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<h2><span class="section-number">1.1. </span>Introduction<a class="headerlink" href="#introduction" title="Permalink to this heading"></a></h2>
<p>This document contains instructions for installing, using, and contributing to Composable Kernel (CK).</p>
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<h3><span class="section-number">1.1.1. </span>Documentation Roadmap<a class="headerlink" href="#documentation-roadmap" title="Permalink to this heading"></a></h3>
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<h1><span class="section-number">4. </span>Supported Primitives Guide<a class="headerlink" href="#supported-primitives-guide" title="Permalink to this heading"></a></h1>
<p>This document contains details of supported primitives in Composable Kernel (CK). In contrast to the API Reference
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<h2><span class="section-number">4.1. </span>Softmax<a class="headerlink" href="#softmax" title="Permalink to this heading"></a></h2>
<p>For vectors <span class="math notranslate nohighlight">\(x^{(1)}, x^{(2)}, \ldots, x^{(T)}\)</span> of size <span class="math notranslate nohighlight">\(B\)</span> we can decompose the softmax of concatenated
<span class="math notranslate nohighlight">\(x = [ x^{(1)}\ | \ \ldots \ | \ x^{(T)} ]\)</span> as,</p>
<div class="math notranslate nohighlight">
\begin{align}
m(x) &amp; = m( [ x^{(1)}\ | \ \ldots \ | \ x^{(T)} ] ) = \max( m(x^{(1)}),\ldots, m(x^{(T)}) ) \\
f(x) &amp; = [\exp( m(x^{(1)}) - m(x) ) f( x^{(1)} )\ | \ \ldots \ | \ \exp( m(x^{(T)}) - m(x) ) f( x^{(T)} )] \\
z(x) &amp; = \exp( m(x^{(1)}) - m(x) )\ z(x^{(1)}) + \ldots + \exp( m(x^{(T)}) - m(x) )\ z(x^{(1)}) \\
\operatorname{softmax}(x) &amp;= f(x)\ / \ z(x)
\end{align}</div><p>where <span class="math notranslate nohighlight">\(f(x^{(j)}) = \exp( x^{(j)} - m(x^{(j)}) )\)</span> is of size <span class="math notranslate nohighlight">\(B\)</span> and
<span class="math notranslate nohighlight">\(z(x^{(j)}) = f(x_1^{(j)})+ \ldots+ f(x_B^{(j)})\)</span> is a scalar.</p>
<p>For a matrix <span class="math notranslate nohighlight">\(X\)</span> composed of <span class="math notranslate nohighlight">\(T_r \times T_c\)</span> tiles, <span class="math notranslate nohighlight">\(X_{ij}\)</span>, of size <span class="math notranslate nohighlight">\(B_r \times B_c\)</span> we can
compute the row-wise softmax as follows.</p>
<p>For <span class="math notranslate nohighlight">\(j\)</span> from <span class="math notranslate nohighlight">\(1\)</span> to <span class="math notranslate nohighlight">\(T_c\)</span>, and <span class="math notranslate nohighlight">\(i\)</span> from <span class="math notranslate nohighlight">\(1\)</span> to <span class="math notranslate nohighlight">\(T_r\)</span> calculate,</p>
<div class="math notranslate nohighlight">
\begin{align}
\tilde{m}_{ij} &amp;= \operatorname{rowmax}( X_{ij} ) \\
\tilde{P}_{ij} &amp;= \exp(X_{ij} - \tilde{m}_{ij} ) \\
\tilde{z}_{ij} &amp;= \operatorname{rowsum}( P_{ij} ) \\
\end{align}</div><p>If <span class="math notranslate nohighlight">\(j=1\)</span>, initialize running max, running sum, and the first column block of the output,</p>
<div class="math notranslate nohighlight">
\begin{align}
m_i &amp;= \tilde{m}_{i1} \\
z_i &amp;= \tilde{z}_{i1} \\
\tilde{Y}_{i1} &amp;= \diag(\tilde{z}_{ij})^{-1} \tilde{P}_{i1}
\end{align}</div><p>Else if <span class="math notranslate nohighlight">\(j&gt;1\)</span>,</p>
<ol class="arabic simple">
<li><p>Update running max, running sum and column blocks <span class="math notranslate nohighlight">\(k=1\)</span> to <span class="math notranslate nohighlight">\(k=j-1\)</span></p></li>
</ol>
<div class="math notranslate nohighlight">
\begin{align}
m^{new}_i &amp;= \max(m_i, \tilde{m}_{ij} ) \\
z^{new}_i &amp;= \exp(m_i - m^{new}_i)\ z_i + \exp( \tilde{m}_{ij} - m^{new}_i )\ \tilde{z}_{ij} \\
Y_{ik} &amp;= \diag(z^{new}_{i})^{-1} \diag(z_{i}) \exp(m_i - m^{new}_i)\ Y_{ik}
\end{align}</div><ol class="arabic simple" start="2">
<li><p>Initialize column block <span class="math notranslate nohighlight">\(j\)</span> of output and reset running max and running sum variables:</p></li>
</ol>
<div class="math notranslate nohighlight">
\begin{align}
\tilde{Y}_{ij} &amp;= \diag(z^{new}_{i})^{-1} \exp(\tilde{m}_{ij} - m^{new}_i ) \tilde{P}_{ij} \\
z_i &amp;= z^{new}_i \\
m_i &amp;= m^{new}_i \\
\end{align}</div></section>
</section>
</div>
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*******************
API Reference Guide
*******************
=================
Introduction
=================
This document contains details of the APIs for the Composable Kernel (CK) library and introduces some of the key design
principles that are used to write new classes that extend CK functionality.
