Application to the case $N=2$

Let us apply the above formal results to the case $N=2$. This means that we can approximate a function $f(x)$ with a polynomial $P_3(x)$ of order $2N-1=3$.

The mesh points are the zeros of $L_2(x)=1/2(3x^2-1)$. These points are $x_0=-1/\sqrt{3}$ and $x_1=1/\sqrt{3}$.

Specializing Eq. (16) $$\begin{equation*} Q_{N-1}(x_k)=\sum_{i=0}^{N-1}\alpha_iL_i(x_k) \hspace{1cm} k=0,1,\dots, N-1. \end{equation*}$$ to $N=2$ yields $$\begin{equation*} Q_1(x_0)=\alpha_0-\alpha_1\frac{1}{\sqrt{3}}, \end{equation*}$$ and $$\begin{equation*} Q_1(x_1)=\alpha_0+\alpha_1\frac{1}{\sqrt{3}}, \end{equation*}$$ since $L_0(x=\pm 1/\sqrt{3})=1$ and $L_1(x=\pm 1/\sqrt{3})=\pm 1/\sqrt{3}$.

• «
• 1
• ...
• 48
• 49
• 50
• 51
• 52
• 53
• 54
• 55
• 56
• 57
• 58
• 59
• 60
• 61
• 62
• 63
• 64
• 65
• ...
• 95
• »