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Topic: Gaussian quadrature

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Numerical integration

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	Integration
		The idea of the Gaussian quadrature is that the abscissas are no longer restricted to be equally spaced, and that they can be chosen to give higher accuracy.
		The argument of Gia applies here: If the number of abscissas used in the Gaussian quadrature exceeds by one the number N in the truncated expansion in equation (3.21), the error introduced by the Gaussian quadrature is totally negligible compared with the error in the truncated expansion itself.
		The Gaussian quadrature method with the weight function chosen here seems to be comparable to the one used by Gia, and greatly superior to the integration methods used otherwise in connection with the AVP.
	trond.hjorteland.com /thesis/node24.html (382 words)

	PlanetMath: quadrature
		The terms “quadrature” and “cubature” are typically used in numerical analysis to denote the approximation of a definite integral, typically by a suitable weighted sum.
		Some such numerical quadrature methods are Simpson's rule, the trapezoidal rule, and Gaussian quadrature.
		This is version 9 of quadrature, born on 2002-01-05, modified 2007-02-15.
	planetmath.org /encyclopedia/Quadrature.html (219 words)

	Quadrature
		The first part of this gives the quadrature formula, and is simply a weighted sum of the values of the function at the interpolation or quadrature points, x
		The second part gives the quadrature error, which is
		Recall that, when f(x) is any polynomial of degree n or less, it is represented exactly by p(x), and so the quadrature error vanishes in this case.
	www.maths.lancs.ac.uk /~gilbert/m243b/node11.html (175 words)

	NIntegrate Integration Rules - Wolfram Mathematica
		Gaussian quadrature uses optimal sampling points (through polynomial interpolation) to form a weighted sum of the integrand values over these points.
		A Gaussian rule G(f, n) of n points for integrand f is exact for polynomials of degree 2n-1.
		Gaussian rules are of open type since the integrand is not evaluated at the end points of the interval.
	reference.wolfram.com /mathematica/tutorial/NIntegrateIntegrationRules.html (4135 words)

	Comparison with Gaussian Quadrature
		When the source distribution is smooth, as in the case of a uniformly bright rectangle, we expect numerical quadrature to yield accurate results efficiently.
		To verify this prediction, we compared the multipole expansion against Gaussian quadrature of the integral in equation 3, for a uniform square source.
		However, Gaussian quadrature applies only if the integrand is smooth, and this tends to rule out light sources that are modeled as rasters of point sources.
	www.cs.princeton.edu /~ah/publications/multipole/html/node17.html (269 words)

	Gaussian quadrature - Wikipédia
		Dina analisis numeris, quadrature rule ngarupakeun salah sahiji "pendekatan" definite integral tina function, biasana "dinyatakan" salaku jumlah beurat tina nilai fungsi dina titik husus dijero domain integration.
		The error of a Gaussian quadrature rule can be stated as follows (theorem 3.6.24 in Stoer and Bulirsch).
		Another approach is to use two Gaussian quadrature rules of different orders, and to estimate the error as the difference between the two results.
	su.wikipedia.org /wiki/Gaussian_quadrature (544 words)

	Amazon.com: quadrature (Site not responding. Last check: )
		Famous problems of elementary geometry: the duplication of the cube; the trisection of an angle; the quadrature of the circle; an authorized translation...
		The quadrature of the circle, the square root of two, and the right-angled triangle, by William Alexander Myers.
		The theory and construction of the quadrature of the circle;: Also, the globe or ball reduced to the cube, and two new measures--the octants, with the inclination of the perpendicular line by John May (Unknown Binding - 1866)
	www.amazon.com /s?ie=UTF8&keywords=quadrature&index=blended&page=1 (771 words)

	Numerical integration Summary
		Gaussian quadratures allow one to pick the optimal abscissas as which to evaluate the function.
		The fundamental theorem of Gaussian quadratures shows that the optimal abscissas are the roots of the orthogonal polynomial for the same interval and weighting function.
		Some commonly used Gaussian quadratures are the Gauss-Legendre formula and the Gauss-Chebyshev formula, which are used for closed, definite integrals, the Gauss-Hermite formula, which is used for integrals that have - and as the limits of integration, and the Gauss-Laguerre formula, which is used for integrals on the interval [0,).
	www.bookrags.com /Numerical_integration (2133 words)

	Gaussian 03 Release Notes
		The reference for the APT charges included in the output for some Gaussian calculations is: J. Cioslowski, JACS 111 (1989) 8333.
		The original implementation of the mPW exchange functional was not consistent with the equations in the paper: the local scaling factor was applied in computing the non-local correction.
		The version of mPW in Gaussian 03 corrects this error, but since the parameters in the original paper were optimized with the incorrect functional, the correct one cannot reproduce the original results.
	www.gaussian.com /g_tech/g03_rel.htm (3167 words)