=================
Using CK API
=================
This section describes how to use the CK library API.
=================
CK Datatypes
=================
-----------------
DeviceMem
-----------------
.. doxygenstruct:: DeviceMem
---------------------------
Kernels For Flashattention
---------------------------
The Flashattention algorithm is defined in :cite:t:`dao2022flashattention`. This sections lists the classes that are
used in the CK GPU implementation of Flashattention.
**Gridwise classes**
.. doxygenstruct:: ck::GridwiseBatchedGemmSoftmaxGemm_Xdl_CShuffle
**Blockwise classes**
.. doxygenstruct:: ck::ThreadGroupTensorSliceTransfer_v4r1
.. doxygenstruct:: ck::BlockwiseGemmXdlops_v2
.. doxygenstruct:: ck::BlockwiseSoftmax
**Threadwise classes**
.. doxygenstruct:: ck::ThreadwiseTensorSliceTransfer_StaticToStatic
.. bibliography::
\ No newline at end of file
===================
Contributor's Guide
===================
Pull-request guidelines
=======================
[TODO]
************
Disclaimer
************
-------------------------------
AMD's standard legal Disclaimer
-------------------------------
The information presented in this document is for informational purposes only and may contain technical inaccuracies, omissions, and typographical errors. The information contained herein is subject to change and may be rendered inaccurate for many reasons, including but not limited to product and roadmap changes, component and motherboard version changes, new model and/or product releases, product differences between differing manufacturers, software changes, BIOS flashes, firmware upgrades, or the like. Any computer system has risks of security vulnerabilities that cannot be completely prevented or mitigated. AMD assumes no obligation to update or otherwise correct or revise this information. However, AMD reserves the right to revise this information and to make changes from time to time to the content hereof without obligation of AMD to notify any person of such revisions or changes. THIS INFORMATION IS PROVIDED 'AS IS." AMD MAKES NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE CONTENTS HEREOF AND ASSUMES NO RESPONSIBILITY FOR ANY INACCURACIES, ERRORS, OR OMISSIONS THAT MAY APPEAR IN THIS INFORMATION. AMD SPECIFICALLY DISCLAIMS ANY IMPLIED WARRANTIES OF NON-INFRINGEMENT, MERCHANTABILITY, OR FITNESS FOR ANY PARTICULAR PURPOSE. IN NO EVENT WILL AMD BE LIABLE TO ANY PERSON FOR ANY RELIANCE, DIRECT, INDIRECT, SPECIAL, OR OTHER CONSEQUENTIAL DAMAGES ARISING FROM THE USE OF ANY INFORMATION CONTAINED HEREIN, EVEN IF AMD IS EXPRESSLY ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. AMD, the AMD Arrow logo, Radeon, Ryzen, Epyc, and combinations thereof are trademarks of Advanced Micro Devices, Inc. Other product names used in this publication are for identification purposes only and may be trademarks of their respective companies. Google(R) is a registered trademark of Google LLC. PCIe(R) is a registered trademark of PCI-SIG Corporation. Linux(R) is the registered trademark of Linus Torvalds in the U.S. and other countries. Ubuntu(R) and the Ubuntu logo are registered trademarks of Canonical Ltd. Other product names used in this publication are for identification purposes only and may be trademarks of their respective companies. (C)2023 Advanced Micro Devices, Inc. All rights reserved.
