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Topic: Chebyshev nodes

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	A Matlab Differentiation Matrix Suite
		Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials.
		cheb4c.m (Chebyshev 4th derivative matrix incorporating clamped boundary conditions)
		polint.m (Barycentric polynomial interpolation on arbitrary distinct nodes)(
	dip.sun.ac.za /~weideman/research/differ.html (472 words)

	Pafnuty Chebyshev
		Chebyshev's inequality says that the probability that the outcome of a random variable is more than a standard deviations away from its mean is no more than 1/a
		Chebyshev's inequality is used to prove the weak law of large numbers and the Bertrand-Chebyshev theorem (1845 1850).
		The Chebyshev polynomials are named in his honor.
	www.abacci.com /wikipedia/topic.aspx?cur_title=Pafnuty_Chebyshev (147 words)

	Encyclopedia :: encyclopedia : Nodes (Site not responding. Last check: 2007-10-30)
		In the mathematical subfield of numerical analysis Chebyshev nodes are the roots of the Chebyshev polynomial of the first kind.
		The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial interpolation.
		The myelin sheath is the fatty tissue layer coating the axon.
	www.hallencyclopedia.com /topic/Nodes.html (503 words)

	Chebyshev nodes - Biocrawler (Site not responding. Last check: 2007-10-30)
		In the mathematical subfield of numerical analysis Chebyshev nodes are the roots of the Chebyshev polynomial of the first kind.
		They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial minimizes the problem of Runge's phenomenon.
		The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial interpolation.
	www.biocrawler.com /encyclopedia/Chebyshev_nodes (268 words)

	NationMaster - Encyclopedia: Polynomial interpolation
		To phrase it in terms of linear algebra: For n+1 interpolation nodes there exists a vector space isomorphism Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions.
		We conclude again that Chebyshev nodes are a very good choice for polynomial interpolation, as the growth in n is exponential for equidistant nodes.
		For better Chebyshev nodes, however, such an example is much harder to find because of the theorem: The red curve is the Runge function, the blue curve is a 5th-degree polynomial, while the green curve is a 9th-degree polynomial.
	www.nationmaster.com /encyclopedia/Polynomial-interpolation (2798 words)

	NationMaster - Encyclopedia: Chebyshev polynomials
		The roots are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation.
		This is version 7 of Chebyshev polynomial, born on 2002-02-19, modified 2005-04-28.
		Chebyshev's polynomial of the second kind can be defined in terms of the differential equation
	www.nationmaster.com /encyclopedia/Chebyshev_polynomials/Chebyshev_roots (358 words)

	Chebyshev_polynomials (Site not responding. Last check: 2007-10-30)
		In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which are easily defined recursively, like Fibonacci or Lucas numbers.
		This relationship is used in the Chebyshev spectral method of solving differential equations.
		The spread polynomials are in a sense equivalent to the Chebyshev polynomials of the first kind, but enable one to avoid square roots and conventional trigonometric functions in certain contexts, notably in rational trigonometry.
	en.filepoint.de /info/Chebyshev_polynomials (923 words)

	Chebyshev Filter Macro - Spring 1998
		In the equations for wr and q0, b1 and b0 parameters are specified.
		The subsequent voltages at node b1 and b0 are then used in the.define statements for wr and q0.
		Chebyshev filters are characterized by their ripple response in the passband and a monotonically decreasing transmission in the stopband.
	www.spectrum-soft.com /news/spring98/cheby.shtm (1038 words)

	Chebyshev polynomials . Chebyshev filter (Site not responding. Last check: 2007-10-30)
		One usually distinguishes between Chebyshev polynomials of the first kind which are denoted T n and Chebyshev polynomials of the second kind which are denoted U n.
		The Chebyshev polynomials T n or U n are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence.
		Chebyshev form is a polynomial p x of the form :p x = \sum_ ^ a_n T_n x where T n is the n th Chebyshev polynomial.
	www.uk.kunsimuna.net /Chebyshev_polynomials_UK_084800_qy (441 words)

