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Fix typo in complex autograd docs #49755

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Fix typo in complex autograd docs #49755

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Update docs
soulitzer Dec 22, 2020
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Merge branch 'master' of https://github.com/pytorch/pytorch into comp…
soulitzer Dec 22, 2020
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soulitzer Dec 22, 2020
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12 changes: 6 additions & 6 deletions docs/source/notes/autograd.rst
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Expand Up @@ -302,8 +302,8 @@ From the above equations, we get:

.. math::
\begin{aligned}
\frac{\partial }{\partial z} &= 1/2 * (\frac{\partial }{\partial x} - 1j * \frac{\partial z}{\partial y}) \\
\frac{\partial }{\partial z^*} &= 1/2 * (\frac{\partial }{\partial x} + 1j * \frac{\partial z}{\partial y})
\frac{\partial }{\partial z} &= 1/2 * (\frac{\partial }{\partial x} - 1j * \frac{\partial }{\partial y}) \\
\frac{\partial }{\partial z^*} &= 1/2 * (\frac{\partial }{\partial x} + 1j * \frac{\partial }{\partial y})
\end{aligned}

which is the classic definition of Wirtinger calculus that you would find on `Wikipedia <https://en.wikipedia.org/wiki/Wirtinger_derivatives>`_.
Expand Down Expand Up @@ -397,17 +397,17 @@ Solving the above equations for :math:`\frac{\partial L}{\partial u}` and :math:

.. math::
\begin{aligned}
\frac{\partial L}{\partial u} = 1/2 * (\frac{\partial L}{\partial s} + \frac{\partial L}{\partial s^*}) \\
\frac{\partial L}{\partial v} = -1/2j * (\frac{\partial L}{\partial s} - \frac{\partial L}{\partial s^*})
\frac{\partial L}{\partial u} = \frac{\partial L}{\partial s} + \frac{\partial L}{\partial s^*} \\
\frac{\partial L}{\partial v} = -j * (\frac{\partial L}{\partial s} - \frac{\partial L}{\partial s^*})
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\end{aligned}
:label: [3]

Substituting :eq:`[3]` in :eq:`[1]`, we get:

.. math::
\begin{aligned}
\frac{\partial L}{\partial z^*} &= 1/2 * (\frac{\partial L}{\partial s} + \frac{\partial L}{\partial s^*}) * \frac{\partial u}{\partial z^*} - 1/2j * (\frac{\partial L}{\partial s} - \frac{\partial L}{\partial s^*}) * \frac{\partial v}{\partial z^*} \\
&= \frac{\partial L}{\partial s} * 1/2 * (\frac{\partial u}{\partial z^*} + \frac{\partial v}{\partial z^*} j) + \frac{\partial L}{\partial s^*} * 1/2 * (\frac{\partial u}{\partial z^*} - \frac{\partial v}{\partial z^*} j) \\
\frac{\partial L}{\partial z^*} &= (\frac{\partial L}{\partial s} + \frac{\partial L}{\partial s^*}) * \frac{\partial u}{\partial z^*} - j * (\frac{\partial L}{\partial s} - \frac{\partial L}{\partial s^*}) * \frac{\partial v}{\partial z^*} \\
&= \frac{\partial L}{\partial s} * (\frac{\partial u}{\partial z^*} + \frac{\partial v}{\partial z^*} j) + \frac{\partial L}{\partial s^*} * (\frac{\partial u}{\partial z^*} - \frac{\partial v}{\partial z^*} j) \\
&= \frac{\partial L}{\partial s^*} * \frac{\partial (u + vj)}{\partial z^*} + \frac{\partial L}{\partial s} * \frac{\partial (u + vj)^*}{\partial z^*} \\
&= \frac{\partial L}{\partial s} * \frac{\partial s}{\partial z^*} + \frac{\partial L}{\partial s^*} * \frac{\partial s^*}{\partial z^*} \\
\end{aligned}
Expand Down