----------------------
Third Party Disclaimer
----------------------
Third-party content is licensed to you directly by the third party that owns the content and is not licensed to you by AMD. ALL LINKED THIRD-PARTY CONTENT IS PROVIDED "AS IS" WITHOUT A WARRANTY OF ANY KIND. USE OF SUCH THIRD-PARTY CONTENT IS DONE AT YOUR SOLE DISCRETION AND UNDER NO CIRCUMSTANCES WILL AMD BE LIABLE TO YOU FOR ANY THIRD-PARTY CONTENT. YOU ASSUME ALL RISK AND ARE SOLELY RESPONSIBLE FOR ANY DAMAGES THAT MAY ARISE FROM YOUR USE OF THIRD-PARTY CONTENT.
=====================
Getting Started Guide
=====================
------------
Introduction
------------
This document contains instructions for installing, using, and contributing to Composable Kernel (CK).
Documentation Roadmap
^^^^^^^^^^^^^^^^^^^^^
The following is a list of CK documents in the suggested reading order:
[TODO]
\ No newline at end of file
==========================
Supported Primitives Guide
==========================
This document contains details of supported primitives in Composable Kernel (CK). In contrast to the API Reference
Guide, the Supported Primitives Guide is an introduction to the math which underpins the algorithms implemented in CK.
------------
Softmax
------------
For vectors :math:`x^{(1)}, x^{(2)}, \ldots, x^{(T)}` of size :math:`B` we can decompose the softmax of concatenated
:math:`x = [ x^{(1)}\ | \ \ldots \ | \ x^{(T)} ]` as,
.. math::
:nowrap:
\begin{align}
m(x) & = m( [ x^{(1)}\ | \ \ldots \ | \ x^{(T)} ] ) = \max( m(x^{(1)}),\ldots, m(x^{(T)}) ) \\
f(x) & = [\exp( m(x^{(1)}) - m(x) ) f( x^{(1)} )\ | \ \ldots \ | \ \exp( m(x^{(T)}) - m(x) ) f( x^{(T)} )] \\
z(x) & = \exp( m(x^{(1)}) - m(x) )\ z(x^{(1)}) + \ldots + \exp( m(x^{(T)}) - m(x) )\ z(x^{(1)}) \\
\operatorname{softmax}(x) &= f(x)\ / \ z(x)
\end{align}
where :math:`f(x^{(j)}) = \exp( x^{(j)} - m(x^{(j)}) )` is of size :math:`B` and
:math:`z(x^{(j)}) = f(x_1^{(j)})+ \ldots+ f(x_B^{(j)})` is a scalar.
For a matrix :math:`X` composed of :math:`T_r \times T_c` tiles, :math:`X_{ij}`, of size :math:`B_r \times B_c` we can
compute the row-wise softmax as follows.
For :math:`j` from :math:`1` to :math:`T_c`, and :math:`i` from :math:`1` to :math:`T_r` calculate,
.. math::
:nowrap:
\begin{align}
\tilde{m}_{ij} &= \operatorname{rowmax}( X_{ij} ) \\
\tilde{P}_{ij} &= \exp(X_{ij} - \tilde{m}_{ij} ) \\
\tilde{z}_{ij} &= \operatorname{rowsum}( P_{ij} ) \\
\end{align}
If :math:`j=1`, initialize running max, running sum, and the first column block of the output,
.. math::
:nowrap:
\begin{align}
m_i &= \tilde{m}_{i1} \\
z_i &= \tilde{z}_{i1} \\
\tilde{Y}_{i1} &= \diag(\tilde{z}_{ij})^{-1} \tilde{P}_{i1}
\end{align}
Else if :math:`j>1`,
1. Update running max, running sum and column blocks :math:`k=1` to :math:`k=j-1`
.. math::
:nowrap:
\begin{align}
m^{new}_i &= \max(m_i, \tilde{m}_{ij} ) \\
z^{new}_i &= \exp(m_i - m^{new}_i)\ z_i + \exp( \tilde{m}_{ij} - m^{new}_i )\ \tilde{z}_{ij} \\
Y_{ik} &= \diag(z^{new}_{i})^{-1} \diag(z_{i}) \exp(m_i - m^{new}_i)\ Y_{ik}
\end{align}
2. Initialize column block :math:`j` of output and reset running max and running sum variables:
.. math::
:nowrap:
\begin{align}
\tilde{Y}_{ij} &= \diag(z^{new}_{i})^{-1} \exp(\tilde{m}_{ij} - m^{new}_i ) \tilde{P}_{ij} \\
z_i &= z^{new}_i \\
m_i &= m^{new}_i \\
\end{align}
\ No newline at end of file
===================
CK docker hub
===================
`Docker hub <https://hub.docker.com/r/rocm/composable_kernel>`_
-------------------------------------
Why do I need this?
-------------------------------------
To make our lives easier and bring Composable Kernel dependencies together, we recommend using docker images.