	A Guide to the JUMBL Statistics (Site not responding. Last check: 2007-10-30)
		Nodes of a model are, in a sense, arbitrary.
		Nodes which have a high probability of occurrence should be automated; nodes which have a low probability of occurrence may be handled manually.
		One can use this to determine whether an important node will be visited during random testing, or if a non-random test should be crafted to visit the specified node, as it is not expected to occur in random testing.
	www.cs.utk.edu /sqrl/esp/jumbl2/jumbl_stats.html (2353 words)

	Chebyshev polynomials
		In mathematics the Chebyshev polynomials,named after Pafnuty Chebyshev (Пафнутий Чебышёв), are special polynomials.
		Chebyshev polynomials of the first kind are very important in numerical approximation because they are the best approximation of a continuous function under the maximum norm.
		A polynomial of degree N in Chebyshev form is a polynomial p(x) of the form
	pedia.newsfilter.co.uk /wikipedia/c/ch/chebyshev_polynomials.html (172 words)

	Polynomial interpolation \| Topic Definition \| Find the Meaning and Define the Answer of Polynomial interpolation
		In the case of equidistant nodes, the Lebesgue constant grows exponentially.
		For any fixed table of nodes there is a continuous function f(x) on an interval [a,b] for which the sequence of interpolating polynomials diverges on [a,b].
		If equidistant points are chosen as interpolation nodes, the function from Runge's phenomenon demonstrates divergence of the interpolation, and it is not only continuous but infinitely differentiable on [-1,1].
	www.thefreeencyclopedia.com /definition/word.aspx?w=Polynomial_interpolation (1416 words)

	Chebyshev polynomials (Site not responding. Last check: 2007-10-30)
		The Chebyshev polynomials are named after Pafnuty Chebyshev (Пафнутий Чебышёв) and compose a polynomial sequence.
		This article refers to what are commonly known as Chebyshev polynomials of the first kind, which are a solution to the Chebyshev differential equation:
		The Chebyshev polynomials can be used in the area of numerical approximation.
	encyclopedia.codeboy.net /wikipedia/c/ch/chebyshev_polynomials.html (132 words)

	Chebyshev Interpolation
		Using the CP applet, observe how the extrema of the Chebyshev polynomials are not evenly distributed and how they cluster around the boundary.
		Chebyshev pseudospectral methods, which are based on the interpolating Chebyshev approximation (12), are well established as powerful methods for the numerical solution of PDEs with sufficiently smooth solutions.
		In these cases, the Chebyshev pseudospectral method produces approximations that are contaminated with Gibbs oscillations and suffer from the corresponding loss of spectral accuracy, just like the Chebyshev interpolation methods that the pseudospectral methods are based on.
	www.joma.org /images/upload_library/4/vol6/Sarra/Chebyshev.html (2857 words)

	Comparison of 1-D Polynomial Interpolation On Different Node Distributions (Site not responding. Last check: 2007-10-30)
		This applet compares 1-D polynomial interpolation when the nodes (or data points) are distributed on an equispaced grid versus when they are distributed on a Chebyshev grid.
		As you may have guessed, the way the nodes are distributed in the independent variable (commonly given as x or t) greatly affects the resulting polynomial.
		A common, somewhat obvious way of distributing the nodes is by spacing them equally over the interval you are interested in approximating the unknown function.
	www.math.utah.edu /~wright/applets/distribution/equi-cheby.html (531 words)

	Chebyshev polynomials (Site not responding. Last check: 2007-10-30)
		The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebyshef or Tschebyscheff.
		are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence.
		The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselves are a special case of the Jacobi polynomials.
	www.toolhost.com /Chebyshev_polynomials.html (515 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.
		be a chebyshev polynomial of degree (n1-1) in x1 and n2 in x2.
	home.uchicago.edu /~chevia/chebyregress.f90 (4529 words)

