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kvedala/LU Decomposition.ipynb

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LU Decomposition
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LU Decomposition.ipynb
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"<a href=\"https://colab.research.google.com/gist/kvedala/b4d26ead2f5c7b3293ba6fbe2adf04f2/LU Decomposition.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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{
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"<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n",
"<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Initializer\" data-toc-modified-id=\"Initializer-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Initializer</a></span></li><li><span><a href=\"#Pure-python\" data-toc-modified-id=\"Pure-python-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Pure python</a></span></li><li><span><a href=\"#Numba\" data-toc-modified-id=\"Numba-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>Numba</a></span></li><li><span><a href=\"#Cython\" data-toc-modified-id=\"Cython-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>Cython</a></span></li><li><span><a href=\"#Analysis\" data-toc-modified-id=\"Analysis-5\"><span class=\"toc-item-num\">5&nbsp;&nbsp;</span>Analysis</a></span></li></ul></div>"
]
},
{
"cell_type": "markdown",
"metadata": {
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"colab_type": "text"
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"source": [
"# Initializer "
]
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{
"cell_type": "code",
"metadata": {
"trusted": true,
"id": "7LBnn7gBGq-p",
"colab_type": "code",
"colab": {}
},
"source": [
"%matplotlib inline\n",
"import numba, timeit, ipywidgets\n",
"from numba import prange\n",
"import numpy as np\n",
"from scipy import linalg\n",
"from scipy.version import version as scipy_version\n",
"from matplotlib import pyplot as plt\n",
"from matplotlib.gridspec import GridSpec\n",
"from platform import python_version\n",
"from cython import __version__ as cpy_version\n",
"from IPython.display import set_matplotlib_formats\n",
"set_matplotlib_formats('svg')\n",
"# %reload_ext line_profiler\n",
"%reload_ext cython"
],
"execution_count": 0,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "pCSknYnqGq-0",
"colab_type": "text"
},
"source": [
"# Pure python"
]
},
{
"cell_type": "code",
"metadata": {
"trusted": true,
"id": "mymJI7GtGq-3",
"colab_type": "code",
"colab": {}
},
"source": [
"def lu_decompose(A):\n",
" assert A.shape[0] == A.shape[1], \"Input should be a square matrix!\"\n",
" \n",
" N = A.shape[0]\n",
" U = np.zeros_like(A)\n",
"# L = np.eye(N)\n",
" L = np.zeros_like(A)\n",
" \n",
" for i in range(N):\n",
" # Upper Triangular\n",
" for k in range(i, N):\n",
" # Summation of L(i, j) * U(j, k)\n",
" lu_sum = 0\n",
" for j in range(i):\n",
" lu_sum += L[i,j] * U[j,k]\n",
" # Evaluating U(i, k)\n",
" U[i,k] = A[i,k] - lu_sum\n",
" \n",
" # Lower Triangular\n",
" for k in range(i, N):\n",
" if i == k:\n",
" L[i,i] = 1 # Diagonal as 1\n",
" else:\n",
" # Summation of L(k, j) * U(j, i)\n",
" lu_sum = 0;\n",
" for j in range(i):\n",
" lu_sum += L[k,j] * U[j,i]\n",
" # Evaluating L(k, i)\n",
" L[k,i] = (A[k,i] - lu_sum) / U[i,i]\n",
"\n",
" return L, U"
],
"execution_count": 0,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "ma0Wlan2Gq_B",
"colab_type": "text"
},
"source": [
"# Numba"
]
},
{
"cell_type": "code",
"metadata": {
"trusted": true,
"id": "CUCR_92dGq_E",
"colab_type": "code",
"colab": {
"base_uri": "https://localhost:8080/",
"height": 68
},
"outputId": "9c52367a-9ab4-4210-9e98-653f31ed7a24"
},
"source": [
"@numba.jit(nopython=True, parallel=True, cache=True)\n",
"def lu_decompose_n(A):\n",
" assert A.shape[0] == A.shape[1], \"Input should be a square matrix!\"\n",
" \n",
" N = A.shape[0]\n",
" U = np.zeros_like(A)\n",
"# L = np.eye(N)\n",
" L = np.zeros_like(A)\n",
" \n",
" for i in prange(N):\n",
" # Upper Triangular\n",
" for k in prange(i, N):\n",
" # Summation of L(i, j) * U(j, k)\n",
" lu_sum = 0\n",
" for j in prange(i):\n",
" lu_sum += L[i,j] * U[j,k]\n",
" # Evaluating U(i, k)\n",
" U[i,k] = A[i,k] - lu_sum\n",
" \n",
" # Lower Triangular\n",
" for k in prange(i, N):\n",
" if i == k:\n",
" L[i,i] = 1 # Diagonal as 1\n",
" else:\n",
" # Summation of L(k, j) * U(j, i)\n",
" lu_sum = 0;\n",
" for j in prange(i):\n",
" lu_sum += L[k,j] * U[j,i]\n",
" # Evaluating L(k, i)\n",
" L[k,i] = (A[k,i] - lu_sum) / U[i,i]\n",
"\n",
" return L, U\n",
"\n",
