Gauss-Chebyshev Quadrature
Description
Gauss-Chebyshev quadrature formulas are used to integrate functions f(x) / sqrt( 1 - x² ) from -1 to 1. Let I[-1,1] ( f(x) / sqrt( 1 - x² ) ) denote the integral of f(x) / sqrt( 1 - x² ) over [-1,1]. With respect to the inner product < f,g > = I[-1,1] (f(x) g(x) w(x)), the Chebyshev polynomials T n(x) = cos( n arccos( x ) ) for n > 0, and T0(x) = 1 form an orthogonal family of polynomials with weight function w(x) = 1 / sqrt( 1 - x² ) on [-1,1].
For integrating f(x) / sqrt( 1 - x ² ) over [-1,1], the n -point Gauss-Chebyshev quadrature formula, GCn ( f(x) ), is given by
GCn ( f(x) ) = A1 f(x1) + ··· + An f(xn),where xi , i = 1,...,n, are the zeros of Tn and Ai = pi / n, i = 1,...,n.
The truncation error is I[-1,1] ( f(x) ) - GCn ( f(x) ) = 2 pi f (2n)( c ) / (22n 2n! ),where c is some unknown number -1 < c < 1.
Function List
Functions: For n = 2,3,4,5,6,7,8,9,10,12,14,16,20,24,28,32,40,48,56,64,82,96 and 100
- double Gauss_Chebyshev_Integration_n pts(
double (*f)(double) )
Integrate (*f)(x) / sqrt( 1 - x² ) from -1 to 1 where (*f)(x) is the user supplied function.
- void Gauss_Chebyshev_Zeros_n pts( double nodes[])
Returns the zeros of the Chebyshev polynomial Tn(x) in the user allocated array nodes of dimension n.
- void Gauss_Chebyshev_Coefs_n pts( double wght[] )
Returns the weight of the Gauss-Chebyshev formula in the user allocated array wght of dimension 1.
C Source