NumMod 1 pset 8

1. Feladat

$$f(x)=L_n(x)+\frac{1}{(n+1)!}{f}^{n+1}(\xi )w_n(x)$$ $$L_n(x)=\frac{x-x_0}{x_0-h-x_0}\cdot \frac{x-(x_0+h)}{x_0-h-(x_0+h)}\cdot f(x_0-h)+\frac{x-(x_0-h)}{x_0-(x_0-h)}\cdot \frac{x-(x_0+h)}{x_0-(x_0+h)}\cdot f(x_0)+\frac{x-(x_0-h)}{x_0+h-(x_0-h)}\cdot \frac{x-x_0}{x_0+h-x_0}\cdot f(x_0+h)$$ $$L_n(x)=\frac{(x-x_0)(x-x_0-h)}{2h^2}\cdot f(x_0-h)-\frac{(x-x_0+h)(x-x_0-h)}{h^2}\cdot f(x_0)+\frac{(x-x_0+h)(x-x_0)}{2h^2}\cdot f(x_0+h)$$ $$L_n(x)=\frac{1}{2h^2}((x-x_0)(x-x_0-h)f(x_0-h))-2(x-x_0+h)(x-x_0-h)f(x_0)+(x-x_0+h)(x-x_0)f(x_0+h)$$ $$f^{\prime }(x_0)\approx L_n^{\prime }(x_0)$$

2. Feladat

$$f^{\prime \prime }(x_0)\approx L_n^{\prime \prime }(x_0)$$

3. Feladat

$$f(x+h)=f(x)+h{f}^{\prime }(x)+\frac{h^2}{2}{f}^{\prime \prime }(x)+\frac{h^3}{6}{f}^{\prime \prime \prime }(x)+O(h^4)$$ $$f(x-h)=f(x)-h{f}^{\prime }(x)+\frac{h^2}{2}{f}^{\prime \prime }(x)-\frac{h^3}{6}{f}^{\prime \prime \prime }(x)+O(h^4)$$ $$f(x)=f(x)$$

(1) + (2) - 2*(3)

$$f(x+h)+f(x-h)-2f(x)=h^2{f}^{\prime \prime }(x)+O(h^4)$$ $$⟹f^{\prime \prime }(x)=\frac{f(x+h)+f(x-h)-2f(x)}{h^2}+O(h^2)$$

4. Feladat

$$f(x+2h)=f(x)+2h{f}^{\prime }(x)+\frac{4h^2}{2}{f}^{\prime \prime }(x)+\frac{8h^3}{6}{f}^{\prime \prime \prime }(x)+O(h^4)$$ $$f(x-h)=f(x)-h{f}^{\prime }(x)+\frac{h^2}{2}{f}^{\prime \prime }(x)-\frac{h^3}{6}{f}^{\prime \prime \prime }(x)+O(h^4)$$ $$f(x)=f(x)$$ $$f^{\prime \prime }(x)\stackrel{?}{\approx }\alpha f(x+2h)-(\alpha +\beta )f(x)+\beta f(x-h)$$

(1) + 2*(2)

$$f(x+2h)+2f(x-h)=3f(x)+3h^2{f}^{\prime \prime }(x)+f^{\prime \prime \prime }(x)+O(h^4)$$

(1) + 2*(2) - 3(3)

$$f(x+2h)+2f(x-h)-3f(x)=3h^2{f}^{\prime \prime }(x)+f^{\prime \prime \prime }(x)+O(h^4)$$ $$f^{\prime \prime }(x)=\frac{f(x+2h)+2f(x-h)-3f(x)-f^{\prime \prime \prime }(x)}{3h^2}+O(h^2)$$

tehát $\alpha =1$ és $\beta =-3$ ekkor $-(\alpha +\beta )=-(1-3)=-(-2)=2$ és így tényleg másodrendű az approximáció

6. Feladat

$$D_0=\left[\begin{array}{cccccc}-2& 1& 0& 0& \dots & 0\\ 1& -2& 1& 0& \dots & 0\\ 0& 1& -2& 1& \dots & 0\\ 0& 0& 1& -2& \dots & 0\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& 0& \dots & -2\end{array}\right]$$

related: NumMod 1