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Topic: Lagrange polynomial

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Newton polynomial

Polynomial interpolation

Bernstein polynomial

Vandermonde matrix

Lagrange multipliers

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	Lab for Lagrange Polynomial Approximation
		The Lagrange coefficient polynomials for degree n are:
		Fourth, construct the Lagrange interpolation polynomial of degree n = 5.
		Investigate the error for the Lagrange polynomial approximations in Exercise 3.
	math.fullerton.edu /mathews/numerical/la.htm (242 words)

	Lagrange polynomial - Wikipedia, the free encyclopedia
		In numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolation polynomial for a given set of data points in the Lagrange form.
		The Lagrange basis polynomials are used in numerical integration to derive the Newton-Cotes formulas.
		Lagrange interpolation is often used in digital signal processing of audio for the implementation of fractional delay FIR filters (e.g., to precisely tune digital waveguides in physical modelling synthesis).
	en.wikipedia.org /wiki/Lagrange_polynomial (457 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-04)
		Taylor polynomials do compute a polynomial that in many cases does approximate the function, but it uses the value of the function and its derivatives at one point, not the value of the function at many points.
		Lagrange multipliers are used in "constrained optimization": you want to find the largest value of a function of several values, e.g.
		Lagrange was a very great and prolific mathematician, and he did a lot of work, so you see his name in many different areas of mathematics.
	mathforum.org /library/drmath/view/63984.html (1266 words)

	Phrases/Polynomials - J Wiki
		Polynomials in J are represented as rank 1 arrays of numbers.
		Lagrange interpolation polynomial is a polynomial that takes specified values y in specified points x.
		The name is slightly misleading -- Lagrange polynomial is not used to interpolate experimental data.
	www.jsoftware.com /jwiki/Phrases/Polynomials (141 words)

	Polynomial Interpolations
		To be more precise, assume that we have found a polynomial that interpolates to the n first data points p0,..., pn-1, and also a polynomial that interpolates to the n last data points p1,..., pn.
		In terms of the Lagrange polynomials, the solution of the interpolation problem is given by with the given interpolation points.
		Polynomial interpolation is not restricted to interpolation to point data: one can also interpolate to other information, such as derivative data.
	www.math.hmc.edu /~gu/math142/mellon/Application_to_CAGD/Interpolations_and_Blossoms/Polynomial_Interpolation.html (717 words)

	M239 Lagrange Polynomials in MATLAB Lab
		The poly2sym command converts a vector of coefficients to a symbolic polynomial.
		Exercise: Use lagran to calculate the third degree Lagrange polynomial for cos(x) at the evenly spaced interpolating points for the x-values (abscissas) of 0.0, 0.4, 0.8, and 1.2 (see page 210-211 text for the computations and page 210 figure 4.12b) comparing the graph of this polynomial to the graph for function cos(x).
		How to do this: Create a new polynomial -- stored in D, whose length will be one more than C (its last entry will be 0 for the constant term) and whoe coefficients are those of the integral of the polynomial stored in C. Then use polyval on D to evaluate integral at the endpoints.
	www.saintjoe.edu /~karend/m439/NumAnal-6.html (1085 words)

	Newton polynomial - Wikipedia, the free encyclopedia
		In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form.
		The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using divided differences.
		Neglecting minor numerical stability problems in evaluating the entries of the table, this polynomial is essentially the same as that obtained with the same function and data points in the Lagrange polynomial article.
	en.wikipedia.org /wiki/Newton_polynomial (464 words)

	Lagrange Interpolating Polynomial -- from Wolfram MathWorld
		Lagrange interpolating polynomials are implemented in Mathematica as
		When constructing interpolating polynomials, there is a tradeoff between having a better fit and having a smooth well-behaved fitting function.
		The more data points that are used in the interpolation, the higher the degree of the resulting polynomial, and therefore the greater oscillation it will exhibit between the data points.
	mathworld.wolfram.com /LagrangeInterpolatingPolynomial.html (332 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		This area of study is used as a tool for further numerical analysis in the area of numerical integration, differential equations, etc. It is also a way of dealing with functions in tabular form.
		Given any function defined and continuous on a closed bounded interval, there exists a polynomial that is as "close" to the given function as desired.
		We do not use the Taylor polynomials since their accuracy is concentrated near the point about which they are expanded.
	www2.cs.uregina.ca /~norma/cs261-3.1.html (568 words)

