Hi I have an exercise that I cannot solve.
Can someone help me?
I have to find the solution to the following Hermitian interpolation problem p(x) under the conditions:
$p(x_0)=-1, \ p'(x_0)=1 \ p(x_1)=1$ for $x_0=0$ and $x_1=1$ using the generalized Lagrangian polynomials.
Can someone help me?
I don't know how to start solving the problem.
Let $L_i(x)$ be the Lagrangian polynomial associated with $x_i$ and $q_i(x)=L_i(x)^2$, then try to see what conditions on $P_i(x)$ can make the polynomial $P = \sum_i q_i(x) P_i(x)$ solve your problem. Here you just have a sum of two products since you have two points but this idea will work for an arbitrary number of points to interpolate
Hint : ${\displaystyle q_{i}(x_{i})=1,\quad q_{i}^{\prime }(x_{i})=\sum _{j=0,j\neq i}^{n}{\frac {2}{x_{i}-x_{j}}}\quad {\rm {and}}\quad \forall j\neq i\quad q_{i}^{\prime }(x_{j})=q_{i}(x_{j})=0}$
Hint 2 : search $P_i$ as a polynomial of degree 1