How to solve a cubic function without knowing the coefficients a b and c?-CodePudding

Apology for this silly question. In the cubie function below (as from a school project), I am given the value for x and f(x) while coefficients a, b ,c and constant of d are unknown.

f(x)=ax^3 bx^2 cx d

In such case, is there a way to find out a, b and c by using any python package? I found good amount of python tutorial for solving cubic function but they seem to mainly focus on solving x while a, b and c value are given.

CodePudding user response：

Here is an approach via sympy, Python's symbolic math library.

As an example, we are trying to find the formula for the sum of the first n triangular numbers. The triangular numbers (formula n*(n 1)/2) are 0, 1, 3, 6, 10, 15, 21, ..... The sums of the first n triangular numbers are thus 0, 1, 4, 10, 20, 35, 56, ....

from sympy import Eq, solve
from sympy.abc import a,b,c,d, x

formula = a*x**3   b*x**2   c*x   d  # general cubic formula
xs = [0, 1, 2, 3]  # some x values
fxs = [0, 1, 4, 10]  # the corresponding function values

sol = solve([Eq(formula.subs(x, xi), fx) for xi, fx in zip(xs, fxs)])
print(sol)  # {a: 1/6, b: 1/2, c: 1/3, d: 0}

You can use more x, fx pairs to check that a cubic formula suffices (this won't work with float values, as sympy needs exact symbolic equations).

Also sympy's interpolate can be interesting. This calculates a polynomial through some given points. Such code could look like:

from sympy import interpolate
from sympy.abc import x

xs = [0, 1, 2, 3]
fxs = [0, 1, 4, 10]
fx_dict = dict(zip(xs, fxs))
sol = interpolate(fx_dict, x)
print(sol)  # x**3/6   x**2/2   x/3
print(sol.factor())  # x*(x   1)*(x   2)/6