What's the difference in the use and precision of fsolve, brentq and root for a system of equations?

问题

I have asked this question Is fsolve good to any system of equations?, from which I got a satisfactory answer. The system I presented there

x = A * exp (x+y)

y = 4 * exp (x+y)

, is just a toy model which is similar with my real case problem, fsolve did the work with (code in the answer below):

from scipy.optimize import fsolve
import matplotlib.pyplot as plt
import numpy as np
def f(p,*args):
  x, y = p
  A = args[0] 

  return (x -A* np.exp(x+y),y- 4* np.exp(x+y))
A = np.linspace(0,4,5)
X = []
Y =[]
for a in A:
  x,y =  fsolve(f,(0.0, 0.0) , args=(a))
  X.append(x)
  Y.append(y)
  print(x,y)

plt.plot(A,X)
plt.plot(A,Y)

However, I read here stackoverflow.com/questions/6519380/… that brenqt is much faster than fsolve. I've tried then to use it but keep getting f(a) and f(b) must have different signs. I understand that f must be continuous. f(a) and f(b) must have opposite signs. So, I believe brenqt is not a good choice for this system. Please correct me if I'm wrong here.

In my real case I'm encountering exactly what the answer here how to solve 3 nonlinear equations in python says, i.e."fsolve()) is quite sensitive to initial conditions" I want to avoid to "firstly minimize the sum-of-squares" as I have many more parameters than the OP of that question. How to use optimize.root to produce a similar result as the one I got with fsolve in my original question?

回答1:

I now understand (thanks to the comment above) that the brentq only works for scalar functions. I did found a good solution with optimize.root and it gives a good solution with some of their available methods, for example:

def f(p,*args):
   x,y = p
   A = args[0] 
   return (x -A* np.exp(x+y),y- 4* np.exp(x+y))
A = np.linspace(0,4,5)
X = []
Y =[]
for a in A: 
   sol=optimize.root(f,[1.0,10.0],args=(a),method='lm')
   sol.message
   x,y= sol.x[0],sol.x[1]
   X.append(x)
   Y.append(y)
   print(x,y)
plt.plot(A,X)
plt.plot(A,Y)

I'm still struggling to get an appropriate method to my system, as the solver is extremely sensitive to it. For example, if I use method='broyden' in the same code above I get a completely different solution. I'll post another question to ask for help.

来源：https://stackoverflow.com/questions/55581064/whats-the-difference-in-the-use-and-precision-of-fsolve-brentq-and-root-for-a