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Topic: Polynomial interpolation

Related Topics

Spline interpolation

Lagrange polynomial

Polynomial

Linear interpolation

Polynomial sequence

Curve fitting

List of mathematical functions

Numerical analysis

Determinant

List of numerical analysis topics

Integral calculus

Numerical stability

Elliptic integral

Richardson extrapolation

Pareto interpolation

	NationMaster - Encyclopedia: Polynomial interpolation
		In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial.
		Polynomial interpolation is also essential to perform sub-quadratic multiplication and squaring such as the Karatsuba multiplication where an interpolation through points on a polynomial which defines the product yields the product itself.
		One method is to write the interpolation polynomial in the Newton form and use the method of divided differences to construct the coefficients.
	www.nationmaster.com /encyclopedia/Polynomial-interpolation (2777 words)

	Polynomial interpolation - ALGLIB
		Polynomial interpolation is the most known one-dimensional interpolation method.
		It is proved that there is a class of functions which cannot be interpolated by the polynomial on an equidistant grid.
		It is a fundamental property of an interpolation polynomial.
	www.alglib.net /interpolation/polynomial.php (923 words)

	Citations: Polynomial interpolation in several variables - Gasca, Sauer (ResearchIndex)
		Interpolation on the sphere and on the ball Polynomials on S that are symmetric with respect to x d 1 are related to polynomials.
		) Polynomial interpolation in the subspace K[X;Y ] n of polynomials of degree n corresponds to a linear map L : K[X;Y ] n Gamma K N (1.
		Finally, we note that a characterization (di erent from Theorem 2.1) of the smoothness of polynomial pieces on adjacent triangles, without using B ezier Bernstein techniques, was proved by Davydov, N urnberger, and Zeilfelder [63] 3 Dimension of spline spaces In this section, we summarize....
	citeseer.ist.psu.edu /context/1569546/486442 (906 words)

	Comparison of 1-D Polynomial Interpolation On Different Node Distributions
		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.
		One of the fundamental theorems about polynomials is that if we are given N+1 distinct nodes then we can find a unique polynomial of degree N that passes through all the nodes.
		Unfortunately, this type of distribution causes the interpolant to exhibit the Runge phenomenon (divergence of the interpolant near the ends of the interval.
	www.math.utah.edu /~wright/applets/distribution/equi-cheby.html (531 words)

	Polynomial Interpolation
		The comparison between the interpolation and the function appear in Figure 30
		Note that the only points where the estimate of the error agrees with the error is at the interpolation points where we know it is zero.
		Figure 32 shows the actual error of the interpolation and this approximation.
	www.physics.arizona.edu /~restrepo/475A/Notes/sourcea/node29.html (401 words)

	Polynomial interpolation
		Polynomial interpolation is the act of fitting a polynomial to a given function with defined values in certain discrete data points.
		Polynomial interpolation is an area of inquiry in numerical analysis.
		Even though these problems can be partially avoided by using for example Chebyshev polynomials, the solution that is mostly preferred in practice is to use several polynomials of a lower degree, connected in chains.
	www.ebroadcast.com.au /lookup/encyclopedia/po/Polynomial_interpolation.html (411 words)

	Polynomial interpolation - Biocrawler (Site not responding. Last check: )
		Here, the interpolant is not a polynomial but a spline: a chain of several polynomials of a lower degree.
		This can be seen as a form of polynomial interpolation with harmonic base functions, see trigonometric interpolation and trigonometric polynomial.
		Birkhoff interpolation is the generalization which allows for some derivatives to be given, without specifying the values of p themselves.
	www.biocrawler.com /encyclopedia/Polynomial_interpolation (1217 words)

	Polynomial interpolation (Site not responding. Last check: )
		This "function" may actually be anydiscrete data (such as obtained by sampling), but itis generally assumed that such data may be described by a function.
		Even though these problems canbe partially avoided by using for example Chebyshevpolynomials, the solution that is generally preferred in practice is to use several polynomials of a lower degree, connectedin chains.
		Using harmonic functions to interpolate a periodic function is usually done using Fourier series, for example in discrete Fourier transform.
	www.therfcc.org /polynomial-interpolation-204779.html (316 words)

	Interpolation methods
		Surface interpolation is any formal technique that uses values at sampled locations to predict values at unsampled locations.
		Local polynomial interpolation creates a surface from many different formulas, each of which is optimized for a neighborhood.
		Interpolation using radial basis functions is shown by the purple surface; think of it as a fairly stiff rubber sheet that bends and folds to fit exactly to the sample data points.
	www.sppsr.ucla.edu /up206b/Interpolation_methods.htm (2464 words)

