On the coefficients of integrated expansions of ultraspherical polynomials
SIAM Journal on Numerical Analysis
An improved pseudospectral method for fluid dynamics boundary value problems
Journal of Computational Physics
Spectral integration and two-point boundary value problems
SIAM Journal on Numerical Analysis
On the errors incurred calculating derivatives using Chebyshev polynomials
Journal of Computational Physics
Numerical recipes in FORTRAN (2nd ed.): the art of scientific computing
Numerical recipes in FORTRAN (2nd ed.): the art of scientific computing
Analysis and Application of Fourier--Gegenbauer Method to Stiff Differential Equations
SIAM Journal on Numerical Analysis
Journal of Computational and Applied Mathematics
Accuracy Enhancement for Higher Derivatives using Chebyshev Collocation and a Mapping Technique
SIAM Journal on Scientific Computing
Spectrally Accurate Solution of Nonperiodic Differential Equations by the Fourier--Gegenbauer Method
SIAM Journal on Numerical Analysis
A MATLAB differentiation matrix suite
ACM Transactions on Mathematical Software (TOMS)
On the A-Stability of Runge--Kutta Collocation Methods Based on Orthogonal Polynomials
SIAM Journal on Numerical Analysis
Parameter Optimization and Reduction of Round Off Error for the Gegenbauer Reconstruction Method
Journal of Scientific Computing
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Determining Analyticity for Parameter Optimization of the Gegenbauer Reconstruction Method
SIAM Journal on Scientific Computing
Integration Preconditioning Matrix for Ultraspherical Pseudospectral Operators
SIAM Journal on Scientific Computing
Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation)
Journal of Computational and Applied Mathematics
Journal of Computational Physics
Integration matrix based on arbitrary grids with a preconditioner for pseudospectral method
Journal of Computational and Applied Mathematics
Pseudospectral integration matrix and boundary value problems
International Journal of Computer Mathematics
Mathematics and Computers in Simulation
Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers
Journal of Computational Physics
Recovery of High Order Accuracy in Radial Basis Function Approximations of Discontinuous Problems
Journal of Scientific Computing
Differentiation by integration with Jacobi polynomials
Journal of Computational and Applied Mathematics
Computational Optimization and Applications
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
On the optimization of Gegenbauer operational matrix of integration
Advances in Computational Mathematics
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This paper treats definite integrations numerically using Gegenbauer quadratures. The novel numerical scheme introduces the idea of exploiting the strengths of the Chebyshev, Legendre, and Gegenbauer polynomials through a unified approach, and using a unique numerical quadrature. In particular, the developed numerical scheme employs the Gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms. For a large-scale number of expansion terms, the numerical quadrature has the advantage of converging to the optimal Chebyshev and Legendre quadratures in the L^~-norm and L^2-norm, respectively. The key idea is to construct the Gegenbauer quadrature through discretizations at some optimal sets of points of the Gegenbauer-Gauss (GG) type in a certain optimality sense. We show that the Gegenbauer polynomial expansions can produce higher-order approximations to the definite integrals @!"-"1^x^"^if(x)dx of a smooth function f(x)@?C^~[-1,1] for the small range by minimizing the quadrature error at each integration point x"i through a pointwise approach. The developed Gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes. Exact integrations are obtained for polynomials of any arbitrary degree n if the number of columns in the developed Gegenbauer integration matrix (GIM) is greater than or equal to n. The error formula for the Gegenbauer quadrature is derived. Moreover, a study on the error bounds and the convergence rate shows that the optimal Gegenbauer quadrature exhibits very rapid convergence rates, faster than any finite power of the number of Gegenbauer expansion terms. Two efficient computational algorithms are presented for optimally constructing the Gegenbauer quadrature. We illustrate the high-order approximations of the optimal Gegenbauer quadrature through extensive numerical experiments, including comparisons with conventional Chebyshev, Legendre, and Gegenbauer polynomial expansion methods. The present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods.