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Topic: Chebyshev equation

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	Cubic equation - Wikipedia, the free encyclopedia
		In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power.
		Every cubic equation with real coefficients has at least one solution x among the real numbers; this is a consequence of the intermediate value theorem.
		Therefore the Chebyshev cube root is in fact an analytic function on the whole of the domain D. An alternative construction of the Chebyshev cube root in terms of hypergeometric functions is sketched in the next subsection.
	en.wikipedia.org /wiki/Cubic_equation (1934 words)

	Chebyshev
		Chebyshev always acknowledged the great influence Brashman had been on him while studying at university, and credited him as the main influence in directing his research interests, referring to their "precious personal talks".
		Chebyshev submitted a paper on The calculation of roots of equations in which he solved the equation y = f(x) by using a series expansion for the inverse function of f.
		Chebyshev continued to aim at international recognition with his second paper, written again in French, appearing in 1844 published by Crelle in his journal.
	www-history.mcs.st-and.ac.uk /~history/Mathematicians/Chebyshev.html (2919 words)

	PlanetMath: Chebyshev equation
		Chebyshev's equation is the second order linear differential equation
		These polynomials are, up to multiplication by a constant, the Chebyshev polynomials.
		This is version 3 of Chebyshev equation, born on 2002-11-21, modified 2002-11-21.
	planetmath.org /encyclopedia/ChebyshevEquation.html (166 words)

	Chebyshev polynomials (Site not responding. Last check: 2007-10-15)
		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)

	Analog Filter Design Demystified - Maxim/Dallas
		Similar to the Chebyshev response, it has ripple in the passband and severe rolloff at the expense of ripple in the stop band.
		To ensure conformance with the generic filter described by Equation 1, and to ensure that the last term equals unity, the first two quadratics have been multiplied by a constant.
		Equation 2 is the transfer function of the highpass filter block.
	www.maxim-ic.com /appnotes.cfm/appnote_number/1795 (2219 words)

	Chebyshev polynomials biography .ms (Site not responding. Last check: 2007-10-15)
		The Chebyshev polynomials of the first kind are defined by the recurrence relation
		The Chebyshev polynomials of the second kind are defined by the recurrence relation
		The Chebyshev polynomials of the first and second kind are closely related by the following equations
	www.biography.ms /Chebyshev_polynomial.html (579 words)

	Dictionary of Meaning www.mauspfeil.net
		In mathematics the '''Chebyshev polynomials''', named after Pafnuty Chebyshev (''Пафнутий Чебышёв''), are a polynomial sequence sequence of orthogonal polynomials which are related to de Moivre's formula and which are easily defined recursively, like Fibonacci numbers Fibonacci or Lucas numbers.
		These equations are special cases of the Sturm-Liouville problem Sturm-Liouville differential equation.
		The '''Chebyshev polynomials of the second kind''' are defined by the recurrence relation :
	www.mauspfeil.net /Chebyshev_polynomials.html (957 words)

	Chebyshev polynomial and the Pascal Triangle
		He is known for his work in the field of probability and statistics.This article refers to what are commonly known as Chebyshev polynomials of the first kind.
		Chebyshev polynomials of the first kind are very important in numerical approximation.
		It is obvious that Pascal Triangle of the Second kind structure is built in these relations, which certainly indicates the existing connection between the Pascal Triangle of the Second Kind and the Chebyshev polynomials of the first kind.
	milan.milanovic.org /math/english/fibo/fibo6.html (230 words)

	Maybe this Explains the Economic Cycle... best Chebyshev Polynomial of t... (Site not responding. Last check: 2007-10-15)
		Algorithms: evaluation of the Chebyshev polynomial Tn(X) by recursion Algorithms: evaluation of the Chebyshev polynomial Tn(X) by recursion G. Galler...
		The normalised co-ordinate for which the Chebyshev polynomial is to be evaluated...
		Chebyshev Polynomial of the First Kind -- from MathWorld Chebyshev Polynomial of the First Kind -- from MathWorld The Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev...
	ascot.pl /th/Fourier3/Chebyshev-Polynomial-of-t....htm (517 words)

	Notes on the Design of Optimal FIR Filters (Site not responding. Last check: 2007-10-15)
		Not only did Chebyshev find such polynomials, he found that one exists for each positive integer value of M, and that they are related through a recursion equation, that is, the polynomial for M can be directly obtained from the polynomials for M–1 and M–2.
		This second form of the definition for Chebyshev polynomials is very useful since it is a closed form and because it involves cosines, a functional form appearing frequently in frequency-domain representations of filters.
		Equation 36 is in fact superfluous given equation 35.
	www.appsig.com /papers/tn070/tn070_appB.html (522 words)

	How to read the JPL Ephemeris and Perform Barycentering
		I recognized that the Chebyshev polynomials are a natural form of approximation, and it is actually quite easy to compute the coefficients for a low-order approximation.
		The Chebyshev approximation is also usually quite "optimal" in the sense that the error rapidly decreases with number of terms, and is never more than the next coefficient.
		The Chebyshev coefficients can be found by direct algebraic manipulation of the Chebyshev polynomials, however, by exploiting the orthogonality conditions, an explicit expression for the coefficients can be found in terms of the function evaluated at four points in the interval.
	lheawww.gsfc.nasa.gov /users/craigm/bary (3923 words)

