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Journal articles on the topic 'Quadrature de Gauss'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Kwon, Young-Doo, Soon-Bum Kwon, Bo-Kyung Shim, and Hyun-Wook Kwon. "Comprehensive Interpretation of a Three-Point Gauss Quadrature with Variable Sampling Points and Its Application to Integration for Discrete Data." Journal of Applied Mathematics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/471731.

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This study examined the characteristics of a variable three-point Gauss quadrature using a variable set of weighting factors and corresponding optimal sampling points. The major findings were as follows. The one-point, two-point, and three-point Gauss quadratures that adopt the Legendre sampling points and the well-known Simpson’s 1/3 rule were found to be special cases of the variable three-point Gauss quadrature. In addition, the three-point Gauss quadrature may have out-of-domain sampling points beyond the domain end points. By applying the quadratically extrapolated integrals and nonlinearity index, the accuracy of the integration could be increased significantly for evenly acquired data, which is popular with modern sophisticated digital data acquisition systems, without using higher-order extrapolation polynomials.

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2

Fee, Greg. "Gauss-Legendre quadrature." ACM SIGSAM Bulletin 33, no. 3 (September 1999): 26. http://dx.doi.org/10.1145/347127.347443.

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3

Pranić, Miroslav S., and Lothar Reichel. "Rational Gauss Quadrature." SIAM Journal on Numerical Analysis 52, no. 2 (January 2014): 832–51. http://dx.doi.org/10.1137/120902161.

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4

FRANJIĆ, IVA, JOSIP PEČARIĆ, and IVAN PERIĆ. "GENERAL THREE-POINT QUADRATURE FORMULAS OF EULER TYPE." ANZIAM Journal 52, no. 3 (January 2011): 309–17. http://dx.doi.org/10.1017/s1446181111000721.

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AbstractGeneral three-point quadrature formulas for the approximate evaluation of an integral of a function f over [0,1], through the values f(x), f(1/2), f(1−x), f′(0) and f′(1), are derived via the extended Euler formula. Such quadratures are sometimes called “corrected” or “quadratures with end corrections” and have a higher accuracy than the adjoint classical formulas, which only include the values f(x), f(1/2) and f(1−x) . The Gauss three-point, corrected Simpson, corrected dual Simpson, corrected Maclaurin and corrected Gauss two-point formulas are recaptured as special cases. Finally, sharp estimates of error are given for this type of quadrature formula.

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5

Reichel, Lothar, Miodrag Spalevic, and Jelena Tomanovic. "Rational averaged gauss quadrature rules." Filomat 34, no. 2 (2020): 379–89. http://dx.doi.org/10.2298/fil2002379r.

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It is important to be able to estimate the quadrature error in Gauss rules. Several approaches have been developed, including the evaluation of associated Gauss-Kronrod rules (if they exist), or the associated averaged Gauss and generalized averaged Gauss rules. Integrals with certain integrands can be approximated more accurately by rational Gauss rules than by Gauss rules. This paper introduces associated rational averaged Gauss rules and rational generalized averaged Gauss rules, which can be used to estimate the error in rational Gauss rules. Also rational Gauss-Kronrod rules are discussed. Computed examples illustrate the accuracy of the error estimates determined by these quadrature rules.

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6

Cao, Ting, Huo-tao Gao, Chun-feng Sun, Yun Ling, and Guo-bao Ru. "Application of Improved Simplex Quadrature Cubature Kalman Filter in Nonlinear Dynamic System." Mathematical Problems in Engineering 2020 (May 14, 2020): 1–13. http://dx.doi.org/10.1155/2020/1072824.

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A novel spherical simplex Gauss–Laguerre quadrature cubature Kalman filter is proposed to improve the estimation accuracy of nonlinear dynamic system. The nonlinear Gaussian weighted integral has been approximately evaluated using the spherical simplex rule and the arbitrary order Gauss–Laguerre quadrature rule. Thus, a spherical simplex Gauss–Laguerre cubature quadrature rule is developed, from which the general computing method of the simplex cubature quadrature points and the corresponding weights are obtained. Then, under the nonlinear Kalman filtering framework, the spherical simplex Gauss–Laguerre quadrature cubature Kalman filter is derived. A high-dimensional nonlinear state estimation problem and a target tracking problem are utilized to demonstrate the effectiveness of the proposed spherical simplex Gauss–Laguerre cubature quadrature rule to improve the performance.

