Computation of Gauss-Chebyshev quadrature rule nodes and weights

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Contents - Numerical integration - Gaussian quadratures - Computation of Gauss-Chebyshev quadrature rule nodes and weights

Computation of Gauss-Chebyshev quadrature rule nodes and weights

The Gauss-Chebyshev quadrature formula is used to numerically calculate the integral integral(div(f(x),sqrt(1-power(x,2)))dx,-1,1) using the formula

With the help of a change of variables (which changes both weights w_i and nodes x_i), we can get onto the arbitrary interval [a, b], but we'll consider only the simplest case here. If you perform a change of variables, you should take into account that the formula of error term E_n is changed along with the nodes and weights (you can get a new form by changing the variables in the formula).

The Gauss-Chebyshev quadrature formula for each number of points is completely defined by the set of nodes x_i and weights w_i. For every n the node and weights are known in closed form, therefore it's not necessary to have a special subroutine to calculate them.

$x_j_ = cos(div(π(j+0.5),N)) j∈{0, ..., N-1}$

$w_j_ = div(π,N) j∈{0, ..., N-1}$

Error term

The formula with n points is accurate for the polynomials of order 2n-1 and lower. Error term E_n is:

E_n_ = div(2π,power(2,2n)(2n)!)power(f,(2n))(ξ) ξ∈[-1,1]