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Topic: Clenshaw algorithm

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	polynomial - Article and Reference from OnPedia.com
		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.
	www.onpedia.com /encyclopedia/polynomial (0 words)

	GNU Scientific Library -- Reference Manual - Numerical Integration
		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).
	jamesthornton.com /gnu/gsl/gsl-ref_16.html (0 words)

	Clenshaw algorithm - Biocrawler (Site not responding. Last check: 2007-10-29)
		In the mathematical subfield of numerical analysis the Clenshaw algorithm is a recursive method to evaluate polynomials in Chebyshev form.
		The Clenshaw algorithm can be used to evaluate a polynomial in the Chebyshev form.
		De Casteljau's algorithm to evaluate polynomials in Bézier form
	www.biocrawler.com /encyclopedia/Clenshaw_algorithm (84 words)

	An Analogue for Szegö Polynomials of the Clenshaw Algorithm (ResearchIndex) (Site not responding. Last check: 2007-10-29)
		An Analogue for Szegö Polynomials of the Clenshaw Algorithm (ResearchIndex)
		An Analogue for Szegö Polynomials of the Clenshaw Algorithm (1991)
		We present an analogous algorithm for the evaluation of a linear combination P n j=0 ff j OE j of polynomials OE j that are orthogonal with respect to an inner product defined on (part of) the unit circle.
	citeseer.ist.psu.edu /ammar91analogue.html (364 words)

	GNU Scientific Library -- Reference Manual - Numerical Integration
		The QNG algorithm is non-adaptive procedure which uses fixed Gauss-Kronrod abscissae to sample the integrand at a maximum of 87 points.
		The adaptive bisection algorithm of QAG is used, with modifications to ensure that subdivisions do not occur at the singular point x = c.
		The QAWS algorithm is designed for integrands with algebraic-logarithmic singularities at the end-points of an integration region.
	nereida.deioc.ull.es /html/gsl/gsl-ref_28.html (2405 words)

	(Barrio R., Berges J.-C.) Perturbation Simulations of Rounding Errors in the Evaluation of Chebyshev Series (Site not responding. Last check: 2007-10-29)
		Following these studies Clenshaw's algorithm is recommended except for x near -1 and 1, where the Reinsch modifications to Clenshaw's algorithm [Gentleman 1969] are preferred.
		In some algorithms the theoretical bounds are quite accurate and yield good estimations of the rounding errors, but in others there are some ``pathological'' problems with a bound of several orders of magnitude bigger than average.
		It may be concluded that it is worthwhile controlling rounding errors via perturbation analysis as a complement to theoretical studies of the error bounds since this control serves as a numerical validation and analysis of the theoretical bounds.
	www.jucs.org /jucs_4_6/perturbation_simulations_of_rounding/Barrio_R.html (0 words)

	GNU Scientific Library -- Reference Manual: Numerical Integration Introduction
		The algorithms are built on pairs of quadrature rules, a higher order rule and a lower order rule.
		The algorithms for general functions (without a weight function) are based on Gauss-Kronrod rules.
		The higher order Kronrod rule is used as the best approximation to the integral, and the difference between the two rules is used as an estimate of the error in the approximation.
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_231.html (0 words)

	Polynomial Encyclopedia (Site not responding. Last check: 2007-10-29)
		As a practical matter, an approximate solution that is accurate to a desired precission may be as useful as an exact solution.
		The difference engine of Charles Babbage was designed to create large tables of approximate values of logarithms and trigonometric functions automatically, by evaluating approximating polynomials at many points using Newton's method.
		The reduction of equations in several unknowns to equations each in one unknown is discussed in the article on the Buchberger's algorithm.
	www.hallencyclopedia.com /topic/Polynomial.html (2760 words)

	[No title]
		Algorithm: the derivative of an n-degree Chebyshev polynomial is an (n-1)-degree !
		Algorithm: let the chebyshev polynomial be of degree n1 in x1, and n2 in x2.
		The algorithm is an extension of the one described in the function chebder2d.
	home.uchicago.edu /~chevia/chebyregress.f90 (0 words)

