diff options
Diffstat (limited to 'notebook')
| -rw-r--r-- | notebook/inspect_opencl_layout.ipynb | 612 | ||||
| -rw-r--r-- | notebook/lbm_codegen.ipynb | 828 | ||||
| -rw-r--r-- | notebook/lbm_codegen_d3q19.ipynb | 1160 |
3 files changed, 2600 insertions, 0 deletions
diff --git a/notebook/inspect_opencl_layout.ipynb b/notebook/inspect_opencl_layout.ipynb new file mode 100644 index 0000000..521f93a --- /dev/null +++ b/notebook/inspect_opencl_layout.ipynb @@ -0,0 +1,612 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Thread layouting in OpenCL\n", + "## Setup PyOpenCL" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "import pyopencl as cl\n", + "import numpy" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "platform = cl.get_platforms()[0]\n", + "context = cl.Context(properties=[(cl.context_properties.PLATFORM, platform)])\n", + "queue = cl.CommandQueue(context)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "<pyopencl.Platform 'NVIDIA CUDA' at 0x28ef860>" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "platform" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Initialize array for storing group and local IDs" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "nX = 16\n", + "nY = 16" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "np_data = numpy.ndarray(shape=(nX*nY, 4), dtype=numpy.int32)" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "np_data[:,:] = 0" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([[0, 0, 0, 0],\n", + " [0, 0, 0, 0],\n", + " [0, 0, 0, 0],\n", + " ...,\n", + " [0, 0, 0, 0],\n", + " [0, 0, 0, 0],\n", + " [0, 0, 0, 0]], dtype=int32)" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "np_data" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [], + "source": [ + "cl_data = cl.Buffer(context, cl.mem_flags.READ_WRITE | cl.mem_flags.USE_HOST_PTR, hostbuf=np_data)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "<pyopencl._cl.Buffer at 0x7efd2d9f2888>" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "cl_data" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Define OpenCL kernel\n", + "\n", + "Simply writes out group and local indices to the given data array." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [], + "source": [ + "program = cl.Program(context, \"\"\"\n", + "__kernel void test(__global int* data)\n", + "{\n", + " const unsigned int gid = 4*(get_global_id(1)*get_global_size(1) + get_global_id(0));\n", + " data[gid + 0] = get_group_id(0);\n", + " data[gid + 1] = get_group_id(1);\n", + " data[gid + 2] = get_local_id(0);\n", + " data[gid + 3] = get_local_id(1);\n", + "}\"\"\").build()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Test output for a 1x1 work group size:" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([[ 0, 0, 0, 0],\n", + " [ 1, 0, 0, 0],\n", + " [ 2, 0, 0, 0],\n", + " ...,\n", + " [13, 15, 0, 0],\n", + " [14, 15, 0, 0],\n", + " [15, 15, 0, 0]], dtype=int32)" + ] + }, + "execution_count": 11, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "program.test(queue, (nX,nY), (1,1), cl_data)\n", + "queue.finish()\n", + "cl.enqueue_copy(queue, np_data, cl_data).wait();\n", + "np_data" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Test various work group sizes" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [], + "source": [ + "def computeWorkgroup(i, j):\n", + " program.test(queue, (nX,nY), (i,j), cl_data)\n", + " queue.finish()\n", + " cl.enqueue_copy(queue, np_data, cl_data).wait();" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([[0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3],\n", + " [0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3]], dtype=int32)" + ] + }, + "execution_count": 13, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "computeWorkgroup(4,4)\n", + "np_data.reshape((nX, nY, 4))[:,:,0] # x-index of work group" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2],\n", + " [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2],\n", + " [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2],\n", + " [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2],\n", + " [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3],\n", + " [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3],\n", + " [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3],\n", + " [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3]], dtype=int32)" + ] + }, + "execution_count": 14, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "np_data.reshape((nX, nY, 4))[:,:,1] # y-index of work group" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [], + "source": [ + "computeWorkgroup(8,2)" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([[0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1]], dtype=int32)" + ] + }, + "execution_count": 16, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "np_data.reshape((nX, nY, 4))[:,:,0]" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2],\n", + " [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2],\n", + " [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3],\n", + " [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3],\n", + " [4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4],\n", + " [4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4],\n", + " [5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5],\n", + " [5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5],\n", + " [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6],\n", + " [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6],\n", + " [7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7],\n", + " [7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7]], dtype=int32)" + ] + }, + "execution_count": 17, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "np_data.reshape((nX, nY, 4))[:,:,1]" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([[0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7],\n", + " [0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7]], dtype=int32)" + ] + }, + "execution_count": 18, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "np_data.reshape((nX, nY, 4))[:,:,2]" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n", + " [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", + " [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]], dtype=int32)" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "np_data.reshape((nX, nY, 4))[:,:,3]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Generate some visuals" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [], + "source": [ + "def plotWorkgroupConfig(i,j):\n", + " computeWorkgroup(i,j)\n", + " data = np_data.reshape((nX, nY, 4))\n", + " for k in range(0,4):\n", + " plt.subplot(1,4,k+1)\n", + " plt.imshow(np_data.reshape((nX, nY, 4))[:,:,k])" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "<Figure size 432x288 with 4 Axes>" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plotWorkgroupConfig(1,1)" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "<Figure size 432x288 with 4 Axes>" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plotWorkgroupConfig(4,4)" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "<Figure size 432x288 with 4 Axes>" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plotWorkgroupConfig(8,8)" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "<Figure size 432x288 with 4 Axes>" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plotWorkgroupConfig(2,16)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/notebook/lbm_codegen.ipynb b/notebook/lbm_codegen.ipynb new file mode 100644 index 0000000..e1593c0 --- /dev/null +++ b/notebook/lbm_codegen.ipynb @@ -0,0 +1,828 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "from sympy import *\n", + "init_printing()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Characteristic constants" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "q = 9\n", + "d = 2" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "c = [Matrix(x) for x in [(-1, 1), ( 0, 1), ( 1, 1), (-1, 0), ( 0, 0), ( 1, 0), (-1,-1), ( 0, -1), ( 1, -1)]]" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$$\\left [ \\left[\\begin{matrix}-1\\\\1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1\\\\1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}-1\\\\0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1\\\\0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}-1\\\\-1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\-1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1\\\\-1\\end{matrix}\\right]\\right ]$$" + ], + "text/plain": [ + "⎡⎡-1⎤ ⎡0⎤ ⎡1⎤ ⎡-1⎤ ⎡0⎤ ⎡1⎤ ⎡-1⎤ ⎡0 ⎤ ⎡1 ⎤⎤\n", + "⎢⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎥\n", + "⎣⎣1 ⎦ ⎣1⎦ ⎣1⎦ ⎣0 ⎦ ⎣0⎦ ⎣0⎦ ⎣-1⎦ ⎣-1⎦ ⎣-1⎦⎦" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "c" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "w = [Rational(*x) for x in [(1,36), (1,9), (1,36), (1,9), (4,9), (1,9), (1,36), (1,9), (1,36)]]" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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\n", + "text/latex": [ + "$$\\left [ \\frac{1}{36}, \\quad \\frac{1}{9}, \\quad \\frac{1}{36}, \\quad \\frac{1}{9}, \\quad \\frac{4}{9}, \\quad \\frac{1}{9}, \\quad \\frac{1}{36}, \\quad \\frac{1}{9}, \\quad \\frac{1}{36}\\right ]$$" + ], + "text/plain": [ + "[1/36, 1/9, 1/36, 1/9, 4/9, 1/9, 1/36, 1/9, 1/36]" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "w" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAHZJREFUKJFjYKACCGFgYJjMwMBwmIGB4RMDA8N/BgaGJYQ0XYAq/MzAwHCdWE2ODAwMqgwMDIwMDAwOuDSxoPH3EzKVgYGBgYkYRaOaBlwTeuQGQDEDAwODBJS2ZGBgWABlv2FgYChBN6SBAZJ0cOEH5LiMzgAA6XoX52TB9a4AAAAASUVORK5CYII=\n", + "text/latex": [ + "$$1$$" + ], + "text/plain": [ + "1" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sum(w)" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "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\n", + "text/latex": [ + "$$\\frac{\\sqrt{3}}{3}$$" + ], + "text/plain": [ + "√3\n", + "──\n", + "3 " + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "c_s = sqrt(Rational(1,3))\n", + "c_s" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Moments" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [], + "source": [ + "rho, tau = symbols('rho tau')" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([f_next_0, f_next_1, f_next_2, f_next_3, f_next_4, f_next_5,\n", + " f_next_6, f_next_7, f_next_8], dtype=object)" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "f_next = symarray('f_next', q)\n", + "f_next" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([f_curr_0, f_curr_1, f_curr_2, f_curr_3, f_curr_4, f_curr_5,\n", + " f_curr_6, f_curr_7, f_curr_8], dtype=object)" + ] + }, + "execution_count": 11, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "f_curr = symarray('f_curr', q)\n", + "f_curr" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$$\\left[\\begin{matrix}u_{0}\\\\u_{1}\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡u₀⎤\n", + "⎢ ⎥\n", + "⎣u₁⎦" + ] + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + |