Gauss and Radau IIA Quadrature Formulas

Gauss Quadrature

Orthogonal Polynomials

Let $ P_n(x) $ be the Legendre polynomials orthogonal on $[-1,1]$ with weight function $ w(x) \equiv 1 $:

\[\int_{-1}^1 P_m(x)P_n(x)dx = \frac{2}{2n+1}\delta_{mn}\]

The Rodrigues’ formula gives:

\[P_n(x) = \frac{1}{2^n n!}\frac{d^n}{dx^n}\left[(x^2-1)^n\right]\]

Quadrature Nodes

The Gauss nodes $x_k$ are the roots of $ P_n(x) $:

\[P_n(x_k) = 0,\quad k=1,\dotsc,n\]

These roots satisfy:

• Symmetry: $ x_k = -x_{n+1-k} $
• All roots are simple and lie in $(-1,1)$

Quadrature Weights

The weights are given by:

\[w_k = \int_{-1}^1 \prod_{\substack{j=1\\j\neq k}}^n \frac{x-x_j}{x_k-x_j} dx = \frac{2}{(1-x_k^2)[P_n'(x_k)]^2}\]

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Radau IIA Quadrature

和Legendre多项式的关系

\[R_n(x) = P_n(x) - P_{n-1}(x)\]

Radau right polynomial

\[\frac{d^n}{dx^n}\left[(x+1)^n (x-1)^{n+1}\right]\]

Quadrature Nodes

The Radau IIA nodes $x_k$ consist of:

• The root $ x=1 $
• The roots of $ R_{n-1}(x) $ in $(-1,1)$

Quadrature Weights

Weights are determined by:

\[w_k = \int_{-1}^1 \ell_k(x)dx,\quad \ell_k(x)\]

Lobatto积分是先分部积分，然后使用Gauss-Legendre求解公式吗？

Lobatto quadrature of function $f(x)$ on interval $[-1,1]$ :

\[\int_{-1}^1 f(x) d x=\frac{2}{n(n-1)}[f(1)+f(-1)]+\sum_{i=2}^{n-1} w_i f\left(x_i\right)+R_n\]

Abscissas: $x_i$ is the $(i-1)$ st zero of $P_{n-1}^{\prime}(x)$, here $P_m(x)$ denotes the standard Legendre polynomial of $m$-th degree and the dash denotes the derivative.

Weights:

\[w_i=\frac{2}{n(n-1)\left[P_{n-1}\left(x_i\right)\right]^2}, \quad x_i \neq \pm 1\]

Properties Comparison

Property	Gauss	Radau IIA
Nodes	All interior	Contains $ x=1 $
Degree of exactness	$ 2n-1 $	$ 2n-2 $
Node symmetry	Yes	No