	Abstract_contents
		Parallel Monte Carlo algorithms for evaluating the largest and the smallest eigenvalue of a given matrix are presented.
		The purely deterministic algorithm is a sequence of quadrature formulas for the Wiener measure, where the knots are piecewise linear functions.
		Blending differences are introduced to calculate the blending rational interpolants recursively, algorithm and the computing complexity are discussed and numerical example is given to illustrate the efficiency of the algorithm.
	www.math.hkbu.edu.hk /complexity/1999/abstract/abstract3.shtml (0 words)

	Method for predicting propagation of sound in an ocean - Patent 6876598
		We shall state the truncated algorithm in terms of new coefficients a.sub.l.sup.j-1 that are defined for j=1,2,.
		Suppose the sequence of differential boundary values.sigma..psi..sub.j (0)/.sigma.y in eq.(6.33) is used as input to the truncated algorithm.
		Clenshaw's algorithm is a stable and accurate method to evaluate the Chebyshev interpolation polynomial ##EQU162##
	www.freepatentsonline.com /6876598.html (10603 words)

	Andre Weideman: ILT M-Files
		With the parameters (sigma,b) = (4.3, 4.4) computed by Algorithm 1 the method manages an accuracy of 5.6e-16 which is essentially full precision.
		But in the author's paper Algorithms 1 and 2 have been applied to realistic problems taken from actual applications.
		Algorithm 1 produces good estimates for the optimal (sigma,b) as may be observed from a color contour plot.
	dip.sun.ac.za /~weideman/research/weeks.html (0 words)

	Dr. Dobb's \| Curve Fitting By Chebyshef And Other Methods \| February 1, 1992
		Since the algorithm is recursive anyway, the additional recurrences will not pose an obstacle to any known compilers, as they might in parallelizable code.
		Clenshaw's algorithm is 30 percent faster than evaluation by more obvious methods on sequential processors; even with careful coding for best performance, the difference drops to 20 percent on typical pipeline processors, but the small advantage in accuracy remains.
		The recursive algorithms for evaluation of Chebyshef polynomials can be speeded up on superscalar architectures by determining which array element is needed first, and writing the code to fetch that element prior to the loop iteration in which it is needed.
	www.ddj.com /dept/cpp/184402488?pgno=2 (0 words)

	TOMS Table of Contents
		Amos, Algorithm 610: A Portable FORTRAN Subroutine for Derivatives of the Psi Function, pp.
		Filippo Aluffi-Pentini and Valerio Parisi and Francesco Zirilli, Algorithm 617: DAFNE: A Differential-Equations Algorithm for Nonlinear Equations, pp.
		Robert J. Renka, Algorithm 623: Interpolation on the Surface of a Sphere, pp.
	www.ima.umn.edu /~arnold/contents/toms.html (0 words)

	[No title]
		The pairs C of high degree of precision are suitable for handling C integration difficulties due to a strongly oscillating C integrand.
		C The algorithm is a modification of that in QAGS.
		It is based on the QAGS algorithm applied C after a transformation of the original interval into (0,1).
	www.netlib.org /quadpack/doc (0 words)

	Polynomial
		Numerically solving a polynomial equation in one unknown is easily done on computer by the Durand-Kerner method or by some other root-finding algorithm.
		To evaluate a polynomial in monomial form one can use the Horner scheme.
		For a polynomial in Chebyshev form the Clenshaw algorithm can be used.
	www.brainyencyclopedia.com /encyclopedia/p/po/polynomial.html (2753 words)

	CLENSHAW_CURTIS - Multidimensional Clenshaw Curtis Quadrature
		It is easy to generate a Clenshaw Curtis grid of any order N. Of special interest are nested grids, particularly those for which the nesting involves repeated divisions of the interval length by 2.
		In such a case, the data computed from the previous step can be reused, and the new data allows for an inexpensive estimate of the rate of error decrease.
		The code includes routines which pack into a single array all the abscissas associated with several Clenshaw Curtis grids of abscissas, with the grids selected in one of two ways.
	people.scs.fsu.edu /~burkardt/f_src/clenshaw_curtis/clenshaw_curtis.html (0 words)

