ÓÖÁËÌÁ

Ôï óöÜëìá óôïí ðñïóåããéóôéêü õðïëïãéóìü ôïõ ïñéóìÝíïõ ïëïêëçñþìáôïò ëüãù ôçò áíôéêáôÜóôáóçò ôçò óõíÜñôçóçò $f(x)$ ìå ôï óõìðôùôéêü ðïëõþíõìï $P(x)$ åßíáé:

$$f(x)-P(x)= \frac{1}{2} s(s-1) h^2 f''(\xi) \quad \mbox{ãéá} \quad x_0\leq \xi \leq x_1$$ (155)

üðïôå ôï óöÜëìá óôçí ðñïçãïýìåíç ðñïóåããéóôéêÞ ïëïêëÞñùóç õðïëïãßæåôáé áðü ôçí ïëïêëÞñùóç ôïõ ðáñáðÜíù óöÜëìáôïò, äçëáäÞ

$$E = \int_{x_0}^{x_1} \frac{1}{2}s(s-1) h^2 f''(\xi) dx = \frac{h^3}{2} \int_{s=0}^{s=1} f''(\xi) ds$$

$$= h^3 f''(\xi_1)\left(\frac{s^3}{6}-\frac{s^2}{4}\right)_0^1 =-\frac{1}{12}h^3 f''(\xi_1)$$ (156)

üðïõ $\xi_1 \in [x_0,x_1]$.

Óõìðôùôéêü ðïëõþíõìï 2ïõ âáèìïý

Áêïëïõèþíôáò ôçí ðñïçãïýìåíç äéáäéêáóßá âñßóêïõìå üôé:

$$\int_{x_0}^{x_2}f(x)dx \rightarrow \int_{x_0}^{x_2}P_2(x_s)dx = \int_{x_0}^{x_2} \left(f_0+s\Delta f_0 +\frac{1}{2}s(s-1)\Delta^2 f_0 \right)dx$$

$$= h\int_{s=0}^{s=2} \left(f_0+s\Delta f_0 +\frac{1}{2}s(s-1)\Delta^2 f_0 \right)ds$$

$$= h\left(2f_0+2\Delta f_0+\frac{1}{3}\Delta^2 f_0 \right)$$

$$= \frac{h}{3}\left(f_0+4f_1+f_2 \right)$$ (157)