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In numerical analysis, Chebyshev nodes are a set of specific real algebraic numbers, used as nodes for polynomial interpolation. They are the projection of equispaced points on the unit circle onto the real interval [,], the diameter of the circle. The Chebyshev nodes of the first kind, also called the Chebyshev zeros, are the zeros of the ...
As a practical matter, high-order numeric integration is rarely performed by simply evaluating a quadrature formula for very large . Instead, one usually employs an adaptive quadrature scheme that first evaluates the integral to low order, and then successively refines the accuracy by increasing the number of sample points, possibly only in ...
The Chebyshev nodes are a common choice for the initial approximation because of their role in the theory of polynomial interpolation. For the initialization of the optimization problem for function f by the Lagrange interpolant L n (f), it can be shown that this initial approximation is bounded by
The Chebyshev PS method is frequently confused with other Chebyshev methods. Prior to the advent of PS methods, many authors [7] proposed using Chebyshev polynomials to solve optimal control problems; however, none of these methods belong to the class of pseudospectral methods.
We conclude again that Chebyshev nodes are a very good choice for polynomial interpolation. However, there is an easy (linear) transformation of Chebyshev nodes that gives a better Lebesgue constant. Let t i denote the i-th Chebyshev node. Then, define
This suggests that we look for a set of interpolation nodes that makes L small. In particular, we have for Chebyshev nodes: (+) + We conclude again that Chebyshev nodes are a very good choice for polynomial interpolation, as the growth in n is exponential for equidistant nodes. However, those nodes are not optimal.
In numerical analysis Chebyshev–Gauss quadrature is an extension of Gaussian quadrature method for approximating the value of integrals of the following kind: ∫ − 1 + 1 f ( x ) 1 − x 2 d x {\displaystyle \int _{-1}^{+1}{\frac {f(x)}{\sqrt {1-x^{2}}}}\,dx}
Chebyshev pseudospectral method — uses Chebyshev polynomials (of the first kind) Flat pseudospectral method — combines Ross–Fahroo pseudospectral method with differential flatness; Gauss pseudospectral method — uses collocation at the Legendre–Gauss points; Legendre pseudospectral method — uses Legendre polynomials