Closed Newton-Cotes
- integrate approximated function which is made by interpolating representative points of the integrand
- useful if the value of the integrand at equally spaced points is given
Theory
For simplicity, we only consider one-dimentional integrand.
- $a, b \in \mathbb{R}$,
- $a < b$
We define $N$ paritition of interval $[a, b]$, denoted $\pi(N; a, b)$,
\[\begin{eqnarray} \pi(N; a, b) & := & \{ x_{0}, \ldots, x_{N + 1} \} \nonumber \\ x_{0} & := & a \nonumber \\ x_{k} & := & \frac{ x_{k} - x_{k - 1} }{ N } \quad (k = 1, \ldots, N) \nonumber \\ x_{N+1} & := & b . \nonumber \end{eqnarray}\]The interpolation polynomial in the Lagrange form is given by
\[\begin{eqnarray} l_{j}(x; \pi(N; a, b)) & := & \prod_{m \in \{0, \ldots, N + 1\}, m \neq j} \frac{ x - x_{m} }{ x_{j} - x_{m} } \nonumber \\ L(x; \pi(N, a, b)) & := & \sum_{j=0}^{N+1} f(x_{j}) l_{j}(x) \nonumber \end{eqnarray}\]$L(x)$ interpolates points \((x_{i}, f(x_{i}))\) by Lagrange Polynomial.
\[\begin{eqnarray} \int_{x_{1}}^{x_{2}} L(x; \pi(1; x_{1}, x_{2})) \ dx & = & \int_{x_{1}}^{x_{2}} L(x) \ dx \end{eqnarray}\]