Created
January 15, 2014 23:07
-
-
Save baldwint/8446568 to your computer and use it in GitHub Desktop.
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| { | |
| "metadata": { | |
| "name": "" | |
| }, | |
| "nbformat": 3, | |
| "nbformat_minor": 0, | |
| "worksheets": [ | |
| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "#Using Sympy's `lambdify` with `IPython.parallel`\n", | |
| "\n", | |
| "For a numerical computation I am working on, I need to define a large Liouvillian matrix. Rather than program it in element by element, which is tedious and error-prone, I have used Sympy to construct it algebraically, and then use `lambdify` to make numpy matrices for use in numerical work.\n", | |
| "\n", | |
| "Here I construct a dumb sympy matrix that doesn't mean anything:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "import sympy as s\n", | |
| "from sympy.abc import x,y\n", | |
| "s.init_printing()\n", | |
| "\n", | |
| "element = lambda n, m : m * x**n if (n+m) % 3 else y\n", | |
| "L = s.Matrix([[element(n,m) for m in range(9)] for n in range(9)])\n", | |
| "L" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$\\left[\\begin{matrix}y & 1 & 2 & y & 4 & 5 & y & 7 & 8\\\\0 & x & y & 3 x & 4 x & y & 6 x & 7 x & y\\\\0 & y & 2 x^{2} & 3 x^{2} & y & 5 x^{2} & 6 x^{2} & y & 8 x^{2}\\\\y & x^{3} & 2 x^{3} & y & 4 x^{3} & 5 x^{3} & y & 7 x^{3} & 8 x^{3}\\\\0 & x^{4} & y & 3 x^{4} & 4 x^{4} & y & 6 x^{4} & 7 x^{4} & y\\\\0 & y & 2 x^{5} & 3 x^{5} & y & 5 x^{5} & 6 x^{5} & y & 8 x^{5}\\\\y & x^{6} & 2 x^{6} & y & 4 x^{6} & 5 x^{6} & y & 7 x^{6} & 8 x^{6}\\\\0 & x^{7} & y & 3 x^{7} & 4 x^{7} & y & 6 x^{7} & 7 x^{7} & y\\\\0 & y & 2 x^{8} & 3 x^{8} & y & 5 x^{8} & 6 x^{8} & y & 8 x^{8}\\end{matrix}\\right]$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAbMAAADgCAMAAAC+XnhYAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nlar7P0jGif8twfCUAAAAAElFTkSuQmCC\n", | |
| "prompt_number": 1, | |
| "text": [ | |
| "\u23a1y 1 2 y 4 5 y 7 8 \u23a4\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a20 x y 3\u22c5x 4\u22c5x y 6\u22c5x 7\u22c5x y \u23a5\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a2 2 2 2 2 2\u23a5\n", | |
| "\u23a20 y 2\u22c5x 3\u22c5x y 5\u22c5x 6\u22c5x y 8\u22c5x \u23a5\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a2 3 3 3 3 3 3\u23a5\n", | |
| "\u23a2y x 2\u22c5x y 4\u22c5x 5\u22c5x y 7\u22c5x 8\u22c5x \u23a5\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a2 4 4 4 4 4 \u23a5\n", | |
| "\u23a20 x y 3\u22c5x 4\u22c5x y 6\u22c5x 7\u22c5x y \u23a5\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a2 5 5 5 5 5\u23a5\n", | |
| "\u23a20 y 2\u22c5x 3\u22c5x y 5\u22c5x 6\u22c5x y 8\u22c5x \u23a5\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a2 6 6 6 6 6 6\u23a5\n", | |
| "\u23a2y x 2\u22c5x y 4\u22c5x 5\u22c5x y 7\u22c5x 8\u22c5x \u23a5\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a2 7 7 7 7 7 \u23a5\n", | |
| "\u23a20 x y 3\u22c5x 4\u22c5x y 6\u22c5x 7\u22c5x y \u23a5\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a2 8 8 8 8 8\u23a5\n", | |
| "\u23a30 y 2\u22c5x 3\u22c5x y 5\u22c5x 6\u22c5x y 8\u22c5x \u23a6" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 1 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "In the case of this example, I could have constructed a numpy matrix directly using the same nested loop, but this isn't true of the matrix in my actual problem. Anyway it is nice to see it written out in algebraic notation before plugging in numbers.\n", | |
| "\n", | |
| "I use `lambdify` to get a Numpy matrix for my numerical work:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "numer_L = s.lambdify((x,y), L, 'numpy')\n", | |
| "numer_L(3,4)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "prompt_number": 2, | |
| "text": [ | |
| "matrix([[ 4, 1, 2, 4, 4, 5, 4, 7, 8],\n", | |
| " [ 0, 3, 4, 9, 12, 4, 18, 21, 4],\n", | |
| " [ 0, 4, 18, 27, 4, 45, 54, 4, 72],\n", | |
| " [ 4, 27, 54, 4, 108, 135, 4, 189, 216],\n", | |
