Python fsolve() complains about shape. Why?

Question

Having the function f(x,y,z), I need to solve the restriction f(x,y,z) = 0 and then plot it. I tried to find for each pair (y,z) the value x for which f(x,y,z) = 0:

from numpy import *
from scipy.optimize import fsolve

def func(x,y,z):
    return x+y+z

y = linspace(0,1,100)
z = linspace(0,1,100)
x0 = zeros((y.size,z.size)) + 0.5 # the initial guess
yz = (y[:,newaxis],z[newaxis,:]) # the other parameters
x, info, iterations, message = fsolve(func,x0,yz)
contour(y,z,x)

Python (2.7.5) says "TypeError: fsolve: there is a mismatch between the input and output shape of the 'func' argument 'func'."

But if I test it myself, it gives the same shape:

func(x0,y[:,newaxis],z[:,newaxis]).shape == x0.shape

returns True.

Why does fsolve() complain?

Answer 1

fsolve expects the x argument and the return value of func to be a scalar or one-dimensional array. You'll have to modify your code to work with flattened x values. Eg

def func(x, y, z):
    x = x.reshape(y.size, z.size)
    return (x + y + z).ravel()

and something like this for the call to fsolve :

sol, info, ier, mesg = fsolve(func, x0.ravel(), args=yz, full_output=True)
x = sol.reshape(y.size, z.size)

Answer 2

Here's a comparison to the krylov method advertized in the scipy.optimize tutorial :

from numpy import linspace, zeros, newaxis
import time
from scipy.optimize import root

def func(x,y,z):
    x = x.reshape(y.size, z.size)
    f = x + y + z
    f = f.ravel()
    return f

n = 50
y = linspace(0,1,n)
z = linspace(0,1,n)
x0 = zeros((y.size,z.size)) + 0.5 # the initial guess
yz = (y[:,newaxis],z[newaxis,:]) # the other parameters

start = time.time()
sol1 = root(func, x0.ravel(), args=yz, method='hybr', tol=1e-7)  # same as fsolve
x1 = sol1.x.reshape(y.size, z.size)
print("(fsolve) time taken (sec): %g" % (time.time() - start,))
print("(fsolve) successful: %r (%s)" % (sol1.success, sol1.message))
print("(fsolve) max error: %g" % (abs(func(x1, *yz)).max(),))

start = time.time()
sol2 = root(func, x0.ravel(), args=yz, method='krylov', tol=1e-9)
x2 = sol2.x.reshape(y.size, z.size)
print("(krylov) time taken (sec): %g" % (time.time() - start,))
print("(krylov) successful: %r (%s)" % (sol2.success, sol2.message))
print("(krylov) max error: %g" % (abs(func(x2, *yz)).max(),))

Prints

(fsolve) time taken (sec): 26.9296
(fsolve) successful: False (The iteration is not making good progress, as measured by the 
  improvement from the last ten iterations.)
(fsolve) max error: 1.52656e-16
(krylov) time taken (sec): 0.0173709
(krylov) successful: True (A solution was found at the specified tolerance.)
(krylov) max error: 1.11022e-16