NationMaster - Encyclopedia: Clenshaw algorithm

In the mathematical subfield of numerical analysis the Clenshaw algorithm (Invented by Charles William Clenshaw) is a recursive method to evaluate polynomials in Chebyshev form. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... A visual form of recursion known as the Droste effect. ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are special polynomials. ...

Polynomial in Chebyshev form

A polynomial of degree N in Chebyshev form is a polynomial p(x) of the form

$p(x) = sum_{n=0}^{N} a_n T_n(x)$

where T_n is the nth Chebyshev polynomial.

Clenshaw algorithm

The Clenshaw algorithm can be used to evaluate a polynomial in the Chebyshev form. Given

$p(x) = sum_{n=0}^{N} a_n T_n(x)$

we define

$b_{N} ,!$	$:= a_{N} ,$
$b_{N-1} ,!$	$:= 2 x b_{N} + a_{N-1} ,$
$b_{N-n} ,!$	$:= 2 x b_{N-n+1} + a_{N-n} - b_{N-n+2} ,,; n=2,ldots,N-1 ,$
$b_{0} ,!$	$:= x b_{1} + a_{0} - b_{2} ,$

then

$p(x) = sum_{n=0}^{N} a_n T_n(x) = b_{0}.$

Category: Numerical analysis In the mathematical subfield of numerical analysis, the Horner scheme or Horner algorithm, named after William George Horner, is an algorithm for the efficient evaluation of polynomials in monomial form. ... In mathematics a monomial basis is a way to uniquely describe a polynomial using a linear combination of monomials. ... In the mathematical subfield of numerical analysis the de Casteljaus algorithm, named after its inventor Paul de Casteljau, is a recursive method to evaluate polynomials in Bernstein form or BÃ©zier curves. ... In the mathematical subfield of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials. ...

Results from FactBites:

polynomial - Article and Reference from OnPedia.com (0 words)
	Which algorithm is used for a given polynomial depends on the form of the polynomial and the chosen x.
	For a polynomial in Chebyshev form the Clenshaw algorithm can be used.
	As there is no general closed formula to calculate the roots of a polynomial of degree 5 and higher, root-finding algorithms are used in numerical analysis to approximate the roots.

GNU Scientific Library -- Reference Manual - Numerical Integration (0 words)
	The QNG algorithm is a non-adaptive procedure which uses fixed Gauss-Kronrod abscissae to sample the integrand at a maximum of 87 points.
	The QAWS algorithm is designed for integrands with algebraic-logarithmic singularities at the end-points of an integration region.
	The QAWO algorithm is designed for integrands with an oscillatory factor, \sin(\omega x) or \cos(\omega x).

More results at FactBites »

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Clenshaw algorithm

See also

Polynomial in Chebyshev form