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7

Hagler, Brian A. "Laurent-Hermite-Gauss Quadrature." Journal of Computational and Applied Mathematics 104, no. 2 (May 1999): 163–71. http://dx.doi.org/10.1016/s0377-0427(99)00054-0.

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8

Villarino, Mark B. "Gauss on Gaussian Quadrature." American Mathematical Monthly 127, no. 2 (January 6, 2020): 125–38. http://dx.doi.org/10.1080/00029890.2020.1680201.

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9

Peherstorfer, Franz. "Gauss-Tchebycheff quadrature formulas." Numerische Mathematik 58, no. 1 (December 1990): 273–86. http://dx.doi.org/10.1007/bf01385625.

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10

Milovanović, G. V., M. M. Spalević, and L. J. Galjak. "Kronrod Extensions of Gaussian Quadratures with Multiple Nodes." Computational Methods in Applied Mathematics 6, no. 3 (2006): 291–305. http://dx.doi.org/10.2478/cmam-2006-0016.

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Abstract In this paper, general real Kronrod extensions of Gaussian quadrature formulas with multiple nodes are introduced. A proof of their existence and uniqueness is given. In some cases, the explicit expressions of polynomials, whose zeros are the nodes of the considered quadratures, are determined. Very effective error bounds of the Gauss — Turán — Kronrod quadrature formulas, with Gori — Micchelli weight functions, for functions analytic on confocal ellipses, are derived.

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11

Kang, Sanggoo, Yin Chao Wu, and Suyun Ham. "Singular Integral Solutions of Analytical Surface Wave Model with Internal Crack." Applied Sciences 10, no. 9 (April 30, 2020): 3129. http://dx.doi.org/10.3390/app10093129.

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In this study, singular integral solutions were studied to investigate scattering of Rayleigh waves by subsurface cracks. Defining a wave scattering model by objects, such as cracks, still can be quite a challenge. The model’s analytical solution uses five different numerical integration methods: (1) the Gauss–Legendre quadrature, (2) the Gauss–Chebyshev quadrature, (3) the Gauss–Jacobi quadrature, (4) the Gauss–Hermite quadrature and (5) the Gauss–Laguerre quadrature. The study also provides an efficient dynamic finite element analysis to demonstrate the viability of the wave scattering model with an optimized model configuration for wave separation. The obtained analytical solutions are verified with displacement variation curves from the computational simulation by defining the correlation of the results. A novel, verified model, is proposed to provide variations in the backward and forward scattered surface wave displacements calculated by different frequencies and geometrical crack parameters. The analytical model can be solved by the Gauss–Legendre quadrature method, which shows the significantly correlated displacement variation with the FE simulation result. Ultimately, the reliable analytic model can provide an efficient approach to solving the parametric relationship of wave scattering.

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12

Jin, Shaobo, and Björn Andersson. "A note on the accuracy of adaptive Gauss–Hermite quadrature." Biometrika 107, no. 3 (February 10, 2020): 737–44. http://dx.doi.org/10.1093/biomet/asz080.

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Summary Numerical quadrature methods are needed for many models in order to approximate integrals in the likelihood function. In this note, we correct the error rate given by Liu & Pierce (1994) for integrals approximated with adaptive Gauss–Hermite quadrature and show that the approximation is less accurate than previously thought. We discuss the relationship between the error rates of adaptive Gauss–Hermite quadrature and Laplace approximation, and provide a theoretical explanation of simulation results obtained in previous studies regarding the accuracy of adaptive Gauss–Hermite quadrature.

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13

PIAUD, BENJAMIN, STÉPHANE BLANCO, RICHARD FOURNIER, VICTOR EUGEN AMBRUŞ, and VICTOR SOFONEA. "GAUSS QUADRATURES – THE KEYSTONE OF LATTICE BOLTZMANN MODELS." International Journal of Modern Physics C 25, no. 01 (December 2, 2013): 1340016. http://dx.doi.org/10.1142/s0129183113400160.