-------------------------------------
So what is Composable Kernel?
-------------------------------------
Composable Kernel (CK) library aims to provide a programming model for writing performance critical kernels for machine learning workloads across multiple architectures including GPUs, CPUs, etc, through general purpose kernel languages, like HIP C++.
To get the CK library::
git clone https://github.com/ROCmSoftwarePlatform/composable_kernel.git
run a docker container::
docker run \
-it \
--privileged \
--group-add sudo \
-w /root/workspace \
-v ${PATH_TO_LOCAL_WORKSPACE}:/root/workspace \
rocm/composable_kernel:ck_ub20.04_rocm5.3_release \
/bin/bash
and build the CK::
mkdir build && cd build
# Need to specify target ID, example below is for gfx908 and gfx90a
cmake \
-D CMAKE_PREFIX_PATH=/opt/rocm \
-D CMAKE_CXX_COMPILER=/opt/rocm/bin/hipcc \
-D CMAKE_CXX_FLAGS="-O3" \
-D CMAKE_BUILD_TYPE=Release \
-D GPU_TARGETS="gfx908;gfx90a" \
..
and::
make -j examples tests
To run all the test cases including tests and examples run::
make test
We can also run specific examples or tests like::
./bin/example_gemm_xdl_fp16
./bin/test_gemm_fp16
For more details visit `CK github repo <https://github.com/ROCmSoftwarePlatform/composable_kernel>`_, `CK examples <https://github.com/ROCmSoftwarePlatform/composable_kernel/tree/develop/example)>`_, `even more CK examples <https://github.com/ROCmSoftwarePlatform/composable_kernel/tree/develop/client_example>`_.
-------------------------------------
And what is inside?
-------------------------------------
The docker images have everything you need for running CK including:
* `ROCm <https://www.amd.com/en/graphics/servers-solutions-rocm>`_
* `CMake <https://cmake.org/>`_
* `Compiler <https://github.com/RadeonOpenCompute/llvm-project>`_
-------------------------------------
Which image is right for me?
-------------------------------------
Let's take a look at the image naming, for example "ck_ub20.04_rocm5.4_release". The image specs are:
* "ck" - made for running Composable Kernel
* "ub20.04" - based on Ubuntu 20.04
* "rocm5.4" - ROCm platform version 5.4
* "release" - compiler version is release
So just pick the right image for your project dependencies and you're all set.
-------------------------------------
DIY starts here
-------------------------------------
If you need to customize a docker image or just can't stop tinkering, feel free to adjust the `Dockerfile <https://github.com/ROCmSoftwarePlatform/composable_kernel/blob/develop/Dockerfile>`_ for your needs.
-------------------------------------
License
-------------------------------------
CK is released under the MIT `license <https://github.com/ROCmSoftwarePlatform/composable_kernel/blob/develop/LICENSE>`_.
============================
Composable Kernel User Guide
============================
.. toctree::
:maxdepth: 5
:caption: Contents:
:numbered:
Linux_Install_Guide
tutorial_hello_world
dockerhub
Supported_Primitives_Guide
API_Reference_Guide
Contributors_Guide
Disclaimer
\ No newline at end of file
===============
CK Hello world
===============
-------------------------------------
Motivation
-------------------------------------
This tutorial is aimed at engineers dealing with artificial intelligence and machine learning who would like to optimize their pipelines and squeeze every performance drop by adding Composable Kernel (CK) library to their projects. We would like to make the CK library approachable so the tutorial is not based on the latest release and doesn't have all the bleeding edge features, but it will be reproducible now and forever.
During this tutorial we will have an introduction to the CK library, we will build it and run some examples and tests, so to say we will run a "Hello world" example. In future tutorials we will go in depth and breadth and get familiar with other tools and ways to integrate CK into your project.
-------------------------------------
Description
-------------------------------------
Modern AI technology solves more and more problems in all imaginable fields, but crafting fast and efficient workflows is still challenging. CK is one of the tools to make AI heavy lifting as fast and efficient as possible. CK is a collection of optimized AI operator kernels and tools to create new ones. The library has components required for majority of modern neural networks architectures including matrix multiplication, convolution, contraction, reduction, attention modules, variety of activation functions, fused operators and many more.
So how do we (almost) reach the speed of light? CK acceleration abilities are based on:
* Layered structure.
* Tile-based computation model.
* Tensor coordinate transformation.
* Hardware acceleration use.
* Support of low precision data types including fp16, bf16, int8 and int4.
If you are excited and need more technical details and benchmarking results - read this awesome `blog post <https://community.amd.com/t5/instinct-accelerators/amd-composable-kernel-library-efficient-fused-kernels-for-ai/ba-p/553224>`_.