"lu_decompose_n(np.array([[1.0,0.5],[0.5,1.0]])) # to compile the function once"
],
"execution_count": 4,
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"(array([[1. , 0. ],\n",
" [0.5, 1. ]]), array([[1. , 0.5],\n",
" [0. , 1. ]]))"
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"metadata": {
"tags": []
},
"execution_count": 4
}
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "W2NV6gvUGq_L",
"colab_type": "text"
},
"source": [
"# Cython"
]
},
{
"cell_type": "code",
"metadata": {
"trusted": true,
"id": "OXDhjbkzGq_N",
"colab_type": "code",
"colab": {}
},
"source": [
"%%cython -c=-flto=thin -c=-Ofast -c=-fopenmp --link-args=-fopenmp\n",
"# cython: boundscheck=False, wraparound=False, cdivision=True\n",
"\n",
"cimport numpy as np\n",
"import numpy as np\n",
"\n",
"from cython.parallel cimport prange\n",
"\n",
"def lu_decompose_c(np.ndarray[double, ndim=2] A):\n",
" cdef:\n",
" int N = A.shape[0]\n",
" int i, j, k\n",
" double lu_sum\n",
" np.ndarray[double, ndim=2] U = np.zeros_like(A)\n",
" np.ndarray[double, ndim=2] L = np.zeros_like(A)\n",
" \n",
" assert A.shape[0] == A.shape[1], \"Input should be a square matrix!\"\n",
" \n",
" for i in prange(N, nogil=True):\n",
" # Upper Triangular\n",
" for k in range(i, N):\n",
" # Summation of L(i, j) * U(j, k)\n",
" lu_sum = 0\n",
" for j in prange(i):\n",
" lu_sum += L[i,j] * U[j,k]\n",
" # Evaluating U(i, k)\n",
" U[i,k] = A[i,k] - lu_sum\n",
" \n",
" # Lower Triangular\n",
" for k in range(i, N):\n",
" if i == k:\n",
" L[i,i] = 1 # Diagonal as 1\n",
" else:\n",
" # Summation of L(k, j) * U(j, i)\n",
" lu_sum = 0\n",
" for j in prange(i):\n",
" lu_sum += L[k,j] * U[j,i]\n",
" # Evaluating L(k, i)\n",
" L[k,i] = (A[k,i] - lu_sum) / U[i,i]\n",
"\n",
" return L, U"
],
"execution_count": 0,
"outputs": []
},
{
"cell_type": "markdown",
"metadata": {
"id": "CXbVo77PGq_S",
"colab_type": "text"
},
"source": [
"# Analysis"
]
},
{
"cell_type": "code",
"metadata": {
"trusted": true,
"id": "Pox_AtGWGq_T",
"colab_type": "code",
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"height": 49,
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"source": [
"sizes = range(2, 70)\n",
"t_py = []\n",
"t_numba = []\n",
"t_scipy = []\n",
"t_cython = []\n",
"\n",
"num_repetitions = 5000\n",
"py_max_size = 10\n",
"\n",
"prg_bar = ipywidgets.IntProgress(min=sizes[0], max=sizes[-1], value=0,\n",
" description='')\n",
"display(prg_bar)\n",
"\n",
"for size in sizes:\n",
" prg_bar.value = size\n",
" prg_bar.description = 'Size: %d' % (size)\n",
" A = np.random.random([size,size])\n",
" if size < py_max_size:\n",
" t = timeit.timeit('lu_decompose(A)', globals=globals(), number=num_repetitions)\n",
" t_py.append(t)\n",
" t = timeit.timeit('lu_decompose_n(A)', globals=globals(), number=num_repetitions)\n",
" t_numba.append(t * 1000)\n",
" t = timeit.timeit('lu_decompose_c(A)', globals=globals(), number=num_repetitions)\n",
" t_cython.append(t * 1000)\n",
" t = timeit.timeit('linalg.lu(A)', globals=globals(), number=num_repetitions)\n",
" t_scipy.append(t * 1000)\n",
" del A\n",
"prg_bar.layout.visibility = 'hidden'"
],
"execution_count": 6,
"outputs": [
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"source": [
"fig = plt.figure(figsize=(12,5))\n",
"fig.suptitle('Comparison of LU Decomposition functions')\n",
"gs = GridSpec(1,4)\n",
"ax1 = plt.subplot(gs[0,0])\n",
"ax2 = plt.subplot(gs[0,1:])\n",
"ax2.set_yticks([])\n",
"ax22 = ax2.twinx()\n",
"ax22.set_xlim([sizes[0], sizes[-1]])\n",
"\n",
"ax1.plot(sizes[:len(t_py)], t_py, '-x', label='python (v%s)'%(python_version()), c='r')\n",
"ax1.set_ylabel('Execution time (in sec)')\n",
"ax22.set_ylabel('Execution time (in sec)')\n",
"# ax22.plot(sizes[:len(t_py)], t_py, '-x', label='python (v%s)'%(python_version()), c='r')\n",
"ax22.plot(sizes, t_numba, '-x', label='numba (v%s)'%(numba.__version__), c='g')\n",
"ax22.plot(sizes, t_scipy, '-s', label='scipy (v%s)'%(scipy_version), c='b')\n",
"ax22.plot(sizes, t_cython, '-x', label='cython (v%s)'%(cpy_version), c='magenta')\n",
"ax22.set_ylabel('Execution time (in millisec)')\n",
"\n",
"for ax in [ax1, ax2, ax22]:\n",
" ax.minorticks_on()\n",
" ax.grid(True, which='major')\n",
" ax.grid(True, which='minor', lw=.2)\n",
" if ax != ax22:\n",
" ax.set_xlabel('Matrix dimension (NxN)')\n",
" if ax != ax2:\n",
" ax.legend()"
],
"execution_count": 7,
"outputs": [
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},
{
"cell_type": "code",
"metadata": {
"id": "-ydyQ_yKHsVJ",
"colab_type": "code",
"colab": {}
},
"source": [
""
],
"execution_count": 0,
"outputs": []
}
]
}
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