An introduction to the numerical analysis of spectral methods
An introduction to the numerical analysis of spectral methods
Spectral integration and two-point boundary value problems
SIAM Journal on Numerical Analysis
Boundary Layer Resolving Pseudospectral Methods for Singular Perturbation Problems
SIAM Journal on Scientific Computing
Integration Preconditioning of Pseudospectral Operators. I. Basic Linear Operators
SIAM Journal on Numerical Analysis
Spectral methods in MatLab
Journal of Global Optimization
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Integration Preconditioning Matrix for Ultraspherical Pseudospectral Operators
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
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
Journal of Computational and Applied Mathematics
Optimal Gegenbauer quadrature over arbitrary integration nodes
Journal of Computational and Applied Mathematics
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The theory of Gegenbauer (ultraspherical) polynomial approximation has received considerable attention in recent decades. In particular, the Gegenbauer polynomials have been applied extensively in the resolution of the Gibbs phenomenon, construction of numerical quadratures, solution of ordinary and partial differential equations, integral and integro-differential equations, optimal control problems, etc. To achieve better solution approximations, some methods presented in the literature apply the Gegenbauer operational matrix of integration for approximating the integral operations, and recast many of the aforementioned problems into unconstrained/constrained optimization problems. The Gegenbauer parameter 驴 associated with the Gegenbauer polynomials is then added as an extra unknown variable to be optimized in the resulting optimization problem as an attempt to optimize its value rather than choosing a random value. This issue is addressed in this article as we prove theoretically that it is invalid. In particular, we provide a solid mathematical proof demonstrating that optimizing the Gegenbauer operational matrix of integration for the solution of various mathematical problems by recasting them into equivalent optimization problems with 驴 added as an extra optimization variable violates the discrete Gegenbauer orthonormality relation, and may in turn produce false solution approximations.