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Legendre-Gauss Quadrature

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Legendre-Gauss quadrature is a numerical integration method also called "the" Gaussian quadrature or Legendre quadrature. A Gaussian quadrature over the interval [-1,1] with weighting function W(x)=1. The abscissas for quadrature order n are given by the roots of the Legendre polynomials P_n(x), which occur symmetrically about 0. The weights are

w_i=-(A_(n+1)gamma_n)/(A_nP_n^'(x_i)P_(n+1)(x_i))
(1)
=(A_n)/(A_(n-1))(gamma_(n-1))/(P_(n-1)(x_i)P_n^'(x_i)),
(2)

where A_n is the coefficient of x^n in P_n(x). For Legendre polynomials,

A_n=((2n)!)/(2^n(n!)^2)
(3)

(Hildebrand 1956, p. 323), so

(A_(n+1))/(A_n)=([2(n+1)]!)/(2^(n+1)[(n+1)!]^2)(2^n(n!)^2)/((2n)!)
(4)
=(2n+1)/(n+1).
(5)

Additionally,

gamma_n=int_(-1)^1[P_n(x)]^2dx
(6)
=2/(2n+1)
(7)

(Hildebrand 1956, p. 324), so

w_i=-2/((n+1)P_(n+1)(x_i)P_n^'(x_i))
(8)
=2/(nP_(n-1)(x_i)P_n^'(x_i)).
(9)

Using the recurrence relation

(1-x^2)P_n^'(x)=-nxP_n(x)+nP_(n-1)(x)
(10)
=(n+1)xP_n(x)-(n+1)P_(n+1)(x)
(11)

(correcting Hildebrand 1956, p. 324) gives

w_i=2/((1-x_i^2)[P_n^'(x_i)]^2)
(12)
=(2(1-x_i^2))/((n+1)^2[P_(n+1)(x_i)]^2)
(13)

(Hildebrand 1956, p. 324).

The weights w_i satisfy

sum_(i=1)^nw_i=2,
(14)

which follows from the identity

sum_(nu=1)^n(1-x_nu^2)/((n+1)^2[P_(n+1)(x_nu)]^2)=1.
(15)

The error term is

E=(2^(2n+1)(n!)^4)/((2n+1)[(2n)!]^3)f^((2n))(xi).
(16)

Beyer (1987) gives a table of abscissas and weights up to n=16, and Chandrasekhar (1960) up to n=8 for n even.

nx_iw_i
2+/-0.577351.000000
300.888889
+/-0.7745970.555556
4+/-0.3399810.652145
+/-0.8611360.347855
500.568889
+/-0.5384690.478629
+/-0.906180.236927

The exact abscissas are given in the table below.

nx_iw_i
2+/-1/3sqrt(3)1
308/9
+/-1/5sqrt(15)5/9
4+/-1/(35)sqrt(525-70sqrt(30))1/(36)(18+sqrt(30))
+/-1/(35)sqrt(525+70sqrt(30))1/(36)(18-sqrt(30))
50(128)/(225)
+/-1/(21)sqrt(245-14sqrt(70))1/(900)(322+13sqrt(70))
+/-1/(21)sqrt(245+14sqrt(70))1/(900)(322-13sqrt(70))

The abscissas for order n quadrature are roots of the Legendre polynomial P_n(x), meaning they are algebraic numbers of degrees 1, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, ..., which is equal to 2|_n/2_| for n>1 (OEIS A052928).

Similarly, the weights for order n quadrature can be expressed as the roots of polynomials of degree 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ..., which is equal to |_n/2_| for n>1 (OEIS A008619). The triangle of polynomials whose roots determine the weights is

x-2
(17)
x-1
(18)
9x-5
(19)
216x^2-216x+49
(20)
45000x^2-32200x+5103
(21)
2025000x^3-2025000x^2+629325x-58564
(22)
142943535000x^3-113071253400x^2+27510743799x-1976763932
(23)
1707698764800000x^4-1707698764800000x^3+606530263046400x^2-88878097916608x+4373849390625
(24)

(OEIS A112734).

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References

Abbott, P. "Tricks of the Trade: Legendre-Gauss Quadrature." Mathematica J. 9, 689-691, 2005.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 462-463, 1987.Chandrasekhar, S. Radiative Transfer. New York: Dover, pp. 56-62, 1960.Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 323-325, 1956.Sloane, N. J. A. Sequences A008619, A052928, and A112734 in "The On-Line Encyclopedia of Integer Sequences."

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Legendre-Gauss Quadrature

Cite this as:

Weisstein, Eric W. "Legendre-Gauss Quadrature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Legendre-GaussQuadrature.html

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