Solving a multiple variable equation with python

Question

I'm trying to create a program to solve this equation:

2.5 x + 3.5 y + 4.5 z + 5.5 t + 6.5 w + 10.5 f = d

I want to be able to set a value for d and get posite and whole numbers as a result for each variable.

import sympy as sp

import numpy as np


x, y, z, t, w, f = sp.var('x y z t w f', Naturals0=True, positive=True)

var = [x, y, z, t, w, f]

d = 14


Eqn = sp.Eq(2.5*x + 3.5*y + 4.5*z + 5.5*t + 6.5*w + 10.5*f, d)

for i in var:
    
    print(sp.solveset(Eqn, i, domain=sp.S.Naturals0))

I'm using the code above but I'm having 2 problems, first it give me back only the relative answers for each variable and I've not found a way to "control" the answer for being only positive and whole.

I know that maybe I get a lot of results depending on the number I set for d, but I need results in therms of numbers, nos equations.

Last but not least, I've already tried doing with numpy and matrix solving, but not suceeded.

Thanks in advance

Answer 1

When you want integer solutions for multiple variables in an equation you might be wanting to use diophantine ; solveset is more for solving for a single variable in terms of the others.

For diophantine give it the equation as an expression with integer coefficients:

>>> from sympy import Add, nsimplify, Eq, symbols, ordered
>>> s = d, f, t, w, x, y, z=symbols('d, f, t, w, x, y, z')
>>> e = Eq(2.5*x + 3.5*y + 4.5*z + 5.5*t + 6.5*w + 10.5*f, d)
>>> nsimplify(e.rewrite(Add), rational=True)
-d + 21*f/2 + 11*t/2 + 13*w/2 + 5*x/2 + 7*y/2 + 9*z/2
>>> eq = _

Now get an integer solution:

 >>> from sympy import diophantine
 >>> isol = diophantine(eq, syms=s); isol
 {(t_0, t_1, t_1 + t_2, t_1 + t_2 + t_3, t_1 + t_2 + t_3 + t_4,
 8*t_0 - 191*t_1 - 107*t_2 - 63*t_3 - 11*t_4 + 9*t_5,
 -6*t_0 + 143*t_1 + 80*t_2 + 47*t_3 + 8*t_4 - 7*t_5)}

This is a set of tuples (in this case 1) of either integers or variables representing integers that are a solution to the equation. In this case, 6 parameters are free to be chosen and then a particular solution can be found. We'll create a function f that can be used to see particular solutions easily:

>>> from sympy import Tuple, Dict, Lambda
>>> tsol = Tuple(*isol.pop())
>>> dsol = Dict(*zip(v, tsol))
>>> p = tuple(ordered(tsol.free_symbols))
>>> f = Lambda(p, dsol)

Now we can see a particular solution and see that it satisfies the original expression:

>>> f(1,2,3,4,5,6)
{d: 1, f: 2, t: 5, w: 9, x: 14, y: -948, z: 706}
>>> eq.subs(_)
0

So there are an infinite number of solutions governed by 6 arbitrary integers from which the other two are determined.