	AERADE subject listing for Vapour pressure. Aromatic and cyclo compounds.
		The coefficients for the equation are given for each compound and three tables of vapour pressure are included: in SI units at 5 K intervals, in atmospheres at 5 degree C intervals, in lbf/sq in at 10 degree F intervals.
		The coefficients of the equation for each compound are given together with tables of the vapour pressures at 5 K intervals.
		The constants of the Chebyshev equation for each compound are given together with values of critical temperature, critical pressure and acentric factor.
	aerade.cranfield.ac.uk /subject-listing/esdu/ES19.html (847 words)

	[No title]
		This improves, by factor $\log n$, the known arithmetic time bound for Chebyshev interpolation and gives an alternative supporting algorihtm for the record estimate of $O(n \log n)$ for Chebyshev evaluation, obtained by Gerasoulis in 1987 and based on a distinct algorithm.
		In the next section, we recall the known correlationsbetween the Chebyshev set and the roots of 1.
		This, however,immediately follows from (\ref{eq5.2}) and from the equation $p(x) = r(z)$, implied by (\ref{eq3.2}) and (\ref{eq5.3}).
	comet.lehman.cuny.edu /vpan/fromVAX/chebyshev98_150.tex (1214 words)

	2.4 Modified Moments from the Chebyshev Semi-iterative Method (Site not responding. Last check: 2007-10-15)
		Equation (18) is equivalent to the continuous integral
		The extraction of moments from iterates can be accelerated by using the recurrence relations for the Chebyshev polynomials defined in Equation (9).
		in Equation (15) associated with the modified moments
	www.cs.utk.edu /~berry/csi/node12.html (129 words)

	Dissertation Abstract, Jodi L. Mead (Site not responding. Last check: 2007-10-15)
		For the grid transformed Chebyshev method with a small number of grid points it is, however, more appropriate to compare its accuracy with that of high-order finite difference methods.
		The efficiency of the Chebyshev pseudospectral method is further improved by developing Runge-Kutta methods for the temporal discretization which maximize imaginary stability intervals.
		Solutions of the second order wave equation, with the absorbing boundary condition imposed either by the matched layer or by the one way wave equations, are compared.
	math.boisestate.edu /~mead/thesis.html (547 words)

	Optimal Frequency Warpings (Site not responding. Last check: 2007-10-15)
		Similarly, the peak error is essentially the same for least squares and weighted equation error, with the Chebyshev case being able to shave almost 0.1 Bark from the maximum error at high sampling rates.
		By forcing equal and opposite peak errors, the Chebyshev case is able to lower the peak error from 0.67 to 0.64 Barks.
		The optimal Chebyshev, least squares, and weighted equation-error cases are almost indistinguishable.
	ccrma-www.stanford.edu /~jos/bbt/Optimal_Frequency_Warpings.html (731 words)

	Personal Home Page (Site not responding. Last check: 2007-10-15)
		A Chebyshev interpolation scheme is proposed for the short-time Liouville-von Neumann propagator.
		The short Chebyshev recursion ensures that the divergence due to the complex eigenvalues of the Liouville superoperator is kept under control.
		A short-time Chebyshev propagator for the Liouville-von Neumann equation.
	www.unm.edu /~hguo/lvn.html (157 words)

	Chebyshev polynomials
		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.
	www.sciencedaily.com /encyclopedia/chebyshev_polynomials (163 words)

	AERADE subject listing for Vapour pressure. Aliphatic compounds containing nitrogen. (Site not responding. Last check: 2007-10-15)
		The Wagner equation was fitted to the data from the melting point to close to the critical point (except for some compounds for which it was cut off around 0.9 critical) and the coefficients are given for each compound.
		Two tables of vapour pressure calculated using the equation are given: one in SI units at 5 K intervals and one in bars at 5 degree C intervals.
		The coefficients of the equation are given for each compound and three tables of vapour pressure values calculated using them are included; in SI units at 5 K intervals, in atmospheres at 5 degree C intervals, in lbf/sq in at 10 degree F intervals.
	www.aerade.aero /subject-listing/esdu/ES21.html (896 words)

	Notes on the Design of Optimal FIR Filters (Site not responding. Last check: 2007-10-15)
		While many different types of variable changes could be employed, this one matches the boundary conditions (an obvious requirement) but happens to employ the cosine function, a member of the same family used to define the Chebyshev polynomials.
		To select K we note that all but one of the ripples in the polynomial’s response are used in the stopband and these are split evenly between the positive and negative frequencies.
		Equation 37 is then used to obtain values for
	www.appsig.com /papers/tn070/tn070_appC.html (376 words)