	polynomial interpolation - Article and Reference from OnPedia.com
		But this is true due to a special property of polynomials of best approximation known from Chebyshev alternance theorem.
		For any table of nodes there is a continuous function f(x) on an interval a,b" title="a,b">a,b for which the sequence of interpolating polynomials diverges on a,b" title="a,b">a,b.
		For better Chebyshev nodes, however, such an example is much harder to find because of the theorem:
	www.onpedia.com /encyclopedia/Polynomial-interpolation (1302 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		The Chebyshev points are the roots of the Chebyshev polynomial Tn where n is the number of points requested for the interpolation.
		The n nodes are equally spaced on an interval between xmin and xmax, the min and max values of the elements in the vector x.
		The outputs are Sx the nodes at which the spline is evaluated; and Sy the value of the spline at the points in Sx.
	www.mts.jhu.edu /~castello/castello/tempzip/spline/spline.txt (1287 words)

	Chebyshev, Pafnuty Lvovich - Hutchinson encyclopedia article about Chebyshev, Pafnuty Lvovich
		He later tackled probability and statistics, fields in which he proved important basic theorems including the law of large numbers.
		Chebyshev was born in Okatovo in the Kaluga region of Russia.
		This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional.
	encyclopedia.farlex.com /Chebyshev%2c+Pafnuty+Lvovich (167 words)

	CS 328 - Program 4
		Chebyshev proved results regarding the errors incurred by polynomial interpolation.
		To simplify his result, he showed that one could minimize the maximum error in polynomial interpolation by spacing the nodes for the interpolating polynomial not using uniform spacing, but using the same relative spacing of the intercepts of a certain class of polynomials.
		Chebyshev node generator as a function of the number of nodes needed
	web.umr.edu /~price/cs328/ws06/prg4.html (1270 words)

	wikien.info: Main_Page (Site not responding. Last check: 2007-10-30)
		In numerical analysis, Chebyshev nodes are the roots of the Chebyshev polynomial of the first kind.
		All Chebyshev nodes are contained in the interval [−1, 1].
		To get nodes over an arbitrary interval [a, b] a linear transformation can be used.
	www.hostingciamca.com /index.php?title=Chebyshev_nodes (351 words)

	Chebyshev Polynomials - Dicy.com
		Chebyshev polynomials, and go on to determine the generating function as a quotient of modified Chebyshev polynomials.
		Chebyshev always acknowledged the great influence Brashman had been on him while...
		Examples of Chebyshev polynomials of matrices are presented, and it is noted that if A is far from...
	www.dicy.com /search.cfm?st=chebyshev%20polynomials (311 words)

	Polynomial Interpolation
		Given a continuous function f defined on the interval [-1,1], and a set of n points (the nodes) in this interval, the (0,1,.....,m) Hermite-Fejér interpolation polynomial for f is the unique polynomial of degree (m+1)n-1 which agrees with f at the nodes, and whose first m derivatives vanish at each node.
		In the above figure, the nodes occur at the 8 points where the two graphs intersect at a horizontal point of inflection of the polynomial.
		The graphs also suggest that for fixed m, the Lebesgue constant for (0,1,.....,2m) HF interpolation on the Chebyshev nodes is an increasing function of n, the number of nodes.
	www.latrobe.edu.au /maths/smith/interp.html (708 words)

	POLYNOM: Chebyshev_poly Class Reference
		Coefficients of the expansion over the Chebyshev polynomials of the orthogonal projection on the space of polynomials of maximum degree N.
		This array must be of size at least N+1 and must have allocated by the user prior to the call of this method.
		This array must be of size at least N+1 and must have been allocated by the user prior to the call of this method.
	www.lorene.obspm.fr /school/Monday_doc/classChebyshev__poly.html (466 words)

	Polynomial interpolation
		In the case of equally spaced interpolation nodes, it follows that the interpolation error is O. However, this does not necessarily mean that the error goes to zero as n → ∞.
		We conclude again that Chebyshev nodes are a very good choice for polynomial interpolation.
		For any function f(x) continuous on a bounded interval [a,b], there exists a table of nodes for which the sequence of interpolating polynomials converges to f(x) uniformly on [a,b].
	www.xasa.com /wiki/en/wikipedia/p/po/polynomial_interpolation.html (1351 words)

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