	Multivariate polynomial interpolation (Site not responding. Last check: )
		This is work in progress, an attempt to record for handy reference various facts concerning multivariate polynomial interpolation as I come across them.
		Here is a very recent bibliography (in postscript) on multivariate polynomial interpolation, which will appear as part of a survey, by Gasca and Sauer, in a special issue of Advances in Computational Mathematics.
		error in linear interpolation at the vertices of a simplex.
	www.cs.wisc.edu /~deboor/multiint/multiint.html (136 words)

	The MATLAB Notebook v1.5
		Interpolate sin(2pix) on a uniform, nine-point partition of [0,1].
		The cubic spline interpolation for the function f(x) on the partition is the piecewise cubic polynomial S(x) which has the following properties.
		Function spline(x,y,z) can be used to create not-a-knot spline interpolants, where x and y are the data to be inperplated, and z be the vector on which the spline takes the values.
	www.shsu.edu /~mth_jxw/Cmpch3.html (993 words)

	Numerical Interpolation
		The bottom line is, no matter how smooth the interpolation is and how close it is to the raw data, the problem is not completely solved unless the physical meaning behind the theme has been captured.
		The method of least squares is most commonly used to calculate the coefficients of the interpolating polynomial.
		Although the polynomial interpolation is probably the most widely used interpolating method, the rational function interpolation stands out when the data or function changes rapidly in some local regions, e.g., poles.
	www.efunda.com /math/num_interpolation/num_interpolation.cfm (224 words)

	Polynomial Interpolation On The Unit Sphere - Xu (ResearchIndex)
		Abstract: The problem of interpolation at (n + 1) points on the unit sphere S by spherical polynomials of degree at most n is studied.
		Yuan Xu, Polynomial Interpolation on the unit sphere, SIAM J. Numer Anal.
		(context) - Bojanov, Xu Polynomial interpolation on the unit ball and on the unit sp..
	citeseer.ist.psu.edu /601074.html (550 words)

	An Interpolation Applet
		New polynomials are entered by first clearing the screen, typing in one pair of numbers per line in the "x y values" column, and then pressing "Process".
		The four different polynomials constructed for this graph are displayed at the bottom of the screen.
		Thus the leaf-like figure in the graph is not the graph of a single polynomial; it was formed by pasting two polynomials together, using the "next" command.
	www.math.ucla.edu /~ronmiech/Interpolation/HTMDOCS/Introduction/Interpolation_Applet.htm (1721 words)

	PHYS 251 - Polynomial Interpolation
		Since Newton's forward interpolation formula uses values of the function at points ahead in the table, it can be used at the beginning of the tabulated values, but not at the end.
		The zero- and first-order terms constitute essential a linear interpolation, while the second-order term is a correction term to that linear interpolation.
		frequently encounter interpolation problem is one in which the data is not equally spaced in the independent variable.
	www.physics.gmu.edu /~amin/phys251/Topics/NumAnalysis/Approximation/polynomialInterp.html (893 words)

	Interpolation
		As mentioned before, interpolation schemes are often used to increase the accuracy of table lookup.
		Interpolation error for a sinusoidal wavetable can have two components: first, the continuous signal (the theoretical result of reading the wavetable continuously in time, as if the output sample rate were infinite) might not be a pure sinusoid; and second, the amplitude might be wrong.
		Figure 2.11 shows the effect of using linear (two-point) interpolation to fill in a sinusoid of period 6.
	www.crca.ucsd.edu /~msp/techniques/latest/book-html/node31.html (1006 words)

	Bivariate polynomial interpolation (Site not responding. Last check: )
		The background of flat field images is usually well modelled by a 2D polynomial of degrees 3 and 4 in variables sample and line respectively.
		A polynomial of lower degree, for example linear in both dimensions or a constant background should be enough.
		Because small errors in the determination of the background are carried through the whole rest of the reduction and are even amplified at the edges of the orders, care should be taken in the background fitting.
	www.arcetri.astro.it /CC/Manuals/midas/volb/node153.html (203 words)