	Lagrange (Site not responding. Last check: 2007-11-04)
		This program builds a Lagrange interpolating polynomial for a given set of data points and a given function or a given set of function values.The function generating the Lagrange polynomial has the form:
		n - is the degree of the polynomial to be generated.
		a -is the value the Lagrange polynomial is to be evaluated at.
	www.math.colostate.edu /~allena/lagrange.htm (146 words)

	Lagrange Interpolation
		Lagrange interpolation is a way to pass a polynomial of degree N-1 through N points.
		Lagrange polynomials are the interpolating polynomials that equal zero in all given points, save one.
		Graph of a Polynomial Defined by Its Roots
	www.cut-the-knot.org /Curriculum/Calculus/LagrangeInterpolation.shtml (179 words)

	The Lagrange Polynomial
		It is so easy to use more and more Lagrange Polynomials to approximate a function that the limits of its accuracy need to be stated at the outset.
		This implies that there is one and only one polynomial of degree N-1 which passes through the N data points, we can write the Lagrange polynomial which accomplishes this almost by inspection. The Lagrange polynomials are defined with respect to x and the data abscissa x
		is a polynomial of degree N-1 and that sums of the L
	www.phys.ufl.edu /~coldwell/interpolation/Lagrange.htm (594 words)

	MA/CS 371 - Lab 3 (Site not responding. Last check: 2007-11-04)
		MATLAB represents a polynomial as a vector, where the first element is the coefficient of the highest power of x and the last element is the constant term.
		It just so happens that if you interpret the two vectors as polynomials, their convolution is the product of the polynomials.
		The polynomial that polyfit returns is NOT going to be the same as the interpolating polynomial.
	www.cs.utk.edu /~berry/cs371/labs/lab3.html (675 words)

	Sums of Consecutive Powers Project
		(n) to be a polynomial for several special cases, we may start to look for patterns in the hope of discovering a general polynomial formula.
		(n) as a polynomial of degree p + 1, assuming that it is a polynomial of this degree.
		There is a hierarchy of mathematical objects that appear in nature, and the Bernoulli polynomials (and especially the Bernoulli numbers) happen to lie outside of any "direct" construction from the integers.
	www.math.rutgers.edu /~erowland/sumsofpowers-project.html (1333 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		% % lagrange - computes Lagrange polynomial for given % interpolation points % % input: % x - vector of interpolation points x0, x1,..., xn % j - index of Lagrange polynomial L_{n,j}(x) to compute.
		% the numerator is (x-x0)(x-x1)...(x-xn), where the factor % (x-xj) is excluded, which means the numerator is a % polynomial whose roots are x0, x1,..., xn except xj.
		% the roots function returns the coefficients of a % polynomial that has the given roots.
	www.stanford.edu /~lambers/math105a/code/lagrange.m (315 words)

	[No title]
		Then there is a polynomial of degree two or less that goes through these three points.
		Choose n to be one less than the number of data points since the Lagrange quotients are indexed from 0 through n.
		We now initialize the Lagrange polynomial for the addition of terms.
	www.cbu.edu /~wschrein/media/ANA/lagrange.mw (386 words)

	LAB #6: Polynomial Interpolation
		There are several methods for defining, constructing, and evaluating the polynomial of a given degree that passes through data values.
		Determine the point x0 at which a polynomial p(x) of degree 1 is equal to zero.
		Remark: xval is a point at which the polynomial is to be evaluated, xdata is the set of abscissas, and d is the divided difference information computed by divdif.
	people.scs.fsu.edu /~burkardt/math2070/lab_06.html (2297 words)

	polynomial — Infoplease.com
		polynomial, mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers.
		The degree of a polynomial in one variable is the highest power of the variable appearing with a nonzero coefficient; in the example given above, the degree is 4.
		A Bayesian approach for predicting with polynomial regression of unknown degree.
	www.infoplease.com /ce6/sci/A0839599.html (168 words)

	Embedded.com - Educated Guessing Games (Site not responding. Last check: 2007-11-04)
		We then fit a Chebyshev polynomial to it; if it is a Lagrange we are fitting, the order is one greater than the Lagrange polynomial.
		The order of the polynomial depends on the length of the table, which can mean-even with pre-computed elements-a lengthy computation when the number of accurate digits you need is large.
		Error is also difficult to calculate in a Lagrange polynomial, and even with Chebyshev it is not a trivial task.
	www.embedded.com /story/OEG20020222S0039 (1633 words)

