One-Dimensional Grids
There are various choices of one dimensional grids for integrating functions over some finite or infinite intervals.
Grid |
Domain |
\(x_i\) |
\(w_i\) |
|---|---|---|---|
\([0,N]\) |
\(i - 1\) |
1 |
|
\([-1,1]\) |
\(\cos\left( \frac{i}{n+1} \pi \right)\) |
\(\frac{\pi}{n+1} \sin^2 \left( \frac{i}{n+1} \pi \right)\) |
|
\([-1,1]\) |
\(\cos\left( \frac{(i-1)}{n-1}\pi \right)\) |
\(w_{1} = w_{n} = \frac{\pi}{2(n-1)}, \quad w_{i\neq 1,n} = \frac{\pi}{n-1}\) |
|
\([-1,1]\) |
\(-1 + 2 \left(\frac{i-1}{n-1}\right)\) |
\(w_1 = w_n = \frac{1}{n}, \quad w_{i\neq 1,n} = \frac{2}{n}\) |
|
\([-1,1]\) |
\(\frac{i}{n+1}\) |
\(\frac{2}{n+1} \sum_{m=1}^n \frac{\sin(m \pi x_i)(1-\cos(m \pi))}{m \pi}\) |
|
|
\([-1,1]\) |
\(\frac{2 i - 1}{2 N_{pts}}\) |
\(\frac{2}{n^2 \pi} \sin(n\pi x_i) \sin^2(n\pi /2) + \frac{4}{n \pi} \sum_{m=1}^{n-1} \frac{\sin(m \pi x_i)\sin^2(m\pi /2)}{m}\) |
\([-1,1]\) |
\(\tanh\left( \frac{\pi}{2} \sinh(i\delta) \right)\) |
\(\frac{\frac{\pi}{2}\delta \cosh(i\delta)}{\cosh^2(\frac{\pi}{2}\sinh(i\delta))}\) |
|
\([-1,1]\) |
\(-1 + 2 \left(\frac{i-1}{N_{pts}-1}\right)\) |
\(w_i = 2 / (3(N - 1)) \text{ if } i = 0, \quad 8 / (3(N - 1)) \text{ if } i \geq 1 \text{ and is odd}, \quad 4 / (3(N - 1)) \text{ if } i \geq 2 \text{ and is even}.\) |
|
\([-1,1]\) |
\(-1 + \frac{2i + 1}{N_{pts}}\) |
\(\frac{2}{n}\) |
|
\([-1,1]\) |
\(\cos (\pi (i - 1) / (N_{pts} - 1))\) |
\(w_i = \frac{c_k}{n} \bigg(1 - \sum_{j=1}^{\lfloor n/2 \rfloor} \frac{b_j}{4j^2 - 1} \cos(2j\theta_i) \bigg), \quad b_j = 1 \text{ if } j = n/2, \quad 2 \text{ if } j < n/2, \quad c_j = 1 \text{ if } k \in \{0, n\}, \quad 2 \text{ else}\) |
|
\((-1,1)\) |
\(\cos\bigg(\frac{(2i - 1)\pi}{2N_{pts}}\bigg)\) |
\(\frac{2}{n}\bigg(1 - 2 \sum_{j=1}^{\lfloor n/2 \rfloor} \frac{\cos(2j \theta_j)}{4 j^2 - 1} \bigg)\) |
|
\((-1,1)\) |
\(\cos(i \pi / N_{pts})\) |
\(\frac{4 \sin(\theta_i)}{n} \sum_{j=1}^{\lfloor n/2 \rfloor} \frac{\sin(2j - 1)\theta_i}{2j - 1}\) |
Grid also includes popular quadrature grids that can integrate polynomials up to degree \(2n - 1\):
Gauss-Legendre(domain \([-1, 1]\)),Gauss-Chebyshev(domain \([-1, 1]\)),Gauss-Laguerre(domain \([0, \infty)\)).
The Trefethen polynomial transformation of various grids on \([-1, 1]\) is also included: