# How to build an analytic discipline?¶

A simple MDODiscipline can be created using analytic formulas, e.g. $$y_1=2x^2$$ and $$y_2=5+3x^2z^3$$, thanks to the class AnalyticDiscipline which is a quick alternative to model a simple analytic MDO problem!

## Create the dictionary of analytic outputs¶

First of all, we have to define the output expressions in a dictionary where keys are output names and values are formula with string format:

expressions = {'y_1': '2*x**2', 'y_2': '5+3*x**2+z**3'}


## Create and instantiate the discipline¶

Then, we create and instantiate the corresponding AnalyticDiscipline inheriting from MDODiscipline by means of the API function create_discipline() with:

• discipline_name="AnalyticDiscipline",

• name="analytic",

• expressions=expr_dict.

In practice, we write:

from gemseo.api import create_discipline

disc = create_discipline("AnalyticDiscipline", name="analytic", expressions=expressions)


Note

GEMSEO takes care of the grammars and MDODiscipline._run() method generation from the expressions argument. In the background, GEMSEO considers that x is a monodimensional float input parameter and y_1 and y_2 are monodimensional float output parameters.

## Execute the discipline¶

Lastly, this discipline can be executed as any other:

from numpy import array
input_data = {"x": array([2.0]), "z": array([3.0])}

out = disc.execute(input_data)
print("y_1 =", out["y_1"])
print("y_2 =", out["y_2"])


which results in:

y_1 = [ 8.]
y_2 = [ 44.]


The discipline will provide analytic derivatives (Jacobian) automatically using the sympy library, by means of the AnalyticDiscipline._compute_jacobian() method.

This can be checked easily using MDODiscipline.check_jacobian():

disc.check_jacobian(input_data,
derr_approx=disc.FINITE_DIFFERENCES,
step=1e-5, threshold=1e-3)


which results in:

   INFO - 10:34:33 : Jacobian:  dp y_2/dp x succeeded!
INFO - 10:34:33 : Jacobian:  dp y_2/dp z succeeded!
INFO - 10:34:33 : Jacobian:  dp y_1/dp x succeeded!
INFO - 10:34:33 : Jacobian:  dp y_1/dp z succeeded!
INFO - 10:34:33 : Linearization of MDODiscipline: AnalyticDiscipline is correct !
True