Table of the zeros of the Legendre polynomials of order 1-16 and the weight coefficients for Gauss’ mechanical quadrature formula

Lowan, Arnold N.; Davids, Norman; Levenson, Arthur

doi:10.1090/s0002-9904-1942-07771-8

Cited by 81 publications

(26 citation statements)

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“…We have also found that the appended program can compute the numerical integration formulas of orders ; 1 ≤ n ≤ 48 which agree with standard tables presented in [4][5][6][7]. We hope that the MATLAB program presented here will be a useful addition to the existing software on numerical integration.…”

Section: Resultssupporting

confidence: 75%

“…However, we succeeded in determining zeros up to 32 digits accuracy by using the powerful built-in function solve (…..) available in the MATLAB programming. For even values of n, we have n/2 pairs of abscissas with opposite signs, while for odd values of n, there are pairs and hence only half the values need to be tabulated in tables of weights and abscissas of Gauss Legendre quadrature formulas, such tables are available in [4][5][6][7]. They are printed with 15-digits of accuracy for orders 2, 3, 4, 5, 6, 7, 8, 9, 10, 12.…”

Section: Theoremmentioning

confidence: 99%

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Application of MATLAB Symbolic Maths with Variable Precision Arithmetic (VPA) to Compute Some High Order Gauss Legendre Quadrature Rules

Rathod

Sathish²,

Islam

et al. 1970

GANIT: J. Bangladesh Math. Soc.

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Gauss Legendre Quadrature rules are extremely accurate and they should be considered seriously when many integrals of similar nature are to be evaluated. This paper is concerned with the derivation and computation of numerical integration rules for the three integrals:which are dependent on the zeros and the squares of the zeros of Legendre Polynomial and is quite well known in the Gaussian Quadrature theory. We have developed the necessary MATLAB programs based on symbolic maths which can compute the sampling points and the weight coefficients and are reported here upto 32 -digits accuracy and we believe that they are reported to this accuracy for the first time. The MATLAB programs appended here are based on symbolic maths. They are very sophisticated and they can compute Quadrature rules of high order, whereas one of the recent MATLAB program appearing in reference [21] can compute Gauss Legendre Quadrature rules upto order twenty, because the zeros of Legendre polynomials cannot be computed to desired accuracy by MATLAB routine roots (……..). Whereas we have used the MATLAB routine solve (……..) to find zeros of polynomials which is very efficient. This is worth noting in the present context.

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Section: Resultssupporting

confidence: 75%

Section: Theoremmentioning

confidence: 99%

Application of MATLAB Symbolic Maths with Variable Precision Arithmetic (VPA) to Compute Some High Order Gauss Legendre Quadrature Rules

Rathod

Sathish²,

Islam

et al. 1970

GANIT: J. Bangladesh Math. Soc.

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“…(1) which, just as the density of the incident radiant flux, is approximated by the Gauss quadrature formula [21] at each point of the computational region (i = 1 ... N p ):…”

Section: Brief Review Of Modern Methods Of Solving the Radiative-tranmentioning

confidence: 99%

Numerical solution of the radiative-transfer equation for an absorbing, emitting, and scattering medium with a complex 3-D geometry

Тимошпольский

German

Гринчук

et al. 2005

J Eng Phys Thermophys

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A method is proposed for numerically solving the integro-differential, radiative-transfer equation with the use of its piecewise-analytic solutions obtained by the discrete-ordinate method and grids constructed by the finiteelement method. The advantages of the method proposed and some results of calculation of the radiativetransfer characteristics for one-, two-, and three-dimensional problems are discussed.Radiative energy transfer is of crucial importance in many natural and technical processes of energy exchange. This pertains equally to high-temperature processes (combustion of organic fuels, thermal treatment of metals, hightemperature synthesis and pyrolysis in chemical technologies, etc.) where the radiative energy transfer accounts for 90% or more of the total energy exchange (see, for example, [1][2][3][4]) and to the processes occurring at lower atmospheric temperatures [5,6]. It is known that an exact estimation of the characteristics of heat and mass transfer in technological processes allows one to obtain a significant economical effect, i.e., to increase the quality and functional characteristics of products and decrease their cost, as well as to make for good environmental conditions and to conserve material, energy, and manpower resources. For example, the temperature fields of many technological processes (several tens of them are described in [1,2]) occurring at temperatures from 125 to 1600 o C should be calculated with an accuracy of ,1-2 o C.Radiative heat exchange plays a dominant role in the total heat exchange in high-temperature processes in gaseous media. The accuracy of estimation of the temperature fields of such media is primarily dependent on the correctness of calculation of the radiative-transfer characteristics. This is also very important for optimization of the heating of steel products having a different geometry in ring furnaces with a moving bottom in which the working temperatures can reach 1200 o C.Mathematical Model. It is difficult to calculate the characteristics of radiative heat transfer in selectively emitting, absorbing, and scattering media because, in this case, it is necessary to take into account the multiple processes of reradiation on solid particles, the selectivity of the radiation of gas components, and the temperature inhomogeneity and complex configuration of the radiating volume. The correctness of estimation of the radiative-heat-exchange characteristics depends, to a large extent, on the correctness of solution of the radiative-transfer equation [7][8][9]. In the case of a local thermodynamic equilibrium, this equation defines the law of conservation of radiant energy in the process of its propagation in an absorbing, emitting, and scattering medium: l⋅∇I λ (r, l) + [χ λ (r) + σ λ (r)] I λ (r, l) = χ λ (r) B λ (T (r)) + σ λ (r) 4π ∫ 4π p λ (r, l, l ′ ) I λ (r, l ′ ) dΩ ′ .(1)The boundary conditions for Eq. (1) are determined by the radiation and reflection processes occurring on the boundary surfaces of the medium and can be written, in the general cas...

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“…Computations necessary for determining the point locations and the weights were done by Lowan, Davids, and Levenson [24]. The result is that…”

Section: Linearly Polarized Incident Wavementioning

confidence: 99%

Light Scattering from an Optically Active Sphere into a Circular Sensor Aperture

Pendleton¹,

Rosen²,

Gillespie³

1997

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Two methods are developed to average V and I Stokes parameters over a circular sensor aperture collecting light scattered from an optically active sphere. One method uses a two-dimensional numerical integration of Bohren's theory in scalar form, which explicitly shows the connection with Mueller matrix elements. The second method uses expansions of vector spherical harmonics in Bohren's theory to integrate over apertures of any size without convergence checks since numerical integration is avoided. Equations simplifying the analytical results for 4π scattering are obtained. Sample computations of average Stokes parameters are performed with both methods, and agreement to six decimal places is obtained.ii

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Table of the zeros of the Legendre polynomials of order 1-16 and the weight coefficients for Gauss’ mechanical quadrature formula

Cited by 81 publications

References 0 publications

Application of MATLAB Symbolic Maths with Variable Precision Arithmetic (VPA) to Compute Some High Order Gauss Legendre Quadrature Rules

Application of MATLAB Symbolic Maths with Variable Precision Arithmetic (VPA) to Compute Some High Order Gauss Legendre Quadrature Rules

Numerical solution of the radiative-transfer equation for an absorbing, emitting, and scattering medium with a complex 3-D geometry

Light Scattering from an Optically Active Sphere into a Circular Sensor Aperture

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