.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "tutorial/polar_transform.py"
.. LINE NUMBERS ARE GIVEN BELOW.
.. only:: html
.. note::
:class: sphx-glr-download-link-note
:ref:`Go to the end <sphx_glr_download_tutorial_polar_transform.py>`
to download the full example code.
.. rst-class:: sphx-glr-example-title
.. _sphx_glr_tutorial_polar_transform.py:
=========================
Polar Transformation
=========================
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As another example, if we were interested in transforming Cartesian coordinates to polar form:
.. math::
\begin{align}
p_r &= \sqrt{x^2 + y^2} \\
p_{\theta} &= \tan^{-1}\left(\frac{y}{x}\right)
\end{align}
We can implement this with an ``ExplicitSystem`` by declaring the inputs and outputs
of this system as follows:
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.. code-block:: Python
import condor as co
from condor.backend import operators as ops
class PolarTransform(co.ExplicitSystem):
x = input()
y = input()
output.r = ops.sqrt(x**2 + y**2)
#output.theta = ops.atan2(y, x)
output.theta = ops.atan(y/x)
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In general, once you've defined any system in Condor, you can just evaulate it
numerically by passing in numbers:
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.. code-block:: Python
p = PolarTransform(x=3, y=4)
print(p)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
<PolarTransform: x=3, y=4>
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The output returned by such a call is designed for inspection to the extent that we
recommend working in an interactive session or debugger, especially when getting
accustomed to Condor features.
For example, the outputs of an explicit system are accessible directly:
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.. code-block:: Python
print(p.r)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
[5.]
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They can also be retrieved collectively:
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.. code-block:: Python
print(p.output)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
PolartransformOutput(r=array([5.]), theta=array([0.92729522]))
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You can of course call it again with different arguments
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.. code-block:: Python
print(PolarTransform(x=1, y=0).output.asdict())
.. rst-class:: sphx-glr-script-out
.. code-block:: none
{'r': array([1.]), 'theta': array([0.])}
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While the *binding* of the results in a datastructure is nice, the real benefit of
constructing condor models is in calling iterative solvers. For example, we could
perform symbolic manipulation to define another ``ExplicitSystem`` with :math:`x =
r\cos\theta` and :math:`y = r\sin\theta`. Or we can we use Condor to
numerically solve this algebraic system of equations using an ``AlgebraicSystem`` by
declaring the input radius and angle as ``parameter``\s and the solving variables for
:math:`x` and :math:`y`. Mathematically, we are defining the system of algebraic
equations
.. math::
r &= p_r (x^*, y^*) \\
\theta &= p_{\theta} (x^*, y^*)
and letting an iterative solver find the solution :math:`x^*,y^*` satisfying both
residual equations given parameters :math:`r` and :math:`\theta`. In Condor,
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.. code-block:: Python
class CartesianTransform(co.AlgebraicSystem):
# r and theta are input parameters
r = parameter()
theta = parameter()
# solver will vary x and y to satisfy the residuals
x = variable(initializer=1)
y = variable(initializer=0)
# get r, theta from solver's x, y
p = PolarTransform(x=x, y=y)
# residuals to converge to 0
residual(r == p.r)
residual(theta == p.theta)
out = CartesianTransform(r=1, theta=ops.pi / 4)
print(out.x, out.y)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
[0.70710678] [0.70710678]
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Note also that passing the inputs (or any intermediates) to plain numeric functions
that can handle symbolic objects as well as pure numerical objects (float or numpy
arrays) could work for this simple example. However, since we *embedded* the
``PolarTransform`` model in this solver, the system evaluated with the solved variable
values is directly accessible if the ``bind_embedded_models`` option is ``True``
(which it is by default), as in:
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.. code-block:: Python
print(out.p.output)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
PolartransformOutput(r=array([1.]), theta=array([0.78539816]))
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Note that this has multiple solutions due to the form of the algebraic relationship of
the polar/rectangular transformation. The :class:`AlgebraicSystem` uses Newton's
method as the solver, so the solution that is found depends on the initial conditions.
The :attr:`initializer` attribute on the :attr:`variable` field determines the initial
position. For example,
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.. code-block:: Python
CartesianTransform.set_initial(x=-1, y=-1)
out = CartesianTransform(r=1, theta=ops.pi / 4)
print(out.variable)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
CartesiantransformVariable(x=array([-0.70710678]), y=array([-0.70710678]))
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An additional :attr:`warm_start` attribute determines whether the initializer is
over-wrriten. Since the default is true, we can inspect the initializer values,
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.. code-block:: Python
print(CartesianTransform.x.initializer, CartesianTransform.y.initializer)
.. rst-class:: sphx-glr-script-out
.. code-block:: none
[-0.70710678] [-0.70710678]
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and re-solve with attr:`warm_start` False
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.. code-block:: Python
CartesianTransform.y.warm_start = False
.. rst-class:: sphx-glr-timing
**Total running time of the script:** (0 minutes 0.025 seconds)
.. _sphx_glr_download_tutorial_polar_transform.py:
.. only:: html
.. container:: sphx-glr-footer sphx-glr-footer-example
.. container:: sphx-glr-download sphx-glr-download-jupyter
:download:`Download Jupyter notebook: polar_transform.ipynb <polar_transform.ipynb>`
.. container:: sphx-glr-download sphx-glr-download-python
:download:`Download Python source code: polar_transform.py <polar_transform.py>`
.. container:: sphx-glr-download sphx-glr-download-zip
:download:`Download zipped: polar_transform.zip <polar_transform.zip>`
.. only:: html
.. rst-class:: sphx-glr-signature
`Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_