyshev technique, we present analytical expressions for first higher order modal field as well as cladding decay parameter in case of graded index fiber. This method utilizes the formulation of a linear relation of the ratio of first- and zero-order modified Bessel functions, with reciprocal of Cladding decay parameter. This leads to the evaluation of the Cladding decay parameter from a simple equation involving a third-order determinant and the first higher order modal field is thereafter found from two simple equations. Thus the paper presents the technique of avoiding complicated calculations of Chebyshev coefficients and therefore, the present formalism will be extremely important for practical engineering problems. Taking step and parabolic index fibers as examples, we show that our estimations of the said propagation characteristics agree excellently with the available exact results. The concerned calculations involve very little computations.