Estoy tratando de entender la Interpolación de Hermite. Aquí está mi pedagógica ejemplo.
Quiero aproximado de $f(x)=e^x$ sobre el dominio $[-1,1]$ el uso de la interpolación de Hermite.
Yo elijo el de Chebyshev ceros de $T_{n+1}(x)$ como la interpolación de puntos, es decir, los puntos de $x_0,x_1,\dots,x_n$, donde, $$ x_i = \text{cos}\left(\left(k+\frac{1}{2}\right)\frac{\pi }{n+1}\right). $$
At each point $x_i$ I construct the list $f(x_i),f'(x_i),...,f^{(m)}(x_i)$ for $i=0,\dots,n$. That is I want to use not only the values of $f$ at the points $x_i$ but also its first $m$ derivatives at these points. The error in the approximation by the Hermite Interpolation Polynomial $P_n(x)$ is given by,
$$ E_{n}(x)=f(x)-P_{n}(x)=\frac{f^{(n+1)}(\xi_{x})}{(n+1)!}\prod_{k=0}^{n}(x-x_{i})^{m} $$
where where $\xi_x \en [-1,1]$. For $f(x)=e^x$ this can be easily bounded,
$$ \left|E_{n}(x)\right|\leq\frac{e}{(n+1)!}\prod_{k=0}^{n}(x-x_{i})^{m} $$
Of course if $m=0$ then the Hermite interpolating polynomial is just the Lagrange Interpolating polynomial. The problem occurs when I increase $m$. The number of derivatives to include in the interpolation procedure at each point $x_i$.
For example suppose $n=15$ and $m=10$, then according to the bound the error at $x=1$ should be less than 10^-35 (or something ridiculously small). However the resulting Hermite polynomial oscillates wildly and grows without bound in the interval $[-1,1]$. This reminds me much of Runges phenomenon, but this contradicts the error bound for the Hermite interpolating polynomial $P_n(x)$ and I am using the chebyshev zeros as the interpolating points. I observed the same behaviour when trying to approximate $tan(x)$ on a closed sub-interval of $(-\pi/2,\pi/2)$ and the inverse error function using Hermite interpolation.
Has anyone else observed this? Is this expected behaviour? Or have I overlooked something. As it stands it would "seem" Hermite interpolation is not very useful, but I would have expected it to be better than Lagrange interpolation. Have I missed the point?? Can anyone shed some light please??
Incidentally I build the Hermite Interpolating polynomials using Mathematica's built in function InterpolatingPolynomial[] as follows.
n = 15; m = 10;
x[k_] := N[Cos[(k + 1/2) \[Pi]/(n + 1)]]
interpolationPoints =
Table[Join[{{N[x[j]]}}, Table[D[Exp[y], {y, k}] /. y -> x[j], {k, 0, m}]], {j, 0, n}] ;
hermiteP[x_] = InterpolatingPolynomial[interpolationPoints, x];
Plot[{Exp[x], hermiteP[x]}, {x, -1, 1}]
Plot[Exp[x] - hermiteP[x], {x, -1, 1}]
If I reduce the number of derivatives $m$ the result gets better. But I thought the result should get better as I increase $m$ no la otra manera alrededor.
