Legendre-Gauss Quadrature

Legendre-Gauss quadrature is a numerical integration method also called "the" Gaussian quadrature or Legendre quadrature. A Gaussian quadrature over the interval  with weighting function . The abscissas for quadrature order  are given by the roots of the Legendre polynomials , which occur symmetrically about 0. The weights are

			(1)
			(2)

where  is the coefficient of  in . For Legendre polynomials,

(3)

(Hildebrand 1956, p. 323), so

			(4)
			(5)

Additionally,

			(6)
			(7)

(Hildebrand 1956, p. 324), so

			(8)
			(9)

Using the recurrence relation

			(10)
			(11)

(correcting Hildebrand 1956, p. 324) gives

			(12)
			(13)

(Hildebrand 1956, p. 324).

The weights  satisfy

(14)

which follows from the identity

(15)

The error term is

(16)

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

2		1.000000
3	0	0.888889
		0.555556
4		0.652145
		0.347855
5	0	0.568889
		0.478629
		0.236927

The exact abscissas are given in the table below.

2		1
3	0
4
5	0

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

Similarly, the weights for order  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  for  (OEIS A008619). The triangle of polynomials whose roots determine the weights is

	(17)
	(18)
	(19)
	(20)
	(21)
	(22)
	(23)
	(24)

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."