import petsc4py, sys
petsc4py.init(sys.argv)
From petsc4py import PETSc

class MyMat:
    def __init__(self):
        self.x_i = None
    def mult(self, A, v, y):
        # J(x)*v = 2*x*v
        y[...] = 2 * self.x_i * v

def fun(snes, x, f):
    # F = x**2
    f[...] = x * x

def jac(snes, x, J, P):
    # J(x)*v = 2*x*v
    msh = J.getPythonContext()
    msh.x_i = x.copy(msh.x_i)
    return False # same nz pattern

N = 1
msh = MyMat()
J = PETSc.Mat().createPython(N, msh)
J.setUp()
x, f = J.getVecs()
b = f.duplicate()

snes = PETSc.SNES().create()

ksp = snes.getKSP()
pc = ksp.getPC()
pc.setType(PETSc.PC.Type.NONE)

snes.setFunction(fun, f)
snes.setJacobian(jac, J, J)
snes.setFromOptions()

x[...] = 1
b[...] = 4

snes.solve(b,x)