For more details visit our `github repo <https://github.com/ROCmSoftwarePlatform/composable_kernel>`_.
-------------------------------------
Hardware targets
-------------------------------------
CK library fully supports "gfx908" and "gfx90a" GPU architectures and only some operators are supported for "gfx1030". Let's check the hardware you have at hand and decide on the target GPU architecture
========== =========
GPU Target AMD GPU
========== =========
gfx908 Radeon Instinct MI100
gfx90a Radeon Instinct MI210, MI250, MI250X
gfx1030 Radeon PRO V620, W6800, W6800X, W6800X Duo, W6900X, RX 6800, RX 6800 XT, RX 6900 XT, RX 6900 XTX, RX 6950 XT
========== =========
There are also `cloud options <https://aws.amazon.com/ec2/instance-types/g4/>`_ you can find if you don't have an AMD GPU at hand.
-------------------------------------
Build the library
-------------------------------------
First let's clone the library and rebase to the tested version::
git clone https://github.com/ROCmSoftwarePlatform/composable_kernel.git
cd composable_kernel/
git checkout tutorial_hello_world
To make our lives easier we prepared `docker images <https://hub.docker.com/r/rocm/composable_kernel>`_ with all the necessary dependencies. Pick the right image and create a container. In this tutorial we use "rocm/composable_kernel:ck_ub20.04_rocm5.3_release" image, it is based on Ubuntu 20.04, ROCm v5.3, compiler release version.
If your current folder is ${HOME}, start the docker container with::
docker run \
-it \
--privileged \
--group-add sudo \
-w /root/workspace \
-v ${HOME}:/root/workspace \
rocm/composable_kernel:ck_ub20.04_rocm5.3_release \
/bin/bash
If your current folder is different from ${HOME}, adjust the line `-v ${HOME}:/root/workspace` to fit your folder structure.
Inside the docker container current folder is "~/workspace", library path is "~/workspace/composable_kernel", navigate to the library::
cd composable_kernel/
Create and go to the "build" directory::
mkdir build && cd build
In the previous section we talked about target GPU architecture. Once you decide which one is right for you, run cmake using the right GPU_TARGETS flag::
cmake \
-D CMAKE_PREFIX_PATH=/opt/rocm \
-D CMAKE_CXX_COMPILER=/opt/rocm/bin/hipcc \
-D CMAKE_CXX_FLAGS="-O3" \
-D CMAKE_BUILD_TYPE=Release \
-D BUILD_DEV=OFF \
-D GPU_TARGETS="gfx908;gfx90a;gfx1030" ..
If everything went well the cmake run will end up with::
-- Configuring done
-- Generating done
-- Build files have been written to: "/root/workspace/composable_kernel/build"
Finally, we can build examples and tests::
make -j examples tests
If everything is smooth, you'll see::
Scanning dependencies of target tests
[100%] Built target tests
---------------------------
Run examples and tests
---------------------------
Examples are listed as test cases as well, so we can run all examples and tests with::
ctest
You can check the list of all tests by running::
ctest -N
We can also run them separately, here is a separate example execution::
./bin/example_gemm_xdl_fp16 1 1 1
The arguments "1 1 1" mean that we want to run this example in the mode: verify results with CPU, initialize matrices with integers and benchmark the kernel execution. You can play around with these parameters and see how output and execution results change.
If everything goes well and you have a device based on gfx908 or gfx90a architecture you should see something like::
a_m_k: dim 2, lengths {3840, 4096}, strides {4096, 1}
b_k_n: dim 2, lengths {4096, 4096}, strides {1, 4096}
c_m_n: dim 2, lengths {3840, 4096}, strides {4096, 1}
launch_and_time_kernel: grid_dim {480, 1, 1}, block_dim {256, 1, 1}
Warm up 1 time
Start running 10 times...
