Equating coefficients using python

Question

This is actually a quite simple task to do manually, given the two equation, but nevertheless I wanted to know if its possible to equate coefficients in python (maybe using numpy or sympy/scipy?). So the equations I have look something like this:

y = 1.0066*x**2-1.8908*x-0.888
U = 0.5kB*(r-r0)**2

Now ideally the second equation should be first multiplied out to take shape like the one above and then kB should be calculated via the three coefficients a (=1.0066), b(=1.8908) and c (0.888) with r0 being a constant aswell (basically giving me 3 values for kB) So the only variables are x and r ,but I am not really interested in them

Is it possible to perform such a task? Bare in mind that I am just a beginner

Thank you

Answer 1

You could use SymPy to represent the right-hand sides as expressions with respect to symbolic variables x , r , r0 and kB :

x, r, r0, kB = sym.symbols('x,r,r0,kB')
y = 1.0066*x**2-1.8908*x-0.888
U = 0.5*kB*(r-r0)**2

Now we can convert y and U into polynomials with respect to x and r :

In [39]: sym.poly(y, x)
Out[39]: Poly(1.0066*x**2 - 1.8908*x - 0.888, x, domain='RR')

In [40]: sym.poly(U, r)
Out[40]: Poly(0.5*kB*r**2 - 1.0*kB*r0*r + 0.5*kB*r0**2, r, domain='RR[r0,kB]')

sym.Poly s have a all_coeffs method which returns a list of the coefficients:

In [41]: sym.poly(y, x).all_coeffs()
Out[41]: [1.00660000000000, -1.89080000000000, -0.888000000000000]

In [42]: sym.poly(U, r).all_coeffs()
Out[42]: [0.5*kB, -1.0*kB*r0, 0.5*kB*r0**2]

We can use zip to pair the coefficients from the two lists:

In [43]: list(zip(sym.poly(y, x).all_coeffs(), sym.poly(U, r).all_coeffs()))
Out[43]: 
[(1.00660000000000, 0.5*kB),
 (-1.89080000000000, -1.0*kB*r0),
 (-0.888000000000000, 0.5*kB*r0**2)]

and then use sympy.Eq to equate the expressions, and use sympy.solve to solve them for kB . The Python construct I'm using here to generate the list is called a list comprehension :

In [44]: [sym.solve(sym.Eq(a, b), [kB]) for a, b in zip(sym.poly(y, x).all_coeffs(), sym.poly(U, r).all_coeffs())]
Out[44]: [[2.01320000000000], [1.8908/r0], [-1.776/r0**2]]

Putting it all together:

import sympy as sym

x, r, r0, kB = sym.symbols('x,r,r0,kB')
y = 1.0066*x**2-1.8908*x-0.888
U = 0.5*kB*(r-r0)**2
result = [sym.solve(sym.Eq(a, b), [kB]) for a, b in 
          zip(sym.poly(U, r).all_coeffs(), sym.poly(y, x).all_coeffs())]
print(result)

prints

[[2.01320000000000], [1.8908/r0], [-1.776/r0**2]]