Finding Roots With Scipy.optimize.root

I am trying to find the root y of a function called f using Python. Here is my code: def f(y): w,p1,p2,p3,p4,p5,p6,p7 = y[:8] t1 = w - 0.500371726*(p1**0.92894164) - (-

Solution 1:

OK, after some fooling around, we focus on another aspect of good optimization/root finding algorithms. In the comments above we went back and forth around which method in scipy.optimize.root() to use. An equally important question for near-bulletproof 'automatic' root finding is zeroing in on good initial guesses. Often times, good initial guesses are, in fact, not near the real answer at all. Instead, they need to be arranged so that they will naturally lead the solver in the right direction.

In your particular case, your guesses were, in fact, sending the algorithm off in strange directions.

My toy reconstruction of your problem is:

import numpy as np
import scipy as sp
import scipy.optimize
def f(y):
    w,p1,p2,p3,p4,p5,p6,p7 = y[:8]
    def t(p,w,a):       
        b = -0.998515304
        e1 = 0.92894164
        e2 = 1.1376649return(w-a*p**e1 - b*(1-p)**e2)
    t1 = t(p1,w,1.0)
    t2 = t(p2,w,4.0) 
    t3 = t(p3,w,16.0)
    t4 = t(p4,w,64.0)
    t5 = t(p5,w,256.0)
    t6 = t(p6,w,512.0)
    t7 = t(p7,w,1024.0)       
    t8 = p1 + p2 + p3 + p4 + p5 + p6 + p7 - 1.0return(t1,t2,t3,t4,t5,t6,t7,t8)
guess = 0.0001
x0 = np.array([-1000.0,guess,guess,guess,guess,guess,guess,guess])
sol = sp.optimize.root(f, x0, method='lm')
print('w=-1000: ', sol.x, sol.success,sol.nfev,np.sum(f(sol.x)))

Note that I did not use your specific prefactors (I wanted to broaden the range I explored), although I kept your particular exponents on the p terms.

The real secret is in the initial guess, which I made the same for all p terms. Having it a 0.1 or above bombed much of the time, since some terms want to go one way and some the other. Reducing it to 0.01 worked well for this problem. (I will note that the w term is very robust - varying from -1000. to +1000. had no effect on the solution). Reducing the initial guess even further has no effect on this particular problem, but it does no harm either. I would keep it very small.

Yes, you know that at least some terms will be much larger. But, you are putting the solver in a position where it can clearly and directly proceed towards the real solution.

Good luck.