I am just wondering if anyone has encountered a similar problem before. Let's consider a differentiable function in 1 dimension for now. Suppose that this function is expensive to compute. In addition, assume that we have the following available information:
1) $f(a)$ and $f'(a)$ at $x=a$
2) $f(b)$ at $x=b$.
Now to approximate $f$, I can perform a linearization about $x=a$ because I have knowledge regarding its derivative. How can I improve on this approximation given information $f(b)$? Assume that the derivative at $x=b$ is very expensive to obtain.
Anyone has ideas/suggestions or has seen this problem before?