Equation solving in Python

Question

I am trying to solve equations such as the following for x:

Here the alpha's and K are given, and N will be upwards of 1,000. Is there a way to specify the LHS given an np.array for the alpha's using sympy? My hope was to define:

eqn = Eq(LHR - K)
solve(eqn,x)

by telling sympy that LHS= sum( a_i + x).

Any tips on solvers which would do this the fastest would also be appreciated. Thanks!

I was hoping for something like:

from sympy import Symbol, symbols, solve, summation, log
import numpy as np
N=10
K=1
alpha=np.random.randn(N, 1)
x = Symbol('x')
i = Symbol('i')
eqn = summation(log(x+alpha[i]), (i, 1, N))
solve(eqn-K,x)

Answer 1

You can't index a NumPy array with a SymPy symbol. Since your sum is finite, just use the Python sum function:

>>> alpha=np.random.randn(1, N)
>>> sum([log(x + i) for i in alpha[0]])
log(x - 1.85289943713841) + log(x - 1.40121781484552) + log(x - 1.21850393539695) + log(x - 0.605693136420962) + log(x - 0.575839713282035) + log(x - 0.105389419698408) + log(x + 0.415055726774043) + log(x + 0.71601559149345) + log(x + 0.866995633213984) + log(x + 1.12521825562504)

But even so, I don't get why you don't just rewrite this as (x - alpha[0])*(x - alpha[1])*...*(x - alpha[N - 1]) - exp(K) , as suggested by Warren Weckesser. You can then use a numerical solver like SymPy's nsolve or something from another library to solve this numerically

>>> nsolve(Mul(*[(x - i) for i in alpha[0]]) - exp(K), 1)
mpf('1.2696755961730152')

You could also solve the log expression numerically, but unless your logs can have negative arguments, these should be the same.