What is an efficient algorithm for finding the minimum of a parabola-shaped function? I have a continuous function f(x) that is bounded on the interval (0, N), where N is a large positive integer (~10,000,000). The function is shaped like an upwards-facing parabola, however, it is slightly skewed (so not exactly parabolic). I am able to calculate values of f(x), however it is quite computationally expensive to sample.
What are some possible algorithms for efficiently and accurately approximating the minimum value of this parabola-shaped function using the smallest number of iterations?

The parabola going through $(a,f(a))$, $(b,f(b))$, $(c,f(c))$ has a minimum at
$$G(a,b,c):=\frac{1}{2}\left( \frac{f(a)(b^2-c^2)+f(b)(c^2-a^2)+f(c)(a^2-b^2)}{f(a)(b-c)+f(b)(c-a)+f(c)(a-b)}\right)$$
So if you start with three reasonable guesses $x_0$, $x_1$ and $x_2$, you can iterate with $x_{n+1}=G(x_{n-2}, x_{n-1}, x_n)$ until you get whatever convergence you desire, and then take $f$ of the limit at the end.
