Suppose that we have a nonlinear $f(x)$ which may be convex/nonconvex. We want to linearize this function in $N$ points over an arbitrary interval $[a,b]$ in such a way that the resultant $N$ linear approximations gives the least deviation from the original function $f(x)$. Then we can say this set of lines provides the best approximation of $f(x)$ over $[a,b]$. Is this a standard problem which has been already investigated? If no, what is the cheapest way to model this problem?
2026-04-23 08:32:57.1776933177
The difference between a function and its linear approximations
130 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
If you place the points on the curve, and make a polyline by connecting these points, then you are doing approximation by a spline of degree 1. Spline approximations (of all degrees) are well understood. See for example deBoor’s book “A Practical Guide to Splines”. The error depends on the maximum value of the second derivative of the original function.
More generally, approximation by polylines has been studied extensively, and you’ll find lots of references by googling that term. However, I don’t recall ever seeing the tangent line technique that you suggested.