Perf: 1.10017 ms, 117.117 TFlops, 87.6854 GB/s, DeviceGemmXdl<256, 256, 128, 4, 8, 32, 32, 4, 2> NumPrefetch: 1, LoopScheduler: Default, PipelineVersion: v1
Meanwhile, running it on a gfx1030 device should result in::
a_m_k: dim 2, lengths {3840, 4096}, strides {4096, 1}
b_k_n: dim 2, lengths {4096, 4096}, strides {1, 4096}
c_m_n: dim 2, lengths {3840, 4096}, strides {4096, 1}
DeviceGemmXdl<256, 256, 128, 4, 8, 32, 32, 4, 2> NumPrefetch: 1, LoopScheduler: Default, PipelineVersion: v1 does not support this problem
But don't panic, some of the operators are supported on gfx1030 architecture, so you can run a separate example like::
./bin/example_gemm_dl_fp16 1 1 1
and it should result in something nice similar to::
a_m_k: dim 2, lengths {3840, 4096}, strides {1, 4096}
b_k_n: dim 2, lengths {4096, 4096}, strides {4096, 1}
c_m_n: dim 2, lengths {3840, 4096}, strides {4096, 1}
arg.a_grid_desc_k0_m0_m1_k1_{2048, 3840, 2}
arg.b_grid_desc_k0_n0_n1_k1_{2048, 4096, 2}
arg.c_grid_desc_m_n_{ 3840, 4096}
launch_and_time_kernel: grid_dim {960, 1, 1}, block_dim {256, 1, 1}
Warm up 1 time
Start running 10 times...
Perf: 3.65695 ms, 35.234 TFlops, 26.3797 GB/s, DeviceGemmDl<256, 128, 128, 16, 2, 4, 4, 1>
Or we can run a separate test::
ctest -R test_gemm_fp16
If everything goes well you should see something like::
Start 121: test_gemm_fp16
1/1 Test #121: test_gemm_fp16 ................... Passed 51.81 sec
100% tests passed, 0 tests failed out of 1
-----------
Summary
-----------
In this tutorial we took the first look at the Composable Kernel library, built it on your system and ran some examples and tests. Stay tuned, in the next tutorial we will run kernels with different configs to find out the best one for your hardware and task.
P.S.: Don't forget to switch out the cloud instance if you have launched one, you can find better ways to spend your money for sure!
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margin: 1em 0 1em 0;
padding: 0.4em;
}
div.genindex-jumpbox {
border-top: 1px solid #ddd;
border-bottom: 1px solid #ddd;
margin: 1em 0 1em 0;
padding: 0.4em;
}
/* -- domain module index --------------------------------------------------- */
table.modindextable td {
padding: 2px;
border-collapse: collapse;
}
/* -- general body styles --------------------------------------------------- */
div.body {
min-width: 360px;
max-width: 800px;
}
div.body p, div.body dd, div.body li, div.body blockquote {
-moz-hyphens: auto;
-ms-hyphens: auto;
-webkit-hyphens: auto;
hyphens: auto;
}
a.headerlink {
visibility: hidden;
}
h1:hover > a.headerlink,
h2:hover > a.headerlink,
h3:hover > a.headerlink,
h4:hover > a.headerlink,
h5:hover > a.headerlink,
h6:hover > a.headerlink,
dt:hover > a.headerlink,
caption:hover > a.headerlink,
p.caption:hover > a.headerlink,
div.code-block-caption:hover > a.headerlink {
visibility: visible;
}
div.body p.caption {
text-align: inherit;
}
div.body td {
text-align: left;
}
.first {
margin-top: 0 !important;
}
p.rubric {
margin-top: 30px;
font-weight: bold;
}
img.align-left, figure.align-left, .figure.align-left, object.align-left {
clear: left;
float: left;
margin-right: 1em;
}
img.align-right, figure.align-right, .figure.align-right, object.align-right {
clear: right;
float: right;
margin-left: 1em;
}
img.align-center, figure.align-center, .figure.align-center, object.align-center {
display: block;
margin-left: auto;
margin-right: auto;
}
img.align-default, figure.align-default, .figure.align-default {
display: block;
margin-left: auto;
margin-right: auto;
}
.align-left {
text-align: left;
}
.align-center {
text-align: center;
}
.align-default {
text-align: center;
}
.align-right {
text-align: right;
}
/* -- sidebars -------------------------------------------------------------- */
div.sidebar,
aside.sidebar {
margin: 0 0 0.5em 1em;
border: 1px solid #ddb;
padding: 7px;
background-color: #ffe;
width: 40%;
float: right;
clear: right;
overflow-x: auto;
}
p.sidebar-title {
font-weight: bold;
}
nav.contents,
aside.topic,
div.admonition, div.topic, blockquote {
clear: left;
}
/* -- topics ---------------------------------------------------------------- */
nav.contents,
aside.topic,
div.topic {
