working example

implicit.py

from openmdao.api import *
import numpy as np 

"""
Test model:
-------------
Find value of state variable "x" such that the line 2*x + 1 has a specific 
value of 'y1'
"""

class MyState(Component):

    def __init__(self): 
        super(MyState, self).__init__()

        self.add_param("y1", np.pi)
        self.add_param("y2", 0.0)
        self.add_state("x", 1.0)

    def solve_nonlinear(self,p,u,r):
        pass

    def apply_nonlinear(self, p, u, r):
        r['x'] = p['y1'] - p['y2']


    def linearize(self, params, unknowns, resids):
        J = {}
        J['x','y1'] = 1.0
        J['x','y2'] = -1.0
        J['x', 'x'] = 0.0
        return J


class Intercept(Component):

    def __init__(self): 
        super(Intercept, self).__init__()
        self.add_param("x", 0.0)
        self.add_output("y2", 1.0)

    def solve_nonlinear(self, p, u, r):
        u['y2'] = 2.0*p['x'] + 1.0

    def linearize(self, p, u, r):
        return {('y2', 'x') : 2.0}


p = Problem()
p.root = Group()

p.root.add("state", MyState(), promotes=["*"])
p.root.add("intercept", Intercept(), promotes=["*"])

params = (('y1', np.pi),)
p.root.add("param", IndepVarComp(params), promotes=["*"])

p.root.ln_solver = ScipyGMRES()
p.root.ln_solver.options['atol'] = 1e-8
p.root.ln_solver.options['maxiter'] = 100
p.root.ln_solver.options['restart'] = 100

p.root.nl_solver = Newton()
p.root.nl_solver.options['rtol'] = 1.4e-8
p.root.nl_solver.options['maxiter'] = 75
p.root.nl_solver.options['iprint'] = 0

p.setup()
p.run()

print "x, y1, y2:", p['x'], p['y1'], p['y2']
p.check_partial_derivatives()
print
p.check_total_derivatives()