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In this paper, we compare two families of Lattice Boltzmann (LB) models derived by means of Gauss quadratures in the momentum space. The first one is the HLB (N;Qx,Qy,Qz) family, derived by using the Cartesian coordinate system and the Gauss–Hermite quadrature. The second one is the SLB (N;K,L,M) family, derived by using the spherical coordinate system and the Gauss–Laguerre, as well as the Gauss–Legendre quadratures. These models order themselves according to the maximum order N of the moments of the equilibrium distribution function that are exactly recovered. Microfluidics effects (slip velocity, temperature jump, as well as the longitudinal heat flux that is not driven by a temperature gradient) are accurately captured during the simulation of Couette flow for Knudsen number (kn) up to 0.25.

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14

Jandrlic, Davorka, Miodrag Spalevic, and Jelena Tomanovic. "Error estimates for certain cubature formulae." Filomat 32, no. 20 (2018): 6893–902. http://dx.doi.org/10.2298/fil1820893j.

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We estimate the errors of selected cubature formulae constructed by the product of Gauss quadrature rules. The cases of multiple and (hyper-)surface integrals over n-dimensional cube, simplex, sphere and ball are considered. The error estimates are obtained as the absolute value of the difference between cubature formula constructed by the product of Gauss quadrature rules and cubature formula constructed by the product of corresponding Gauss-Kronrod or corresponding generalized averaged Gaussian quadrature rules. Generalized averaged Gaussian quadrature rule ?2l+1 is (2l + 1)-point quadrature formula. It has 2l + 1 nodes and the nodes of the corresponding Gauss rule Gl with l nodes form a subset, similar to the situation for the (2l + 1)-point Gauss-Kronrod rule H2l+1 associated with Gl. The advantages of bG2l+1 are that it exists also when H2l+1 does not, and that the numerical construction of ?2l+1, based on recently proposed effective numerical procedure, is simpler than the construction of H2l+1.

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15

Hagler, Brian A. "d-fold Hermite–Gauss quadrature." Journal of Computational and Applied Mathematics 136, no. 1-2 (November 2001): 53–72. http://dx.doi.org/10.1016/s0377-0427(00)00575-6.

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16

Bokhari, M. A., Asghar Qadir, and H. Al-Attas. "On Gauss-Type Quadrature Rules." Numerical Functional Analysis and Optimization 31, no. 10 (September 16, 2010): 1120–34. http://dx.doi.org/10.1080/01630563.2010.510981.

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17

CHAN, A. H. C. "THE MAGIC OF GAUSS QUADRATURE." Engineering Computations 8, no. 2 (February 1991): 189–90. http://dx.doi.org/10.1108/eb023833.

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18

Alpert, Bradley K. "Hybrid Gauss-Trapezoidal Quadrature Rules." SIAM Journal on Scientific Computing 20, no. 5 (January 1999): 1551–84. http://dx.doi.org/10.1137/s1064827597325141.

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19

Pranić, Miroslav S., and Lothar Reichel. "Generalized anti-Gauss quadrature rules." Journal of Computational and Applied Mathematics 284 (August 2015): 235–43. http://dx.doi.org/10.1016/j.cam.2014.11.016.

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20

Milovanović, Gradimir V., and Aleksandar S. Cvetković. "Gauss–Hermite interval quadrature rule." Computers & Mathematics with Applications 54, no. 4 (August 2007): 544–55. http://dx.doi.org/10.1016/j.camwa.2007.01.027.

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21

Milovanović, Gradimir V., and Aleksandar S. Cvetković. "Gauss–Laguerre interval quadrature rule." Journal of Computational and Applied Mathematics 182, no. 2 (October 2005): 433–46. http://dx.doi.org/10.1016/j.cam.2004.12.021.

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22

Pejcev, Aleksandar, and Ljubica Mihic. "Errors of Gauss-Radau and Gauss-Lobatto quadratures with double end point." Applicable Analysis and Discrete Mathematics 13, no. 2 (2019): 463–77. http://dx.doi.org/10.2298/aadm180408011p.