	Deriving quadrature rules from Gaussian processes (Site not responding. Last check: )
		Quadrature rules are often designed to achieve zero error on a small set of functions, e.g.
		It is also shown how classical Gaussian quadrature rules, trigonometric lattice rules, and spline rules can be extended to the average-case and to multiple dimensions by deriving them from Gaussian processes.
		In addition to being more robust, multidimensional quadrature rules designed for the average-case are found to be much less ambiguous than those designed for a given polynomial degree.
	research.microsoft.com /~minka/papers/quadrature.html (168 words)

	Professor Shu Chang
		Among them, the integration rules of Gaussian quadrature are extensively used, where both the weighting coefficients and the coordinates of the grid points are taken as unknowns.
		Another drawback of Gaussian quadrature is that, since the coordinates are determined by the roots of some functions, they cannot be arbitrarily chosen.
		In the case of solving a differential-integral equation, the grid points, where the unknowns are set, are usually generated in advance, and may not be the same as the coordinates required for Gaussian quadrature to be performed.
	serve.me.nus.edu.sg /shuchang/Research/GIQ.htm (561 words)

	Gaussian Quadrature
		In Gaussian quadrature we choose as quadrature points the m roots to P
		The quadrature is exact when y is a polynomial of degree 2m1 in x.
		Because there are m weights and m Gauss points, we have 2m parameters which are chosen to exactly represent a polynomial of degree 2m-1, which has 2m parameters.
	faculty.washington.edu /finlayso/ebook/quadrature/methods/Gauss.htm (104 words)

	Quadrature.NET
		One of my favorite explanations of the theory of quadrature can be found here.
		The order of Gaussian quadrature 'n' necessary to the exact solution is given by 2n-1 = q.
		Accumulate() assembles the quadrature by accumulating the results of function evaluation at each sampling point.
	engineeringobjects.com /DotNet/Quadrature/Quadrature.htm (452 words)

	Module scipy.integrate.quadrature (Site not responding. Last check: )
		Outputs: (val, None) val -- Gaussian quadrature approximation to the integral.
		Description: Integrate func from a to b using Gaussian quadrature with absolute tolerance tol.
		Outputs: (val, err) val -- Gaussian quadrature approximation (within tolerance) to integral.
	www.scipy.org /doc/api_docs/scipy.integrate.quadrature.html (574 words)

	Gaussian Quadrature
		The simplest form of Gaussian Integration is based on the use of an optimally chosen polynomial to approximate the integrand f(t) over the interval [-1,+1].
		Gaussian quadrature formulae are evaluating using abscissae and weights from a table like that included here.
		The choice of value of n is not always clear, and experimentation is useful to see the influence of choosing a different number of points.
	pathfinder.scar.utoronto.ca /~dyer/csca57/book_P/node44.html (634 words)

	Papers on quadrature
		Diethelm, Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals, Numer.
		K.J. Förster and K. Petras, On the variance of Gaussian quadrature formulae in the ultraspherical case, Calcolo 31 (1996), no. 1-2, 1-34.
		Petras, On the universality of the Gaussian quadrature formula, East Journal on Approximations 2 (1996), 427-438.
	www.cs.kuleuven.ac.be /~ronald/Publi/PUB96/node4.html (293 words)

	Diffusivity Approximation
		These angular integrals may be reduced to a calculation of the intensity at certain quadrature points (zenith angles) in each hemisphere with surprising accuracy.
		For two point Gaussian quadrature the quadrature weights happen to be unity, but not so for three point (or higher order) Gaussian quadrature.
		What is often used to evaluate (48) instead of Gaussian quadrature is a diffusivity approximation.
	dust.ess.uci.edu /facts/rt/node20.html (291 words)

	6.2 Calculation of Element Matrices
		We evaluate the integral expressions for the matrices (12), (14), and (16) numerically using Gaussian quadrature [27].
		The integral is obtained by applying Gaussian quadrature on the 1-D interval twice.
		Generally speaking, for integrands that are polynomial of degree 2N-1 or less, Gaussian quadrature evaluates the integral exactly with N weights and abscissas.
	www.cs.ucla.edu /~dt/papers/tog94/node19.html (165 words)

	Equation: Spread option using Gaussian quadrature - SITMO
		This equation uses the Gauss-Legendre quadrature to approximate the value of a spread option.
		The Gauss-Legendre quadrature abscissas (Xi) are rescaled in the range -4 to +4.
		The equation is unbiased and gives very accurate results, typical 6 digit accuracy with 16 quadrature points.
	www.sitmo.com /eq/541 (114 words)