	Chapter 13 Numerical Methods
		When the table is complete, the coefficients for the interpolating polynomial are the divided differences at the far left of each row, beginning with the second row (shown in light gray in Figure 13-2).
		In the discussion of polynomial interpolation, we stated that we need to choose enough points to give an accurate impression of the function we are interpolating.
		Interpolating a function with not enough points, or poorly placed points, leads to an interpolating polynomial that does not accurately reflect the function we think we are interpolating.
	www.oreilly.com /catalog/masteralgoc/chapter/ch13.html (4033 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.
		Polynomial interpolation is not restricted to interpolation to point data: one can also interpolate to other information, such as derivative data.
		The interpolant then becomes where the are defined through their cardinal properties: To satisfy these requirements, the new must differ from the original.
	www.math.hmc.edu /~gu/math142/mellon/Application_to_CAGD/Interpolations_and_Blossoms/Polynomial_Interpolation.html (717 words)

	LAB #6: Polynomial Interpolation (Site not responding. Last check: )
		Interpolating means that the function passes through all the given points and does its best between them.
		Assuming we can determine these magical polynomials, this is a second way to define the interpolating polynomial to a set of data.
		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.
	www.math.pitt.edu /~troy/math2070/lab_06.html (2876 words)

	AMTH247 Lecture 11 Interpolation I
		Interpolation, as we will see later, is the basis of variety of other numerical algorithms.
		The choice of class of interpolating functions to use for any particular set of data, depends on the data; we choose a class of functions which seems to give a good representation of the data.
		of coefficients of a polynomial and a vector
	turing.une.edu.au /~amth247/Lectures_2003/Lecture_11/lecture (1123 words)

	LAB #7: Polynomial Interpolation and Beyond (Site not responding. Last check: )
		The interpolation error will be computed by picking a much larger number of points than are used for interpolation and computing the maximum error on this larger set of test points.
		And the interpolation error is probably going to involve the size of the intervals, and the norm of the second derivative of the function, both of which are easy to understand.
		(x) of linear interpolants to f with the property that hmax, the size of the maximum interval, goes to zero as n increases, then the linear interpolants converge to f both pointwise, and uniformly.
	www.math.pitt.edu /~troy/math2070/lab_07.html (2384 words)

	Polynomial Interpolation
		The major focus of the work in polynomial interpolation at the Bendigo campus of La Trobe University has been on the Lagrange and Hermite-Fejér interpolation processes, and on their generalizations.
		An important part of the research has involved studies of the Lebesgue function and Lebesgue constant, which are of crucial importance in determining the convergence behaviour of the (0,1,.....,m) HF interpolation polynomials as the number n of nodes increases.
		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)

	Connecting the Dots
		We will mostly tour the classical world of polynomial interpolation, but will end with an excursion to the more modern land of piecewise polynomial interpolation and finite elements, and glimpse an application to the simulation of colliding fl holes.
		An amusing example of spline interpolation is the matlab file plotsig.m, which uses cubic splines to interpolate President Clinton's signature.
		The end of the talk concentrated on piecewise polynomial interpolation in three dimensions, and particularly on some recent work concerning the adaptive construction of tetrahedral meshes for computing initial data for fl hole collisions.
	www.ima.umn.edu /~arnold/interpolation/interpolation.html (917 words)

	CISC-271
		This emphasized the difference between local interpolation methods, such as straight line and Hermite interpolation, and interpolation where data points effect the interpolation curve globally, such as full degree polynomial interpolation and cubic splines.
		Let me emphasize that for material on piecewise polynomial interpolation the notation Pi(x) the i in the subscript refers to the polynomial interpolating the ith piece, and not the degree of the polynomial.
		The Lagrange basis is not likely to cause numerical errors as is the case when using the monomial basis for polynomial interpolation.
	www.cs.queensu.ca /home/daver/B271 (1327 words)

	Aitken's and Neville's Methods
		Usually there is an abundance of nodes (think 50, 100,...) and the degree of the interpolating polynomial is small (think 2, 3, 4, 5 or 6).
		The accuracy of interpolation increases with the degree of the polynomial.
		Since the polynomial constructions are unique the following theorem applies for the Lagrange Polynomial, Newton polynomial and the polynomials constructed with both Aitken's method and Neville's method too.
	math.fullerton.edu /mathews/n2003/NevilleAlgorithmMod.html (600 words)

	Polynomial interpolation (Site not responding. Last check: )
		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)

	Multivariate polynomial interpolation
		I am interested in bounding the p-norm of the error in a multivariate polynomial interpolation scheme by norms of derivatives which kill the interpolating space (and require the function interpolated to be no smoother than is necessary).
		These are the estimates that numerical analysts use to show that a numerical scheme based on multivariate polynomial interpolation has the highest order of accuracy that its polynomial reproduction allows.
		The error in linear interpolation at the vertices of a simplex, Waldron 1994).
	www.math.auckland.ac.nz /~waldron/Multivariate/multivariate.html (864 words)

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