See the book OPTIMIZATION: Algorithms and Applications by Rajesh Kumar Arora, Page-$44$.
2.3.4 Cubic Polynomial Fit
In this method, the function f(x) to be minimized is approximated by a cubic polynomial $P(x)$ as
$$P(x) = a_0 + a_1x + a_2x^2 + a_3x^3 \qquad (2.7)$$
If the function $f(x)$ is evaluated at four different points, then the polynomial coefficients $a0$, $a1$, $a2$, and $a4$ can be evaluated by solving four simultaneous linear equations. Alternatively, if the value of the function and its derivatives are available at two points, the polynomial coefficients can still be evaluated. Once a polynomial is approximated for the function, the minimum point can be evaluated using the polynomial coefficients.
The first step in this search method is to bracket the minimum of the func- tion between two points, x1 and x2, such that the following conditions hold:
$$ f′(x1)f′(x2) < 0 \qquad (2.8) $$
Using the information of $f(x_1), f′(x_1), f(x_2)$, and $f′(x_2)$, the minimum point of the approximating cubic polynomial can be given as
I am trying to learn this algorithm. I am searching this algorithm on the Internet but isn't finding anything. I have a feeling that this algorithm probably has other name(s).
Does this method/algorithm have any other name according to Numerical Analysis?
Perhaps cubic interpolation will work?