	Communications of the ACM
		D. March ACM Algorithm 434: Exact Probabilities for ${R\times{C}}$ Contingency Tables 991--992 W. Fullerton ACM Algorithm 435: Modified Incomplete Gamma Function.
		393--395 Henk Koppelaar and Peter Molenaar Remark on ``Algorithm 486: Numerical Inversion of Laplace Transform [D5]'' 395--396 Linda Kaufman Remark on ``Algorithm 496: The LZ Algorithm to Solve the Generalized Eigenvalue Problem for Complex Matrices [F2]''.
		Algorithm 516: An Algorithm for Obtaining Confidence Intervals and Point Estimates Based on Ranks in the Two Sample Location Problem [G1].
	www.math.utah.edu /ftp/pub/tex/bib/toc/toms.html (0 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		The pairs of high degree of precision are suitable for handling integration difficulties due to a strongly oscillating integrand.
		- QAGS : Is an integrator based on globally adaptive interval subdivision in connection with extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).
		The algorithm is a modification of that in QAGS.
	netlib2.cs.utk.edu /quadpack/readme (0 words)

	The numeric package
		If MuPAD cannot compute the numerical value of the definite integral, the function call itself is returned with evaluated arguments.
		Note, that in this case only finite intervals can be handled and no endpoint singularities and no discontinuity in the interior of the interval of integration must exist.
		Algorithm 424: Clenshaw-Curtis Quadrature [D-1].Communications of the ACM, Vol.
	math.berkeley.edu /~mgu/mupad/mupad_html_help/numeric11.html (163 words)

	Table of Mathematical functions
		Algorithms for performing numerical integration of a function in one dimension
		Algorithms for finding the minimum (or maximum) of a function in one or more dimensions
		Algorithms for finding the root of a function in one or more dimensions
	seal.web.cern.ch /seal/work-packages/mathlibs/mathTable.html (0 words)

	Unitary Hessenberg matrices and the generalized Parker-Forney-Traub algorithm for inversion of Szegö-Vandermonde ... (Site not responding. Last check: 2007-10-29)
		Abstract: It is well-known that the Horner polynomials (sometimes called the associated polynomials) describe the structure of the inverses of Vandermonde matrices V (x) = j 1.
		23 The QR algorithm for unitary Hessenberg matrices (context) - Gragg - 1986 ACM
		2 polynomials of the Clenshaw algorithm (context) - Ammar, Gragg et al.
	citeseer.ist.psu.edu /olshevsky03unitary.html (677 words)

	Intergation algorithm
		Several years ago I managed to get a copy of an article that described the algorithm used by HP calculators to do numerical integration.
		The article of which you speak is very likely this one written by William H. Kahan, the numerical analyst who consulted with HP on many of their calculator algorithms.
		I've read Kahan's article on the matter (you can find it in the same place as the integration article) but there is still lots of holes to be filled in.
	www.hpmuseum.org /cgi-sys/cgiwrap/hpmuseum/archv013.cgi?read=37847 (0 words)

	Download Fast Clenshaw-Curtis Quadrature - Fast Clenshaw-Curtis Quadrature computes Clenshaw Curtis weights and nodes ...
		This extremely fast and efficient algorithm uses MATLAB's ifft routine to compute the Clenshaw-Curtis nodes and weights in linear time.
		Fast Fourier Transform code implements the O(n log n) Cooley-Tukey FFT algorithm as simply as possible.
		SAGE is a distribution of Python and Pyrex for use as a computer algebra system and plotting system.
	webscripts.softpedia.com /script/Scientific-Engineering-Ruby/Mathematics/Fast-Clenshaw-Curtis-Quadrature-34235.html (0 words)

