Calculate the interpolation polyonmial $p$ of the Hermitian interpolation problem
(i) to the data
$$x_0=0, x_1=1, y_0^{(0)}=1, y_1^{(0)}=2, y_0^{(1)}=1, y_1^{(1)}=2$$
and check the derivative values.
(ii) to the data
$$x_0=0=1, y_0^{(0)}=2, y_0^{(1)}=3, y_0^{(2)}=4, y_0^{(3)}=5$$
and check the derivative values.
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Could you give me a hint with the formula because I am confused about how to fing the desird polynomial? Do we maybe use the Lagrange or Newton polynomial?
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I read the example in the links and I tried to do that we the divided difference method :
$$\begin{matrix}z_0=0 & f[z_0]=1 & & & \\ & & \frac{f'(z_0)}{1}=1 & & & f[z_2,z_1,z_0]=0\\ z_1=0 & f[z_1]=1 & & & \\ & & f[z_2,z_1]=\frac{f(z_2)-f(z_1)}{z_2-z_1}=1 & & & & f[z_3,z_2,z_1,z_0]=1\\ z_2=1 & f[z_2]=2 & & & \\ & & \frac{f'(z_3)}{1}=2 & & & f[z_3,z_2,z_1]=1\\ z_3=1 & f[z_3]=2 & & & \end{matrix}$$
Then do we use the formula $$p(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2}f''(x_0)(x-x_0)^2+\frac{f(x_1)-f(x_0)-f'(x_0)(x-x_0)-\frac{1}{2}f''(x_0)(x_1-x_0)^2}{(x_1-x_0)^3}(x-x_0)^3$$ ?
Best Answer
We are given the data
$$x_0=0, x_1=1, y_0^{(0)}=1, y_1^{(0)}=2, y_0^{(1)}=1, y_1^{(1)}=2$$
We create the divided difference table.
Using your results (table looks skewed, it should be a nice diagonal like the notes)
$$\begin{matrix}0 & 1 & & & \\ & & 1 & & & 0\\ 0 & 1 & & &\\ & & 1 & & & & 1\\ 1 & 2 & & & \\ & & 2 & & & 1\\ 1 & 2 & & & \end{matrix}$$
The Hermite Interpolating Polynomial using the Divided Difference form is given by (see linked notes for an alternate view)
$H_{2n-1}(x) = f(x_0) + (x-x_0)f[z_0,z_1] + (x-x_0)^2f[z_0, z_1,z_2] + \ldots + (x-x_0)^2\ldots (x-x_{n-1})^2 (x-x_n)f[z_0,z_1,\ldots,z_n]$
We can write the Hermite Interpolating Polynomial using the values across the topmost diagonal and this formula.
$$P_3(x) = 1 + (x-0)(1) + (x-0)^2(0) + (x-0)^2(x-1)(1) = x^3-x^2+x+1$$
You should try examples $2a$ and $2b$ to make sure you follow in these notes and the example that calculates $H_5(1.5)$ in the linked notes above.
Lastly, it is important to understand that the derivation here includes the Lagrange interpolating Polynomial and derivative information and it can be calculated that way (a little lengthier, but maybe less error prone). See this worked out example and try it.