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Plot of the first five T n Chebyshev polynomials (first kind). The Chebyshev polynomials of the first kind are obtained from the recurrence relation: = = + = (). The recurrence also allows to represent them explicitly as the determinant of a tridiagonal matrix of size :
The recurrence relation for is (+) = (), making the coefficients in the recursion relation = , = and the evaluation of the series is given by + = + =, = + + + (), The final step is made particularly simple because () = =, so the end of the recurrence is simply () (); the term is added separately: = + .
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 ...
Chebyshev's equation is the second order linear differential equation + = where p is a real (or complex) constant. The equation is named after Russian mathematician Pafnuty Chebyshev. The solutions can be obtained by power series:
The mathematical basis of Chebfun is numerical algorithms involving piecewise polynomial interpolants and Chebyshev polynomials, and this is where the name "Cheb" comes from. The package aims to combine the feel of symbolic computing systems like Maple and Mathematica with the speed of floating-point numerics.
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation.
Because of this, expansion of functions in terms of Chebyshev polynomials is sometimes used for polynomial approximations in computer math libraries. Some authors use versions of these polynomials that have been shifted so that the interval of orthogonality is [0, 1] or [−2, 2].
The Dickson polynomials with parameter α = 1 are related to Chebyshev polynomials T n (x) = cos (n arccos x) of the first kind by [1] (,) = (). Since the Dickson polynomial D n (x,α) can be defined over rings with additional idempotents, D n (x,α) is often not related to a Chebyshev polynomial.