border: 1px solid #ccc;
padding: 7px;
margin: 10px 0 10px 0;
}
p.topic-title {
font-size: 1.1em;
font-weight: bold;
margin-top: 10px;
}
/* -- admonitions ----------------------------------------------------------- */
div.admonition {
margin-top: 10px;
margin-bottom: 10px;
padding: 7px;
}
div.admonition dt {
font-weight: bold;
}
p.admonition-title {
margin: 0px 10px 5px 0px;
font-weight: bold;
}
div.body p.centered {
text-align: center;
margin-top: 25px;
}
/* -- content of sidebars/topics/admonitions -------------------------------- */
div.sidebar > :last-child,
aside.sidebar > :last-child,
nav.contents > :last-child,
aside.topic > :last-child,
div.topic > :last-child,
div.admonition > :last-child {
margin-bottom: 0;
}
div.sidebar::after,
aside.sidebar::after,
nav.contents::after,
aside.topic::after,
div.topic::after,
div.admonition::after,
blockquote::after {
display: block;
content: '';
clear: both;
}
/* -- tables ---------------------------------------------------------------- */
table.docutils {
margin-top: 10px;
margin-bottom: 10px;
border: 0;
border-collapse: collapse;
}
table.align-center {
margin-left: auto;
margin-right: auto;
}
table.align-default {
margin-left: auto;
margin-right: auto;
}
table caption span.caption-number {
font-style: italic;
}
table caption span.caption-text {
}
table.docutils td, table.docutils th {
padding: 1px 8px 1px 5px;
border-top: 0;
border-left: 0;
border-right: 0;
border-bottom: 1px solid #aaa;
}
th {
text-align: left;
padding-right: 5px;
}
table.citation {
border-left: solid 1px gray;
margin-left: 1px;
}
table.citation td {
border-bottom: none;
}
th > :first-child,
td > :first-child {
margin-top: 0px;
}
th > :last-child,
td > :last-child {
margin-bottom: 0px;
}
/* -- figures --------------------------------------------------------------- */
div.figure, figure {
margin: 0.5em;
padding: 0.5em;
}
div.figure p.caption, figcaption {
padding: 0.3em;
}
div.figure p.caption span.caption-number,
figcaption span.caption-number {
font-style: italic;
}
div.figure p.caption span.caption-text,
figcaption span.caption-text {
}
/* -- field list styles ----------------------------------------------------- */
table.field-list td, table.field-list th {
border: 0 !important;
}
.field-list ul {
margin: 0;
padding-left: 1em;
}
.field-list p {
margin: 0;
}
.field-name {
-moz-hyphens: manual;
-ms-hyphens: manual;
-webkit-hyphens: manual;
hyphens: manual;
}
/* -- hlist styles ---------------------------------------------------------- */
table.hlist {
margin: 1em 0;
}
table.hlist td {
vertical-align: top;
}
/* -- object description styles --------------------------------------------- */
.sig {
font-family: 'Consolas', 'Menlo', 'DejaVu Sans Mono', 'Bitstream Vera Sans Mono', monospace;
}
.sig-name, code.descname {
background-color: transparent;
font-weight: bold;
}
.sig-name {
font-size: 1.1em;
}
code.descname {
font-size: 1.2em;
}
.sig-prename, code.descclassname {
background-color: transparent;
}
.optional {
font-size: 1.3em;
}
.sig-paren {
font-size: larger;
}
.sig-param.n {
font-style: italic;
}
/* C++ specific styling */
.sig-inline.c-texpr,
.sig-inline.cpp-texpr {
font-family: unset;
}
.sig.c .k, .sig.c .kt,
.sig.cpp .k, .sig.cpp .kt {
color: #0033B3;
}
.sig.c .m,
.sig.cpp .m {
color: #1750EB;
}
.sig.c .s, .sig.c .sc,
.sig.cpp .s, .sig.cpp .sc {
color: #067D17;
}
/* -- other body styles ----------------------------------------------------- */
ol.arabic {
list-style: decimal;
}
ol.loweralpha {
list-style: lower-alpha;
}
ol.upperalpha {
list-style: upper-alpha;
}
ol.lowerroman {
list-style: lower-roman;
}
ol.upperroman {
list-style: upper-roman;
}
:not(li) > ol > li:first-child > :first-child,
:not(li) > ul > li:first-child > :first-child {
margin-top: 0px;
}
:not(li) > ol > li:last-child > :last-child,
:not(li) > ul > li:last-child > :last-child {
margin-bottom: 0px;
}
ol.simple ol p,
ol.simple ul p,
ul.simple ol p,
ul.simple ul p {
margin-top: 0;
}
ol.simple > li:not(:first-child) > p,
ul.simple > li:not(:first-child) > p {
margin-top: 0;
}
ol.simple p,
ul.simple p {
margin-bottom: 0;
}
aside.footnote > span,
div.citation > span {
float: left;
}
aside.footnote > span:last-of-type,
