# Simple coupled example


```python
'''
Simple coupled example:
'''

import csdl_alpha as csdl
import numpy as np

import time as time
start = time.time()
recorder = csdl.build_new_recorder(inline = True)
recorder.start()

# x = csdl.ImplicitVariable(shape=(1,), name='x', value=0.6180339)
# y = csdl.ImplicitVariable(shape=(1,), name='y', value=0.7861513777)
x = csdl.ImplicitVariable(shape=(1,), name='x', value=0.1)
y = csdl.ImplicitVariable(shape=(1,), name='y', value=0.1)
param = csdl.Variable(shape=(1,), name='param', value=np.ones((1,))*1.0)
test = param+param # should be ignored
test.name = 'ignore'

# simple 2d root finding problem: https://balitsky.com/teaching/phys420/Nm4_roots.pdf
residual_1 = csdl.square(y)*(param - x) - x*x*x
residual_2 = csdl.square(x) + csdl.square(y) - param*param

residual_1.name = 'residual_1'
residual_2.name = 'residual_2'

# sum of solved states
sum_states = x + y
sum_states.name = 'states_sum'

# apply coupling:
x_update = x-residual_1/(-csdl.square(y)-3.0*x*x)
y_update = y-residual_2/(2.0*y)

# ONE SOLVER COUPLING:
# solver = csdl.GaussSeidel('gs_xy')
# solver.add_state(x, residual_1, state_update=x_update)
# solver.add_state(y, residual_2, state_update=y_update)
# solver.run()

# NESTED (x) SOLVER COUPLING:
# solver = csdl.GaussSeidel('gs_x')
# solver.add_state(x, residual_1, state_update=x_update)
# solver.run()

# solver = csdl.GaussSeidel('gs_y')
# solver.add_state(y, residual_2, state_update=y_update)
# solver.run()

# NESTED (y) SOLVER COUPLING:
solver = csdl.nonlinear_solvers.GaussSeidel('gs_y')
solver.add_state(y, residual_2, state_update=y_update)
solver.run()

solver = csdl.nonlinear_solvers.GaussSeidel('gs_x')
solver.add_state(x, residual_1, state_update=x_update)
solver.run()


print('x Value:', x.value)
print('y Value:', y.value)
print('param Value:', param.value)

# A solution: 
x_sol = (np.sqrt(5)-1)/2
y_sol = np.sqrt((-1+np.sqrt(5))/2)
print('an x solution:', x_sol)
print('a y solution:', y_sol)

recorder.active_graph.visualize('FINAL_top_level')

```