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This approximation leads directly to the method of Clenshaw–Curtis quadrature. ... The Chebyshev polynomials of the second kind are defined by the recurrence ...
One can obtain polynomials very close to the optimal one by expanding the given function in terms of Chebyshev polynomials and then cutting off the expansion at the desired degree. This is similar to the Fourier analysis of the function, using the Chebyshev polynomials instead of the usual trigonometric functions.
A typical example of a Chebyshev space is the subspace of Chebyshev polynomials of order n in the space of real continuous functions on an interval, C[a, b]. The polynomial of best approximation within a given subspace is defined to be the one that minimizes the maximum absolute difference between the polynomial and the function.
Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the integrand in terms of Chebyshev polynomials. Equivalently, they employ a change of variables = and use a discrete cosine transform (DCT) approximation for the cosine series.
An enhancement to the Chebyshev pseudospectral method that uses a Clenshaw–Curtis quadrature was developed. [18] The LPM uses Lagrange polynomials for the approximations, and Legendre–Gauss–Lobatto (LGL) points for the orthogonal collocation. A costate estimation procedure for the Legendre pseudospectral method was also developed. [19]
The Chebyshev nodes of the second kind, also called the Chebyshev extrema, are the extrema of the Chebyshev polynomials of the first kind, which are also the zeros of the Chebyshev polynomials of the second kind. Both of these sets of numbers are commonly referred to as Chebyshev nodes in literature. [1] Polynomial interpolants constructed from ...
The Chebyshev pseudospectral method for optimal control problems is based on Chebyshev polynomials of the first kind. It is part of the larger theory of pseudospectral optimal control , a term coined by Ross . [ 1 ]
Plot of the absolute value of the seventh-order (n = 7) Chebyshev rational function for 0.01 ≤ x ≤ 100.Note that there are n zeroes arranged symmetrically about x = 1 and if x 0 is a zero, then 1 / x 0 is a zero as well.