div.citation > span:last-of-type {
padding-right: 0.5em;
}
aside.footnote > p {
margin-left: 2em;
}
div.citation > p {
margin-left: 4em;
}
aside.footnote > p:last-of-type,
div.citation > p:last-of-type {
margin-bottom: 0em;
}
aside.footnote > p:last-of-type:after,
div.citation > p:last-of-type:after {
content: "";
clear: both;
}
dl.field-list {
display: grid;
grid-template-columns: fit-content(30%) auto;
}
dl.field-list > dt {
font-weight: bold;
word-break: break-word;
padding-left: 0.5em;
padding-right: 5px;
}
dl.field-list > dd {
padding-left: 0.5em;
margin-top: 0em;
margin-left: 0em;
margin-bottom: 0em;
}
dl {
margin-bottom: 15px;
}
dd > :first-child {
margin-top: 0px;
}
dd ul, dd table {
margin-bottom: 10px;
}
dd {
margin-top: 3px;
margin-bottom: 10px;
margin-left: 30px;
}
dl > dd:last-child,
dl > dd:last-child > :last-child {
margin-bottom: 0;
}
dt:target, span.highlighted {
background-color: #fbe54e;
}
rect.highlighted {
fill: #fbe54e;
}
dl.glossary dt {
font-weight: bold;
font-size: 1.1em;
}
.versionmodified {
font-style: italic;
}
.system-message {
background-color: #fda;
padding: 5px;
border: 3px solid red;
}
.footnote:target {
background-color: #ffa;
}
.line-block {
display: block;
margin-top: 1em;
margin-bottom: 1em;
}
.line-block .line-block {
margin-top: 0;
margin-bottom: 0;
margin-left: 1.5em;
}
.guilabel, .menuselection {
font-family: sans-serif;
}
.accelerator {
text-decoration: underline;
}
.classifier {
font-style: oblique;
}
.classifier:before {
font-style: normal;
margin: 0 0.5em;
content: ":";
display: inline-block;
}
abbr, acronym {
border-bottom: dotted 1px;
cursor: help;
}
/* -- code displays --------------------------------------------------------- */
pre {
overflow: auto;
overflow-y: hidden; /* fixes display issues on Chrome browsers */
}
pre, div[class*="highlight-"] {
clear: both;
}
span.pre {
-moz-hyphens: none;
-ms-hyphens: none;
-webkit-hyphens: none;
hyphens: none;
white-space: nowrap;
}
div[class*="highlight-"] {
margin: 1em 0;
}
td.linenos pre {
border: 0;
background-color: transparent;
color: #aaa;
}
table.highlighttable {
display: block;
}
table.highlighttable tbody {
display: block;
}
table.highlighttable tr {
display: flex;
}
table.highlighttable td {
margin: 0;
padding: 0;
}
table.highlighttable td.linenos {
padding-right: 0.5em;
}
table.highlighttable td.code {
flex: 1;
overflow: hidden;
}
.highlight .hll {
display: block;
}
div.highlight pre,
table.highlighttable pre {
margin: 0;
}
div.code-block-caption + div {
margin-top: 0;
}
div.code-block-caption {
margin-top: 1em;
padding: 2px 5px;
font-size: small;
}
div.code-block-caption code {
background-color: transparent;
}
table.highlighttable td.linenos,
span.linenos,
div.highlight span.gp { /* gp: Generic.Prompt */
user-select: none;
-webkit-user-select: text; /* Safari fallback only */
-webkit-user-select: none; /* Chrome/Safari */
-moz-user-select: none; /* Firefox */
-ms-user-select: none; /* IE10+ */
}
div.code-block-caption span.caption-number {
padding: 0.1em 0.3em;
font-style: italic;
}
div.code-block-caption span.caption-text {
}
div.literal-block-wrapper {
margin: 1em 0;
}
code.xref, a code {
background-color: transparent;
font-weight: bold;
}
h1 code, h2 code, h3 code, h4 code, h5 code, h6 code {
background-color: transparent;
}
.viewcode-link {
float: right;
}
.viewcode-back {
float: right;
font-family: sans-serif;
}
div.viewcode-block:target {
margin: -1px -10px;
padding: 0 10px;
}
/* -- math display ---------------------------------------------------------- */
img.math {
vertical-align: middle;
}
div.body div.math p {
text-align: center;
}
span.eqno {
float: right;
}
span.eqno a.headerlink {
position: absolute;
z-index: 1;
}
div.math:hover a.headerlink {
visibility: visible;
}
/* -- printout stylesheet --------------------------------------------------- */
@media print {
div.document,
div.documentwrapper,
div.bodywrapper {
margin: 0 !important;
width: 100%;
}
div.sphinxsidebar,
div.related,
div.footer,
#top-link {
display: none;
}
}
\ No newline at end of file
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\ No newline at end of file
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