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Starting from the explicit expression of the corresponding kernels, derived by Gautschi and Li (W. Gautschi, S. Li: The remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadrature rules with multiple end points, J. Comput. Appl. Math. 33 (1990) 315{329), we determine the exact dimensions of the minimal ellipses on which the modulus of the kernel starts to behave in the described way. The effective error bounds for Gauss- Radau and Gauss-Lobatto quadrature formulas with double end point(s) are derived. The comparisons are made with the actual errors.

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23

Zinser, Brian, Wei Cai, and Duan Chen. "Quadrature Weights on Tensor-Product Nodes for Accurate Integration of Hypersingular Functions over Some Simple 3-D Geometric Shapes." Communications in Computational Physics 20, no. 5 (November 2016): 1283–312. http://dx.doi.org/10.4208/cicp.oa-2015-0005.

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AbstractIn this paper, we present accurate and economic integration quadratures for hypersingular functions over three simple geometric shapes in ℝ3 (spheres, cubes, and cylinders). The quadrature nodes are made of the tensor-product of 1-D Gauss nodes on [–1,1] for non-periodic variables or uniform nodes on [0,2π] or [0,π] for periodic ones. The quadrature weights are converted from a brute-force integration of the hypersingular function through interpolating the smooth component of the integrand. Numerical results are presented to validate the accuracy and efficiency of computing hypersingular integrals, as in the computations of Cauchy principal values, with a minimum number of quadrature nodes. The pre-calculated quadrature tables can be then readily used to implement Nyström collocation methods of hypersingular volume integral equations such as the one for Maxwell equations.

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24

FUKUKAWA, K., Y. FUJIWARA, and Y. SUZUKI. "GAUSSIAN NONLOCAL POTENTIALS FOR THE QUARK-MODEL BARYON–BARYON INTERACTIONS." Modern Physics Letters A 24, no. 11n13 (April 30, 2009): 1035–38. http://dx.doi.org/10.1142/s021773230900053x.

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Gaussian nonlocal potentials for the quark-model baryon–baryon interactions are derived by using the Gauss-Legendre quadrature for the special functions. The reliability of the approximation is examined with respect to the phase shifts and the deuteron binding energy. The potential is accurate enough if one uses seven-point Gauss-Legendre quadrature.

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25

Ribičić Penava, Mihaela. "Hermite–Hadamard–Fejér-Type Inequalities and Weighted Three-Point Quadrature Formulae." Mathematics 9, no. 15 (July 22, 2021): 1720. http://dx.doi.org/10.3390/math9151720.

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The goal of this paper is to derive Hermite–Hadamard–Fejér-type inequalities for higher-order convex functions and a general three-point integral formula involving harmonic sequences of polynomials and w-harmonic sequences of functions. In special cases, Hermite–Hadamard–Fejér-type estimates are derived for various classical quadrature formulae such as the Gauss–Legendre three-point quadrature formula and the Gauss–Chebyshev three-point quadrature formula of the first and of the second kind.

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26

Jia, Bin, Ming Xin, and Yang Cheng. "Anisotropic Sparse Gauss-Hermite Quadrature Filter." Journal of Guidance, Control, and Dynamics 35, no. 3 (May 2012): 1014–22. http://dx.doi.org/10.2514/1.55364.

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27

Liu, Qing, and Donald A. Pierce. "A Note on Gauss-Hermite Quadrature." Biometrika 81, no. 3 (August 1994): 624. http://dx.doi.org/10.2307/2337136.

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28

Laurie, Dirk P. "Computation of Gauss-type quadrature formulas." Journal of Computational and Applied Mathematics 127, no. 1-2 (January 2001): 201–17. http://dx.doi.org/10.1016/s0377-0427(00)00506-9.

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29

OMLADIČ, MATJAŽ. "Average quadrature formulas of Gauss type." IMA Journal of Numerical Analysis 12, no. 2 (1992): 189–99. http://dx.doi.org/10.1093/imanum/12.2.189.

