Minimax approximation of $\sqrt{x^2+1}$ on $[0,1]$

Question

How do I find the linear minimax approximation of $\sqrt{x^2+1}$ on $[0,1]$? Should I choose points to check signs, which?

Answer 1

You want to choose real numbers $a,b$ to minimize $\max_{0\le x \le 1} \left| ax + b - \sqrt{x^2+1}\right|$. Try for the max to occur at three points: the two endpoints and one inside the interval, with $ax + b = \sqrt{x^2+1} - r$ at $x=0$ and at $x=1$.

Answer 2

By the Equioscillation Theorem and the convexity of the function, the linear minimax approximation will have the same error at the end points of the interval. It follows that the line will be parallel to the straight $PQ$ connecting the points $P,Q$ on the graph of the function at the end points where $x = 0, 1$. To get the minimax line, shift the line down by half the maximum change in $y$ between the straight $PQ$ and the graph of the function. [Down, because the graph of the function is concave up.]