745:. The direct use of the expansion in such polynomials is computationally demanding because you have to evaluate all polynomials and then sum them. Moreover, their coefficients are often large with opposite signs so that there is high risk of round-off errors. The Clenshaw formula allows you to evaluate the function value by a recurrence relation without evaluating the polynomials. It is therefore much faster and almost always numerically stable. The equation given here is just a special form for evaluating the series of Chebyshev polynomials of the first kind, but the algorithm can be used for any polynomial that obeys some recurrence relation. When I have some more time, I will rewrite this chapter generally including detailed derivation and references to literature. --
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725:. In practice you never obtain Chebyshev expansion that way. There are several kinds of polynomials that can be defined by recurrence relations. When determining polynomial series of such polynomials, you often take advantage of their orthogonality or you obtain them as a solution of a differential equation of some kind etc. Now you need to evaluate the function value of such an expansion for some
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238:. As a matter of fact, the Clenshaw algorithm is a general method for evaluation of a linear combination of functions that can be expressed using a recurrence formula. If you like, I will describe the general algorithm including the references to relevant literature. --
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There's an error in the
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568:{\displaystyle p\left(x\right)=\sum \limits _{n=0}^{N}a_{n}T_{n}\left(x\right)}
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451:{\displaystyle p\left(x\right)=\sum \limits _{n=0}^{N}b_{n}x^{n}}
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This article is like an answer from the
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