	SLATEC Keylist Index
		The routine determines the limit of a given sequence of approximations, by means of the Epsilon algorithm of P.Wynn.
		The routine determines the limit of a given sequence of approximations, by means of the Epsilon algorithm of P.
		An easy-to-use code which minimizes the sum of the squares of M nonlinear functions in N variables by a modification of the Levenberg-Marquardt algorithm.
	www.saao.ac.za /~olivier/SlatecDoc/keylist.index.html (11408 words)

	Atlas: Stability analysis of parallel evaluation of finite series of orthogonal polynomials by Roberto Barrio (Site not responding. Last check: 2007-10-29)
		In this communication we analyse the stability of parallel algorithms for the evaluation of polynomials written as a finite series of a general family of orthogonal polynomials.
		For every method the first algorithm uses parallel techniques in matrix methods by means of the matrix formulation of the sequential algorithm, whereas the second algorithm is based on a recurrence matrix product formulation of the sequential recurrence.
		The analysis shows that the parallel algorithms are almost as stable as their sequential counterparts for practical applications.
	atlas-conferences.com /c/a/e/b/96.htm (232 words)

	toms
		ACM Collected Algorithms The Collected Algorithms (CALGO) is part of a family of publications produced by the ACM.
		Subscribers receive quarterly notification of the appearance of new algorithms, as well as copies of research papers describing them in loose-leaf binder form.
		Use of ACM Algorithms is subject to the ACM Software Copyright and License Agreement Contact For further information about CALGO contact its Editor-in-Chief: Tim Hopkins Computing Laboratory The University of Kent Canterbury Kent CT2 7NF United Kingdom +44-122-776-4000 ext.
	www.netlib.org /toms (0 words)

	[No title]
		%%% %%% This bibliography includes ACM Algorithms %%% 493 -- 735 (or the latest), including %%% Algorithm 568, published in ACM %%% Transactions on Programming Languages and %%% Systems (TOPLAS).
		For ACM Algorithms 1 -- %%% 492, see the companion bibliographies, %%% cacm1960.bib and cacm1970.bib.
		%%% %%% Algorithms published in Communications of %%% the ACM, prior to the founding of TOMS in %%% 1975, are also included in this %%% bibliography, if a TOMS paper contains %%% Remarks or Corrigenda for them.
	math.nist.gov /toms/bibindex/toms.bib (0 words)

	numerical analysis lab 6
		The following function implements the natural spline algorithm described in the textbook.
		You may use the code for Neville algorithm from lab 2.
		The Clenshaw-Curtis algorithm computes the integral of f over [-1,1] by integrating the Chebyshev interpolant.
	math.tut.fi /~piche/numa1/lab6.html (558 words)

	Communications of the ACM
		I. Hill and M. Pike ACM Algorithm 299: Chi-Squared Integral 243--244 J. Gunn ACM Algorithm 300: Coulomb Wave Functions.
		B. Witte ACM Algorithm 332: Jacobi Polynomials 436--437 R. Salazar and S. Sen ACM Algorithm 333: Minit Algorithm For Linear Programming.
		Kolm and T. Dahlstrand Remark on ``Algorithm 333: Minit Algorithm For Linear Programming ([H])'' 50--50 K. Redish Algorithms: Comment on London's Certification of Algorithm 245.
	www.math.utah.edu /ftp/pub/tex/bib/toc/cacm1970.html (0 words)

	[No title]
		%%% %%% This bibliography includes ACM Algorithms %%% 493 -- 735 (or the latest), including %%% Algorithm 568, published in ACM %%% Transactions on Programming Languages and %%% Systems (TOPLAS).
		For ACM Algorithms 1 -- %%% 492, see the companion bibliography, %%% cacm.bib.
		%%% %%% Algorithms published in Communications of %%% the ACM, prior to the founding of TOMS in %%% 1975, are also included in this %%% bibliography, if a TOMS paper contains %%% Remarks or Corrigenda for them.
	elib.cs.sfu.ca /Collections/CMPT/MajorBibs/beebe-bib/toms.bib (790 words)

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