Iteration-Free Computation of Gauss-Legendre Quadrature Nodes and Weights

Gauss-Laguerre Quadrature Evaluation Points and Weights — MATLAB {amp} Simulink Example

19

[5]N. Hale and L. Trefethen, “Chebfun and Numerical Quadrature,” Science China Mathematics,

vol. 55, no. 9, pp. 1749–1760, 2012.

[6]H. Wang andS. Xiang, “On the Convergence Rates of Legendre Approximation,” Mathematics

of Computation, vol. 81, pp. 861–877, 2012.

[7]W.C. Chew, J. Jin, E. Michielssen, andJ. Song, Fast and Efficient Algorithms in Computational

Electromagnetics.Artech House, 2001.

[8]N. Hale and A. Townsend, “Fast and Accurate Computation of Gauss-Legendre and Gauss-

Jacobi Quadrature Nodes and Weights,” SIAM Journal on Scientific Computing, vol. 35,

no. 2, pp. A652–A674, 2013.

[9]G. H. Golub and J. H. Welsch, “Calculation of Gauss Quadrature Rules,”Mathematics of

Computation, vol. 23, pp. 221–230, 1969.

[10]I. Bogaert, B. Michiels, and J. Fostier, “O(1) Computation of Legendre Polynomials and Gauss-

Legendre Nodes and Weights for Parallel Computing,” SIAM Journal on Scientific Com-

puting, vol. 34, no. 3, pp. 83–101, 2012.

[11]A. Glaser, X. Liu,and V. Rokhlin, “A Fast Algorithm for the Calculation of the Roots of

Special Functions,” SIAM Journal on Scientific Computing, vol. 29, no. 4, pp. 1420–1438,

2007.

[12]B. Michiels, I. Bogaert, J. Fostier, and D. De Zutter, “A Weak Scalability Study of the Parallel

Computation of the Translation Operator in the MLFMA,” in Proceedings of the Inter-

national Conference on Electromagnetics in Advanced Applications, Turin, Italy, 9–13

September 2013.

[13]——, “A Well-Scaling Parallel Algorithm for the Computation of the Translation Operator in

the MLFMA,” Accepted for publication in IEEE Transactions on Antennas and Propaga-

tion.

[14] I.Bogaert,“FastGauss-LegendreQuadratureRules,”

http://sourceforge.net/projects/fastgausslegendrequadrature, [Online].

[15]G.Szeg¨o,¨

Uber Einige AsymptotischeEntwicklungen derLegendreschen Funktionen,” Proc.

London Math. Soc., vol. s2-36, no. 1, pp. 427–450, 1934.

[16]F. W. J. Olver, Asymptotics and Special Functions.New York: Academic, 1974.

[17]G. Szeg¨o, Orthogonal Polynomials.Providence, Rhode Island: American Mathematical Soci-

ety, 1978.

[18]R. Piessens, “Chebyshev Series Approximations for the Zeros of the Bessel Functions,” Journal

of Computational Physics, vol. 53, no. 1, pp. 188–192, 1984.

[19]R. C. Li, “Near Optimality of Chebyshev Interpolation for Elementary Function Computa-

tions,” Computers, IEEE Transactions on, vol. 53, no. 6, pp. 678–687, june 2004.

[20]L. N. Trefethen, “Computing Numerically with Functions Instead of Numbers,” Mathematics

in Computer Science, vol. 1, pp. 9–19, 2007.

[21]N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed.SIAM, 2002.

[22]J.-P. Berrut and L. N. Trefethen, “Barycentric Lagrange Interpolation,” SIAM Review, vol. 46,

no. 3, pp. 501–517, 2004.

[23]N. J. Higham, “The Numerical Stability of Barycentric Lagrange Interpolation,” IMA Journal

of Numerical Analysis, vol. 24, pp. 547–556, 2004.

[1]  2015/02/15 22:47   Male / 50 years old level / A teacher / A researcher / Very /

Comment/Request
Is teh code public ? I would like to see it as I am interested to see if it is ok for n{amp}gt;100 and possibliy adapt it.
Thnaks

[2]  2015/02/11 09:38   Male / 60 years old level or over / A teacher / A researcher / Very /

Purpose of use
numerically solving an initial value problem of a hyperbolic 4×4-system of PDE-s

[3]  2013/10/18 20:26   Male / 60 years old level or over / A teacher / A researcher / Very /

Purpose of use
Checking precision of gauss-laguerre nodes and weights at 38 digits.
They are good at quad-precision( 35 digits)
Comment/Request
the display on screen truncates the third column of this calculation due
to lack of page space. In such cases can you extend the page or shift some
columns to another page?
However I tried «copy and paste» of the table to another editor and there I
was able to see all the columns in full.
Thank you

[4]  2012/02/28 01:52   Male / 30 years old level / A student / Very /

Comment/Request
I can integrate the Gauss-Laguerre Integral with an alpha {amp}lt; -1 (for example alpha = -1.8) but I cannot obtain weights!

[5]  2011/08/24 13:11   Male / 30 years old level / A teacher / A researcher / Very /

Purpose of use
Lecture for students.
Comment/Request
That is a great page.
alpha = 

(-σ3 σ15 σ2-σ6-σ12-σ11-σ10 6root(σ16,z,1)-root(σ16,z,2) σ1 σ4-σ2-root(σ16,z,1)2σ1 σ4 σ2-σ7-σ13-σ11-σ10 6root(σ16,z,2)-root(σ16,z,1) root(σ16,z,2)-root(σ16,z,3) root(σ16,z,2)-root(σ16,z,4)σ5 σ4 σ15-σ8-σ13-σ12-σ10 6root(σ16,z,3)-root(σ16,z,4) σ5-σ1-σ3 root(σ16,z,3)2-σ5 σ1 σ3-σ9-σ13-σ12-σ11 6σ14 root(σ16,z,1) σ14 root(σ16,z,2) σ14 root(σ16,z,3)-root(σ16,z,4)3 σ9-σ8-σ7-σ6)where  σ1=root(σ16,z,1) root(σ16,z,3)  σ2=root(σ16,z,3) root(σ16,z,4)  σ3=root(σ16,z,2) root(σ16,z,3)  σ4=root(σ16,z,1) root(σ16,z,4)  σ5=root(σ16,z,1) root(σ16,z,2)  σ6=root(σ16,z,2) root(σ16,z,3) root(σ16,z,4)  σ7=root(σ16,z,1) root(σ16,z,3) root(σ16,z,4)  σ8=root(σ16,z,1) root(σ16,z,2) root(σ16,z,4)  σ9=root(σ16,z,1) root(σ16,z,2) root(σ16,z,3)  σ10=2 root(σ16,z,4)  σ11=2 root(σ16,z,3)  σ12=2 root(σ16,z,2)  σ13=2 root(σ16,z,1)  σ14=root(σ16,z,4)2  σ15=root(σ16,z,2) root(σ16,z,4)  σ16=z4-16 z3 72 z2-96 z 24

Exponentially-fitted Gauss–Laguerre quadrature rule for integrals over an unbounded interval

New quadrature formulae are introduced for the computation of integrals over the whole positive semiaxis when the integrand has an oscillatory behaviour with decaying envelope. The new formulae are derived by exponential fitting, and they represent a generalization of the usual Gauss–Laguerre formulae. Their weights and nodes depend on the frequency of oscillation in the integrand, and thus the accuracy is massively increased. Rules with one up to six nodes are treated with details. Numerical illustrations are also presented.

Понравилась статья? Поделиться с друзьями:
Website Name