	Numerical Differential Equation Analysis Package - Wolfram Mathematica
		This gives the additional trees that are necessary for a fourth-order method assuming that the quadrature conditions through order 4 and the row and column simplifying assumptions of order 1 hold.
		The Gaussian quadrature functionality provided in NumericalDifferentialEquationAnalysis allows you to easily study some of the theory behind ordinary Gaussian quadrature which is a little less sophisticated.
		The basic idea behind Gaussian quadrature is to approximate the value if an integral as a linear combination of values of the integrand evaluated at specific points:
	reference.wolfram.com /mathematica/NumericalDifferentialEquationAnalysis/tutorial/NumericalDifferentialEquationAnalysis.html (2450 words)

	[math/0611057] Gaussian Summation: An Exponentially Converging Summation Scheme
		Gaussian Quadrature is a well known technique for numerical integration.
		Recently Gaussian quadrature with respect to discrete measures corresponding to finite sums have found some new interest.
		The Gaussian summation achieves a given accuracy with a number of points proportional to the sqrt of the large scale whereas other summation schemes require at least a number of function evaluations proportional to the scale.
	arxiv.org /abs/math.NA/0611057 (230 words)

	DUNAVANT - Quadrature Rules for the Triangle
		FEKETE is a FORTRAN90 library of routines for defining Fekete rules for quadrature or interpolation on a triangle.
		STROUD is a FORTRAN90 library containing quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.
		WANDZURA is a FORTRAN90 library of routines for defining Wandzura rules for quadrature on a triangle.
	people.scs.fsu.edu /~burkardt/f_src/dunavant/dunavant.html (581 words)

	Numerical integration on the 11C
		Gaussian quadrature methods are among the theoretically very best existing algorithms for numerical integration.
		Even so, standard Gaussian quadrature methods are not the cutting edge in numerical quadrature nowadays, there are other advanced, more powerful methods that will cope with just about any integral automatically, even if improper, to any desired accuracy, the caveat being they are much more complex and delicate to correctly implement.
		The link to Gaussian quadrature shows the points and weighting factors for respective numbers of points per division.
	www.hpmuseum.org /cgi-sys/cgiwrap/hpmuseum/archv016.cgi?read=94446 (4762 words)

	Quadrature over tetrahedra
		For example, to extend the k-point Gaussian quadrature formula to the tetrahedron, one uses k^3 quadrature points arranged in a distorted cube inside the tetrahedron with appropriate polynomial weights.
		But this solution seems inefficient because many quadrature points are clustered (apparently needlessly) near one of the four vertices of the tetrahedron.
		In the case of quadrature for the purpose of computing stiffness matrix entries for finite element analysis (which was the motivation for my question) some more specialized techniques are available.
	www.cs.cornell.edu /home/vavasis/quad.html (681 words)

	NCL Documentation: Description of Gaussian, fixed, fixed offset, regular, curvilinear grids (Site not responding. Last check: )
		A Gaussian grid is one where each grid point can be uniquely accessed by one-dimensional latitude and longitude arrays (i.e.
		The longitudes are equally spaced while the latitudes are unequally spaced according to the Gaussian quadrature.
		Gaussian grids do not have points at the poles.
	www.ncl.ucar.edu /Document/Functions/sphpk_grids.shtml (308 words)

	Systat Software Inc. - AutoSignal - HTML Help
		The Romberg and Adaptive Quadrature algorithms are generally much slower than the Gaussian Quadrature procedure.
		For numerically intense integrations, you may wish to restrict the integration to the Gaussian Quadrature procedure.
		Gaussian noise with a standard deviation of this % of the data series standard deviation is added at the conclusion of the signal generation.
	www.systat.com /products/AutoSignal/help/?sec=1142 (911 words)

	Module scipy.special.orthogonal (Site not responding. Last check: )
		These calculations are done by finding the eigenvalues of a tridiagonal matrix whose entries are dependent on the coefficients in the recursion formula for the orthogonal polynomials with the corresponding weighting function over the interval.
		Returns the roots (x) of the nth order Chebyshev (of the first kind) polynomial, C_n(x), and weights (w) to use in Gaussian Quadrature over [-2,2] with weighting function (1-(x/2)2)(-1/2).
		Returns the roots (x) of an nth order orthogonal polynomail, and weights (w) to use in appropriate Gaussian quadrature with that orthogonal polynomial.
	www.scipy.org /doc/api_docs/scipy.special.orthogonal.html (855 words)

	Quadrature solutions for 3D capacitance extraction (US6314545)
		A Gaussian quadrature is computed for the pairs allocated to the far field while a high order quadrature is computed for those allocated in the near field.
		The component simulation is arrived after combining information derived from the Gaussian quadrature and the high order quadrature into a matrix which is then solved to obtain the charge distribution.
		Summation of the charges thus obtained yields the capacitance of the element.The high order quadrature is computed using a plurality of basis functions.
	www.delphion.com /details?pn=US06314545__ (556 words)

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