| " [ 0, 81, 4, 243, 324, 4, 486, 567, 4],\n", | |
| " [ 0, 4, 486, 729, 4, 1215, 1458, 4, 1944],\n", | |
| " [ 4, 729, 1458, 4, 2916, 3645, 4, 5103, 5832],\n", | |
| " [ 0, 2187, 4, 6561, 8748, 4, 13122, 15309, 4],\n", | |
| " [ 0, 4, 13122, 19683, 4, 32805, 39366, 4, 52488]])" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 2 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Say I want to do a computation involving this matrix (determinant, say) evaluated at multiple values of $y$:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# in series\n", | |
| "import numpy\n", | |
| "s_result = list(map(lambda y: numpy.linalg.det(numer_L(3,y)), range(30)))\n", | |
| "s_result" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$\\begin{bmatrix}0.0, & 9.9284346199e-51, & 1.17528889921e-49, & 0.0, & 0.0, & 0.0, & 0.0, & 0.0, & 0.0, & 1.32486805867e-62, & -1.46681392565e-30, & 0.0, & 0.0, & -7.24249763767e-49, & 0.0, & 2.48513164374e-45, & 0.0, & 0.0, & 6.83960894448e-29, & 0.0, & 0.0, & 2.98238783422e-29, & -2.62584900178e-45, & 0.0, & 0.0, & 0.0, & 0.0, & 1.22565792285e-25, & 0.0, & 0.0\\end{bmatrix}$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "prompt_number": 3, | |
| "text": [ | |
| "[0.0, 9.9284346199e-51, 1.17528889921e-49, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.324\n", | |
| "86805867e-62, -1.46681392565e-30, 0.0, 0.0, -7.24249763767e-49, 0.0, 2.4851316\n", | |
| "4374e-45, 0.0, 0.0, 6.83960894448e-29, 0.0, 0.0, 2.98238783422e-29, -2.6258490\n", | |
| "0178e-45, 0.0, 0.0, 0.0, 0.0, 1.22565792285e-25, 0.0, 0.0]" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 3 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# set up parallel environment. 2 engines started with `ipcluster start -n 2`\n", | |
| "from IPython.parallel import Client\n", | |
| "rc = Client()\n", | |
| "dview = rc[:]" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 4 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# in parallel\n", | |
| "# do imports and push our lambda function over\n", | |
| "dview.execute('import numpy')\n", | |
| "dview.push(dict(numer_L=numer_L))\n", | |
| "p_result = dview.map_sync(lambda y: numpy.linalg.det(numer_L(3,y)), range(30))\n", | |
| "p_result" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "ename": "CompositeError", | |
| "evalue": "one or more exceptions from call to method: <lambda>\n[0:apply]: NameError: global name 'ImmutableMatrix' is not defined\n[1:apply]: NameError: global name 'ImmutableMatrix' is not defined", | |
| "output_type": "pyerr", | |
| "traceback": [ | |
| "[0:apply]: ", | |
| "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", | |
| "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)\u001b[0;32m<string>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m", | |
| "\u001b[0;32m<ipython-input-5-1f431230550c>\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(y)\u001b[0m", | |
| "\u001b[0;32m/Users/tkb/.virtualenvs/sympy/lib/python2.7/site-packages/numpy/__init__.pyc\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(x, y)\u001b[0m", | |
| "\u001b[0;31mNameError\u001b[0m: global name 'ImmutableMatrix' is not defined", | |
| "", | |
| "[1:apply]: ", | |
| "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", | |
| "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)\u001b[0;32m<string>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m", | |
| "\u001b[0;32m<ipython-input-5-1f431230550c>\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(y)\u001b[0m", | |
| "\u001b[0;32m/Users/tkb/.virtualenvs/sympy/lib/python2.7/site-packages/numpy/__init__.pyc\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(x, y)\u001b[0m", | |
| "\u001b[0;31mNameError\u001b[0m: global name 'ImmutableMatrix' is not defined", | |