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30

Jetter, Kurt. "Uniqueness of Gauss–Birkhoff Quadrature Formulas." SIAM Journal on Numerical Analysis 24, no. 1 (February 1987): 147–54. http://dx.doi.org/10.1137/0724012.

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31

Laurie, Dirk P. "Calculation of Gauss-Kronrod quadrature rules." Mathematics of Computation 66, no. 219 (July 1, 1997): 1133–46. http://dx.doi.org/10.1090/s0025-5718-97-00861-2.

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32

Cali{ò, Franca, Walter Gautschi, and Elena Marchetti. "On computing Gauss-Kronrod quadrature formulae." Mathematics of Computation 47, no. 176 (1986): 639. http://dx.doi.org/10.1090/s0025-5718-1986-0856708-8.

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33

Calio, Franca, Walter Gautschi, and Elena Marchetti. "On Computing Gauss-Kronrod Quadrature Formulae." Mathematics of Computation 47, no. 176 (October 1986): 639. http://dx.doi.org/10.2307/2008178.

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34

Gautschi, Walter, Laura Gori, and Francesca Pitolli. "Gauss Quadrature for Refinable Weight Functions." Applied and Computational Harmonic Analysis 8, no. 3 (May 2000): 249–57. http://dx.doi.org/10.1006/acha.1999.0306.

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35

LIU, QING, and DONALD A. PIERCE. "A note on Gauss—Hermite quadrature." Biometrika 81, no. 3 (1994): 624–29. http://dx.doi.org/10.1093/biomet/81.3.624.

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36

Rathod, H. T., B. Venkatesudu, and K. V. Nagaraja. "Gauss Legendre Quadrature Formulae for Tetrahedra." International Journal for Computational Methods in Engineering Science and Mechanics 6, no. 3 (July 2005): 179–86. http://dx.doi.org/10.1080/15502280590923711.

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37

Calvetti, D., G. H. Golub, W. B. Gragg, and L. Reichel. "Computation of Gauss-Kronrod quadrature rules." Mathematics of Computation 69, no. 231 (February 17, 2000): 1035–53. http://dx.doi.org/10.1090/s0025-5718-00-01174-1.

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38

Gil, Amparo, Javier Segura, and Nico M. Temme. "Noniterative Computation of Gauss--Jacobi Quadrature." SIAM Journal on Scientific Computing 41, no. 1 (January 2019): A668—A693. http://dx.doi.org/10.1137/18m1179006.

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39

Djukić, Dušan Lj, Lothar Reichel, and Miodrag M. Spalević. "Truncated generalized averaged Gauss quadrature rules." Journal of Computational and Applied Mathematics 308 (December 2016): 408–18. http://dx.doi.org/10.1016/j.cam.2016.06.016.

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40

Kim, Kyung Joong. "Two-frequency-dependent Gauss quadrature rules." Journal of Computational and Applied Mathematics 174, no. 1 (February 2005): 43–55. http://dx.doi.org/10.1016/j.cam.2004.03.020.

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41

Alqahtani, Hessah, and Lothar Reichel. "Generalized block anti-Gauss quadrature rules." Numerische Mathematik 143, no. 3 (July 30, 2019): 605–48. http://dx.doi.org/10.1007/s00211-019-01069-z.

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42

O’Leary, Dianne P., Zdeněk Strakoš, and Petr Tichý. "On sensitivity of Gauss–Christoffel quadrature." Numerische Mathematik 107, no. 1 (April 14, 2007): 147–74. http://dx.doi.org/10.1007/s00211-007-0078-x.

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43

Fenu, C., D. Martin, L. Reichel, and G. Rodriguez. "Block Gauss and Anti-Gauss Quadrature with Application to Networks." SIAM Journal on Matrix Analysis and Applications 34, no. 4 (January 2013): 1655–84. http://dx.doi.org/10.1137/120886261.

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44

Rathod, H. T., B. Venkatesudu, K. V. Nagaraja, and Md Shafiqul Islam. "Gauss Legendre–Gauss Jacobi quadrature rules over a tetrahedral region." Applied Mathematics and Computation 190, no. 1 (July 2007): 186–94. http://dx.doi.org/10.1016/j.amc.2007.01.014.

