SymPy

SymPy is a Python library for symbolic mathematics. To give you a feel for how it works, lets extrapolate the extremum location for $f (x)$ given a quadratic model:

(1)

f (x) = A x^{2} + B x + C

and three known values:

(2)

\begin{aligned} f (a) & = A a^{2} + B a + C \\ f (b) & = A b^{2} + B b + C \\ f (c) & = A c^{2} + B c + C \end{aligned}

Rephrase as a matrix equation:

(3)

(\begin{matrix} f (a) \\ f (b) \\ f (c) \end{matrix}) = (\begin{matrix} a^{2} & a & 1 \\ b^{2} & b & 1 \\ c^{2} & c & 1 \end{matrix}) \cdot (\begin{matrix} A \\ B \\ C \end{matrix})

So the solutions for $A$ , $B$ , and $C$ are:

(4)

(\begin{matrix} A \\ B \\ C \end{matrix}) = {(\begin{matrix} a^{2} & a & 1 \\ b^{2} & b & 1 \\ c^{2} & c & 1 \end{matrix})}^{- 1} \cdot (\begin{matrix} f (a) \\ f (b) \\ f (c) \end{matrix}) = (\begin{matrix} long \\ complicated \\ stuff \end{matrix})

Now that we've found the model parameters, we need to find the $x$ coordinate of the extremum.

(5)

\frac{d f}{d x} = 2 A x + B,

which is zero when

(6)

\begin{aligned} 2 A x & = - B \\ x & = \frac{- B}{2 A} \end{aligned}

Here's the solution in SymPy:

>>> from sympy import Symbol, Matrix, factor, expand, pprint, preview
>>> a = Symbol('a')
>>> b = Symbol('b')
>>> c = Symbol('c')
>>> fa = Symbol('fa')
>>> fb = Symbol('fb')
>>> fc = Symbol('fc')
>>> M = Matrix([[a**2, a, 1], [b**2, b, 1], [c**2, c, 1]])
>>> F = Matrix([[fa],[fb],[fc]])
>>> ABC = M.inv() * F
>>> A = ABC[0,0]
>>> B = ABC[1,0]
>>> x = -B/(2*A)
>>> x = factor(expand(x))
>>> pprint(x)
 2       2       2       2       2       2   
a *fb - a *fc - b *fa + b *fc + c *fa - c *fb
---------------------------------------------
 2*(a*fb - a*fc - b*fa + b*fc + c*fa - c*fb) 
>>> preview(x, viewer='pqiv')

Where pqiv is the executable for pqiv, my preferred image viewer. With a bit of additional factoring, that is:

(7)

x = \frac{a^{2} [f (b) - f (c)] + b^{2} [f (c) - f (a)] + c^{2} [f (a) - f (b)]}{2 \cdot {a [f (b) - f (c)] + b [f (c) - f (a)] + c [f (a) - f (b)]}}