	Lagrange Polynomial Interpolation (Site not responding. Last check: 2007-11-04)
		Returns the value of the Lagrange interpolating polynomial of a column vector valued function at a point.
		The interpolating polynomial has degree that is one less than the column dimension of xd and is equal to the j-th column of yd at the j-th element of xd.
		The row vector xd specifies the argument values that the polynomial interpolates.
	www.omatrix.com /manual/lagrange.htm (134 words)

	Lagrange Interpolation Demo (Site not responding. Last check: 2007-11-04)
		This demo implements interpolation of data points using parametric Lagrange polynomials.
		A set of N points can be interpolated by a polynomial of degree at most N-1.
		The curve is constructed by taking a linear combination of the points to be interpolated with pre-computed Lagrange basis functions varying in time.
	www.math.ucla.edu /~baker/java/hoefer/Lagrange.htm (263 words)

	cs261-lect13 (Site not responding. Last check: 2007-11-04)
		If we are using the Lagrange method and we add one more point, it is not easy to move from the Lagrange interpolation polynomial of degree n-1 to the Lagrange interpolating polynomial of degree n.
		Each Lagrange polynomial must be modified and a new Lagrange polynomial calculated.
		If the values of x are arranged in the order smallest to the largest and they are equally spaced, we can derive a simpler formula for the polynomials.
	www2.cs.uregina.ca /~norma/cs261-3.2.html (454 words)

	AMTH247 Lecture 12 Interpolation II
		Although the coefficients for Lagrange interpolation are trivial to compute, the evaluation of the interpolating polynomial requires the evaluation of each of the
		The Lagrange basis has easily calculated coefficients, but evaluation of the interpolating polynomial is not so easy.
		On the other hand in the monomial basis the interpolating polynomial is easily evaluated by Horner's rule, but computation of the coefficients requires the solution of linear system, which, as we saw in Lecture 11, may be ill-conditioned.
	turing.une.edu.au /~amth247/Lectures_2003/Lecture_12/lecture (736 words)

	Constrained, non-linear, derivative-free parallel optimization of continuous, high computing load, noisy objective ...
		Inside the optimization program, we only use polynomials of degree lower or equal to 2.
		We are ignoring the dependence of the other Newton polynomials in the hope of finding a useful technique which can be implemented cheaply.
		The precision of a polynomial interpolation is better when all the interpolation points are close to the center of the space (
	iridia.ulb.ac.be /~fvandenb/mythesis/node17.html (741 words)

	DIVDIF - Divided Difference Polynomials
		Divided difference polynomials are a systematic method of computing polynomial approximations to scattered data.
		The representations are compact, and may easily be updated with new data, rebased at zero, or analyzed to produce the standard form polynomial, integral or derivative polynomials.
		This is a natural way to determine the coefficients of the polynomial that interpolates a given set of data, for instance.
	people.scs.fsu.edu /~burkardt/f_src/divdif/divdif.html (585 words)

	Differentiation
		In Lagrange polynomial approximation, the data is approximated by a polynomial which passes through all the data points.
		These data should be smoothened by fitting a statistically valid polynomial to the data set.
		Using such a polynomial to calculate the derivative at the same temperature gives d(Cp)/d(T)=0.0132692, which is very different from the spline function value, indeed.
	www.polymath-software.com /PolymathHelp/int2.htm (368 words)

	Lagrange Polynomials
		The above algorithm is sufficient for understanding and/or constructing the Lagrange polynomial.
		Form several Lagrange polynomials of degree n = 2, 3, 4, and 5 for the function
		Investigate the error for the Lagrange polynomial approximations of degree n = 2, 3, 4, and 5 in Example 2.
	math.fullerton.edu /mathews/n2003/LagrangePolyMod.html (225 words)

	LAB #6: Polynomial Interpolation (Site not responding. Last check: 2007-11-04)
		We may then want to examine the graph of the polynomial, evaluate it at other points, determine its integral or derivative, or do other things with it.
		Divide that polynomial by its value at 1, which cannot be 0 (Why?).
		The ones it computes are just the ones needed in the algorithm to evaluate the interpolating polynomial that is described in the next section.
	www.math.pitt.edu /~troy/math2070/lab_06.html (2876 words)

	Polynomial interpolation (Site not responding. Last check: 2007-11-04)
		This suggests that polynomial approximation is a useful avenue for study.
		Because there are three points, a quadratic is the appropriate polynomial to use.
		A different approach with similar ends is to generate the Lagrange interpolation polynomial.
	www.math.buffalo.edu /~pitman/courses/mth437/na5/node2.html (174 words)

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