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Chebyshev Quadrature

Classical Chebyshev quadrature is a quadrature rule for numerical integration, using weighting function on the closed interval and equal weights , where is a positive integer. The abscissas are chosen to integrate polynomials of degree at most exactly. The formula with its error term is

(1)

where is the error and the abscissas are found by taking terms through in the Maclaurin series of

$s_n(y)=exp{1/2n[-2+ln(1-y)(1-1/y)+ln(1+y)(1+1/y)]}.$

(2)

The expression has a removable singularity at , with . If denotes the truncated Maclaurin series, define

(3)

The roots of then give the abscissas. The first few values are

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)

The polynomial sequence includes the formal initial value . The least common multiple of the denominators of the coefficients of is given by OEIS A002680, starting at . Multiplying each by this integer gives the coefficient number triangle OEIS A101270, read in increasing powers, including zero coefficients.

For , the roots are all real only for , 2, ..., 7 and , which are the permissible orders for this rule (Hildebrand 1956; Gautschi 1976, p. 101). The rule is exact for polynomials of degree at most for odd and for even . For these orders, if is real-valued and has continuous derivatives through order for odd or for even on , the error term is

$E_n={c_n(f^((n+1))(xi))/((n+1)!) n odd; c_n(f^((n+2))(xi))/((n+2)!) n even$

(14)

for some (Gautschi 1976, pp. 115-116), where

$c_n={int_(-1)^1xG_n(x)dx n odd; int_(-1)^1x^2G_n(x)dx n even.$

(15)

The first few values of are 2/3, 8/45, 1/15, 32/945, 13/756, and 16/1575 (Hildebrand 1956). Beyer (1987) gives abscissas up to and Hildebrand (1956) up to .


2
3	0

4

5	0


6


7	0



9	0

The abscissas can be computed analytically for small .


2
3	0

4

5	0

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 466, 1987.Gautschi, W. "Advances in Chebyshev Quadrature." In Numerical Analysis: Proceedings of the Dundee Conference on Numerical Analysis, 1975 (Ed. G. A. Watson). Berlin, Germany: Springer-Verlag, pp. 100-121, 1976. https://doi.org/10.1007/BFb0080118.Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 345-351, 1956.Salzer, H. E. "Tables for Facilitating the Use of Chebyshev's Quadrature Formula." J. Math. Phys. 26, 191-194, 1947.Sloane, N. J. A. Sequences A002680/M2261 and A101270 in "The On-Line Encyclopedia of Integer Sequences."

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Chebyshev Quadrature

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Weisstein, Eric W. "Chebyshev Quadrature." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChebyshevQuadrature.html

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Chebyshev Quadrature

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