Gauss-Chebyshev Quadrature

Gauss-Chebyshev quadrature formulas are used to integrate functions like \(\frac{f(x)}{\sqrt{1- x^2}}\) and \(f(x) \cdot \sqrt{1- x^2}\) from \(-1\) to \(1\).

Chebyshev Polynomials of the First Kind

Example

use integrate::gauss_quadrature::gauss_first_kind_chebyshev_rule;

let f = |x: f64| 1.0;

let n:usize = 100;

let integral = gauss_first_kind_chebyshev_rule(f, n);
println!("{}",integral);

Understanding Chebyshev Polynomials of the First Kind

With respect to the inner product

\[ \langle f,g \rangle = \int_{-\infty}^{+\infty} f(x) \cdot g(x) \cdot w(x) dx \]

the Chebyshev polynomials are defined by

\[ T_n(x) = \cos(n \cdot \arccos(x)) \quad \text{for} \quad n>0 \]

and \(T_0(x)=1\) form an orthogonal family of polynomials with weight function \(w(x)=\frac{1}{\sqrt{1 - x^2}}\) on \([-1, 1]\).

The \(n\)-point Gauss-Chebyshev quadrature formula, \(GC_n(f(x))\), for approximating the integral of \(\frac{f(x)}{\sqrt{1 - x^2}}\) over \([-1, 1]\), is given by

\[ GC_n ( f(x) ) = A_1 f(x_1) + \cdots + A_n f(x_n) \]

where \(x_i\), \(i = 1,\dots,n\), are the zeros of \(T_n\) and \(A_i = \frac{\pi}{n}\), \(i = 1,\dots,n\).

Chebyshev Polynomials of the Second Kind

Example

use integrate::gauss_quadrature::gauss_second_kind_chebyshev_rule;

fn f(x: f64) -> f64 {
    1.0
}

let n:usize = 100;

let integral = gauss_second_kind_chebyshev_rule(f, n);
println!("{}",integral);

Understanding Chebyshev Polynomials of the Second Kind

With respect to the inner product

\[ \langle f,g \rangle = \int_{-\infty}^{+\infty} f(x) \cdot g(x) \cdot w(x) dx \]

the Chebyshev polynomials are defined by

\[ U_n(x) \cdot \sin(\arccos(x)) = \sin((n+1) \cdot \arccos(x)) \quad \text{for} \quad n>0 \]

and \(U_0(x)=1\) form an orthogonal family of polynomials with weight function \(w(x)=\sqrt{1 - x^2}\) on \([-1, 1]\).

The \(n\)-point Gauss-Chebyshev quadrature formula, \(GC_n(f(x))\), for approximating the integral of \(f(x) \cdot \sqrt{1 - x^2}\) over \([-1, 1]\), is given by

\[ GC_n ( f(x) ) = A_1 f(x_1) + \cdots + A_n f(x_n) \]

where \(x_i\), \(i = 1,\dots,n\), are the zeros of \(U_n\) and

\[ A_i = \frac{\pi}{n + 1} \cdot \sin^2\left(\frac{i\pi}{n + 1} \right) \quad \text{for} \quad i = 1,\dots,n. \]