	New Page 1
		The series term b[3] must be 0 since the power series is a solution of the differential equation.
		We examine the differential equation and see that at most three terms of the power series expansion of the solution are involved in the recurrence relation of the coefficents.
		where the latter is a normalizing factor chosen to make the Chebyshev polynomials constructed here coincide with the Chebyshev polynomials obtained from the recurrence relation.
	www.iyte.edu.tr /~unalufuktepe/WEB/special_equations.htm (698 words)

	An ordinary differential equation (Site not responding. Last check: 2007-10-15)
		An ordinary differential equation (frequently called an "ODE" or a "diffy Q") is an equality involving a function and its derivatives.
		Simple theories exist for first-order (integrating factor) and second-order (Sturm-Liouville theory) ordinary differential equations, and arbitrary ODEs with linear constant coefficients can be solved when they are of certain factorable forms.
		A vast amount of research and huge numbers of publications have been devoted to the numerical solution of differential equations, both ordinary and partial (PDEs) as a result of their importance in fields as diverse as physics, engineering, economics, and electronics.
	tdbasics.net /ode.html (482 words)

	MATHFUNC
		The equations are encountered in a variety of different areas including optics, quantum mechanics, laser physics, electromagnetic wave propagation, solid and fluid mechanics and heat transfer.
		This equation differs from the others in that it has a periodic coefficient and thus it seems natural to ask if there are not solutions to this equation which are also periodic.
		Although the heat conduction equation does not yield a wave solution in the standard form, it does allow the existence of a highly damped and dispersive temperature signal into a conductor when the surface temperature is varied periodically.
	aemes.mae.ufl.edu /%7Euhk/MATHFUNC.htm (15408 words)

	Ordinary Differential Equation (Site not responding. Last check: 2007-10-15)
		An ordinary differential equation (frequently abbreviated ODE) is an equality involving a function and its
		The following are examples of important ordinary differential equations which commonly arise in problems of mathematical physics.
		Forsyth, A. Theory of Differential Equations, 6 vols.
	www.math.sdu.edu.cn /mathency/math/o/o095.htm (357 words)

	normalform.html
		Procedure "NORMA" to obtain normalized form of the linear differential equation.
		outputs is a normal form of the linear differential equation..
		take as input a linear fuchsian equation, the dependent variable, and independent variable and it's output is Riemann P-function:.
	adept.maplesoft.com /categories/mathematics/desolving/html/normalform.html (162 words)

	Chebychev Polynomials (of the First Kind) (Site not responding. Last check: 2007-10-15)
		Here are the first seven Chebyshev Polynomials, along with a plot.
		The eigenvalues are n^2 and the eigenfunctions are the Chebyshev polynomials.
		By one of the major Sturm-Liouville theorems, the Chebyshev polynomials are orthogonal with respect to the weight 1/Sqrt[1-x^2].
	www.ma.iup.edu /projects/CalcDEMma/sturm/sl011.html (65 words)

	Efficient Spectral-Galerkin Method II. Direct Solvers of Second and Fourth Order Equations by Using Chebyshev ...
		They are based on appropriate base functions for the Galerkin formulation which lead to discrete systems with special structured matrices which can be efficiently inverted.
		The last example is a two dimensional diffusion equation: 8...
		7 The spectrum of the Chebyshev collocation operator for the h..
	citeseer.ist.psu.edu /464183.html (399 words)

	2. Derivation of CSI-MSVD Method (Site not responding. Last check: 2007-10-15)
		If the Chebyshev semi-iterative method [13] is used to solve systems defined by Equation (4), the iteration in Equation (3) will be of the form
		In this section, a procedure (CSI-MSVD) for estimating the eigenvalues of M (corresponding to the largest singular values of A) using Equation (5) with the method of modified moments is discussed.
		A more formal review of the theory of iterative methods which addresses issues such as convergence criteria and rates of convergence to establish the optimality of the Chebyshev semi-iterative method is given in [22].
	www.cs.utk.edu /~berry/csi/node8.html (170 words)

	Citations: The accurate solution of Poisson's equation by expansion in Chebyshev polynomials - Haidvogel, Zang ... (Site not responding. Last check: 2007-10-15)
		....the domain, N is the cutoff number of the Chebyshev series in each direction) with an accuracy comparable to that of the Legendre Galerkin approximation and significantly better than that of the Chebyshev tau approximation.
		It is also more efficient than the Legendre Galerkin algorithm in [12] for solving equations with multiple right hand sides, thanks to the FFT.
		These techniques perform well but have the disadvantage that an expensive matrix multiplication must be performed to transform from eigenvectors of the operators back to physical variables which is necessary, e.g.
	citeseer.lcs.mit.edu /context/82499/0 (754 words)

	Differential Equations (Site not responding. Last check: 2007-10-15)
		Bessel's Equation (the material for the lecture on 16.11.2004-18.11.2004,updated 14.11.2004,16.11.2004,18.11.2004)
		Some differential equations, which can be solved by means of Bessel's equation (updated 18.11.2004, 23.11.2004)
		Solution of non-homogeneous heat and wave equations with homogeneous boundary conditions (inserted 14.12.2004, updated 16.12.2004)
	www.hait.ac.il /staff/BenzionS/Differential.Eqn.html (283 words)

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