Lobatto quadrature

Lobatto积分类似于Guass积分，但有如下差异：

1. 积分节点包括积分区间的端点.

2. 积分具有2n–3次代数精度，其中n是节点数.
公式如下：

Lobatto quadrature of function f(x) on interval [–1, +1]:

\int_{-1}^1 {f(x) \, dx} = \frac {2} {n(n-1)}[f(1) + f(-1)] + \sum_{i = 2} ^{n-1} {w_i f(x_i)} + R_n.

Abscissas: x_i is the (i-1)st zero of P_{n}(x). Here Pn(x) are Legendre polynomials.

Weights:

w_i = \frac{2}{n(n-1)[P_{n-1}(x_i)]^2} \quad (x_i \ne \pm 1).

Remainder:  R_n = \frac {- n (n-1)^3 2^{2n-1} [(n-2)!]^4} {(2n-1) [(2n-2)!]^3} f^{(2n-2)}(\xi), \quad (-1 < \xi < 1)

Some of the weights are:


Number of points, n Points, xi Weights, wi
3 0 \frac{4}{3}
\pm 1 \frac{1}{3}
4 \pm \sqrt{\frac {1} {5}} \frac{5}{6}
\pm 1 \frac{1}{6}
5 0 \frac{32}{45}
\pm\sqrt{\frac {3} {7}} \frac{49}{90}
\pm 1 \frac{1}{10}

摘自：http://en.wikipedia.org/w/index.php?title=Gaussian_quadrature



posted on 2012-11-06 22:44  seventhsaint  阅读(389)  评论(0)    收藏  举报