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45

Mihic, Ljubica, Aleksandar Pejcev, and Miodrag Spalevic. "Error bounds for Gauss-Lobatto quadrature formula with multiple end points with Chebyshev weight function of the third and the fourth kind." Filomat 30, no. 1 (2016): 231–39. http://dx.doi.org/10.2298/fil1601231m.

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For analytic functions the remainder terms of quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points -+1, for Gauss-Lobatto quadrature formula with multiple end points with Chebyshev weight function of the third and the fourth kind. Starting from the explicit expression of the corresponding kernel, derived by Gautschi and Li, we determine the locations on the ellipses where maximum modulus of the kernel is attained. The obtained values confirm the corresponding conjectured values given by Gautschi and Li in paper [The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple end points, Journal of Computational and Applied Mathematics 33 (1990) 315-329.]

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46

Masjed-Jamei, Mohammad. "New error bounds for Gauss-Legendre quadrature rules." Filomat 28, no. 6 (2014): 1281–93. http://dx.doi.org/10.2298/fil1406281m.

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It is well-known that the remaining term of any n-point interpolatory quadrature rule such as Gauss-Legendre quadrature formula depends on at least an n-order derivative of the integrand function, which is of no use if the integrand is not smooth enough and requires a lot of differentiation for large n. In this paper, by defining a specific linear kernel, we resolve this problemand obtain new bounds for the error of Gauss-Legendre quadrature rules. The advantage of the obtained bounds is that they do not depend on the norms of the integrand function. Some illustrative examples are given in this direction.

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47

Ehrich, Sven. "On stratified extensions of Gauss–Laguerre and Gauss–Hermite quadrature formulas." Journal of Computational and Applied Mathematics 140, no. 1-2 (March 2002): 291–99. http://dx.doi.org/10.1016/s0377-0427(01)00407-1.

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48

Díaz de Alba, Patricia, Luisa Fermo, and Giuseppe Rodriguez. "Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules." Numerische Mathematik 146, no. 4 (November 18, 2020): 699–728. http://dx.doi.org/10.1007/s00211-020-01163-7.

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AbstractThis paper is concerned with the numerical approximation of Fredholm integral equations of the second kind. A Nyström method based on the anti-Gauss quadrature formula is developed and investigated in terms of stability and convergence in appropriate weighted spaces. The Nyström interpolants corresponding to the Gauss and the anti-Gauss quadrature rules are proved to furnish upper and lower bounds for the solution of the equation, under suitable assumptions which are easily verified for a particular weight function. Hence, an error estimate is available, and the accuracy of the solution can be improved by approximating it by an averaged Nyström interpolant. The effectiveness of the proposed approach is illustrated through different numerical tests.

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49

Xiao, Qing, and Shaowu Zhou. "Comparing unscented transformation and point estimate method for probabilistic power flow computation." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 37, no. 3 (May 8, 2018): 1290–303. http://dx.doi.org/10.1108/compel-09-2017-0393.

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Purpose Unscented transformation (UT) and point estimate method (PEM) are two efficient algorithms for probabilistic power flow (PPF) computation. This paper aims to show the relevance between UT and PEM and to derive a rule to determine the accuracy controlling parameters for UT method. Design/methodology/approach The authors derive the underlying sampling strategies of UT and PEM and check them in different probability spaces, where quadrature nodes are selected. Findings Gauss-type quadrature rule can be used to determine the accuracy controlling parameters of UT. If UT method and PEM select quadrature nodes in two probability spaces related by a linear transform, these two algorithms are equivalent. Originality/value It shows that UT method can be conveniently extended to (km + 1) scheme (k = 4; 6; : : : ) by Gauss-type quadrature rule.

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50

Pozza, Stefano, Miroslav S. Pranić, and Zdeněk Strakoš. "The Lanczos algorithm and complex Gauss quadrature." ETNA - Electronic Transactions on Numerical Analysis 50 (2018): 1–19. http://dx.doi.org/10.1553/etna_vol50s1.

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