牛顿-柯特斯公式 Newton-Cotes Formula

Newton-Cotes formula is a general class of formulas that estimate the integral of a function with the integral of a Lagrange interpolation polynomial of the function.

Say we want to estimate \(\int_{a}^{b}f(x)\,\mathrm{d}x\).
Let \(h=\frac{b-a}{n}, x_i=a+ih, f_i=f(x_i),\quad i=0,1,2,\cdots,n\).
The result polynomial of Lagrange interpolation will be

\[g(x)=\sum_{i=0}^{n}{f_i\cdot l_i(x)}, l_i(x)=\prod_{j=0,j\neq i}^{n}{\frac{x-x_j}{x_i-x_j}}. \]

We estimate the integral of \(f\) with that of \(g\):

\[\int_{a}^{b}{f(x)\,\mathrm{d}x}\approx \int_{a}^{b}{g(x)\,\mathrm{d}x}= \sum_{i=0}^{n}({f_i\cdot \int_{a}^{b}{l_i(x)\,\mathrm{d}x}}).\]

Now the problem boils down to the integral of \(l_i\).
By transformation we can show that \(\int_{a}^{b}{l_i(x)\,\mathrm{d}x}=(b-a)\int_{0}^{1}{l_i(x)\,\mathrm{d}x}\).
(Note: the \(l_i\)s are based on different interpolation nodes.)
Let \(\alpha_{i}=\int_{0}^{1}{l_i(x)\,\mathrm{d}x}\) and \(t_i=\frac{i}{n},\quad i=0,1,2,\cdots,n\).
Then

\[\int_{0}^{1}{x^k\,\mathrm{d}x} \approx \sum_{i=0}^{n}{\alpha_{i}\cdot {t_i}^k}=\frac{1}{k+1},\quad k=0,1,2,\cdots,n. \]

Namely

\[\begin{bmatrix} 1 & 1 & 1 & \cdots & 1 \\ t_0 & t_1 & t_2 & \cdots & t_n \\ t_0^2 & t_1^2 & t_2^2 & \cdots & t_n^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ t_0^n & t_1^n & t_2^n & \cdots & t_n^n \end{bmatrix} \begin{bmatrix} \alpha_0 \\[2pt] \alpha_1 \\[2pt] \alpha_2 \\[2pt] \vdots \\[2pt] \alpha_n \end{bmatrix} = \begin{bmatrix} \frac{1}{1} \\[4pt] \frac{1}{2} \\[4pt] \frac{1}{3} \\[4pt] \vdots \\[4pt] \frac{1}{n+1} \end{bmatrix}. \]

Solving this system gives \(\alpha_{i}\).
Newton-Cotes formula is then

\[\int_{a}^{b}{f(x)\,\mathrm{d}x}\approx (b-a)\sum_{i=0}^{n}{\alpha_{i}f_{i}}. \]

Appendix I. For Newton-Cotes formulas, the degree of exactness is:

\[\text{exactness} = \begin{cases} n+1, & \text{if $n$ is even}, \\ n, & \text{if $n$ is odd}. \end{cases} \]

Appendix II. Newton-Cotes formulas for \(n=1,2,3\).

\[\int_a^b f(x)\,\mathrm{d}x \approx \frac{b-a}{2} \,[f_0 + f_1] \]

\[\int_a^b f(x)\,\mathrm{d}x \approx \frac{b-a}{6} \,[f_0 + 4f_1 + f_2] \]

\[\int_a^b f(x)\,\mathrm{d}x \approx \frac{b-a}{8} \,[f_0 + 3f_1 + 3f_2 + f_3] \]