| "" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 5 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "That didn't work because apparently the lambda function needs `ImmutableMatrix` defined, which I hadn't ever heard of, and is not even the type of the matrix we lambdified:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "type(L)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "prompt_number": 6, | |
| "text": [ | |
| "sympy.matrices.dense.MutableDenseMatrix" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 6 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "In any case, I don't want my engines running any Sympy code. The task I want to distribute is numerical, not algebraic, and hopefully lambdify has generated numpy code that can run on its own." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "##My current workaround\n", | |
| "\n", | |
| "I inspected the `lambdify` source code and found a `lambdastr` function that ought to show me the code being generated:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "from sympy.utilities.lambdify import lambdastr\n", | |
| "lstr = lambdastr((x,y), L, dummify=True)\n", | |
| "lstr" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "prompt_number": 7, | |
| "text": [ | |
| "'lambda x,y: (ImmutableMatrix([[y, 1, 2, y, 4, 5, y, 7, 8], [0, x, y, 3*x, 4*x, y, 6*x, 7*x, y], [0, y, 2*x**2, 3*x**2, y, 5*x**2, 6*x**2, y, 8*x**2], [y, x**3, 2*x**3, y, 4*x**3, 5*x**3, y, 7*x**3, 8*x**3], [0, x**4, y, 3*x**4, 4*x**4, y, 6*x**4, 7*x**4, y], [0, y, 2*x**5, 3*x**5, y, 5*x**5, 6*x**5, y, 8*x**5], [y, x**6, 2*x**6, y, 4*x**6, 5*x**6, y, 7*x**6, 8*x**6], [0, x**7, y, 3*x**7, 4*x**7, y, 6*x**7, 7*x**7, y], [0, y, 2*x**8, 3*x**8, y, 5*x**8, 6*x**8, y, 8*x**8]]))'" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 7 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "It looks like this ought to work if I just import `numpy.matrix` as `ImmutableMatrix`, but no dice:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# in parallel\n", | |
| "# do imports and push our lambda function over\n", | |
| "dview.execute('import numpy')\n", | |
| "dview.execute('from numpy import matrix as ImmutableMatrix')\n", | |
| "dview.push(dict(numer_L=numer_L))\n", | |
| "p_result = dview.map_sync(lambda y: numpy.linalg.det(numer_L(3,y)), range(30))\n", | |
| "p_result" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "ename": "CompositeError", | |
| "evalue": "one or more exceptions from call to method: <lambda>\n[0:apply]: NameError: global name 'ImmutableMatrix' is not defined\n[1:apply]: NameError: global name 'ImmutableMatrix' is not defined", | |
| "output_type": "pyerr", | |
| "traceback": [ | |
| "[0:apply]: ", | |
| "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", | |
| "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)\u001b[0;32m<string>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m", | |
| "\u001b[0;32m<ipython-input-8-f4a5fb6c80d1>\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(y)\u001b[0m", | |
| "\u001b[0;32m/Users/tkb/.virtualenvs/sympy/lib/python2.7/site-packages/numpy/__init__.pyc\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(x, y)\u001b[0m", | |
| "\u001b[0;31mNameError\u001b[0m: global name 'ImmutableMatrix' is not defined", | |
| "", | |
| "[1:apply]: ", | |
| "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", | |
| "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)\u001b[0;32m<string>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m", | |
| "\u001b[0;32m<ipython-input-8-f4a5fb6c80d1>\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(y)\u001b[0m", | |
| "\u001b[0;32m/Users/tkb/.virtualenvs/sympy/lib/python2.7/site-packages/numpy/__init__.pyc\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(x, y)\u001b[0m", | |
| "\u001b[0;31mNameError\u001b[0m: global name 'ImmutableMatrix' is not defined", | |
| "" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 8 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "I got it to work by pushing the generated code as a string, and doing the eval myself:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# in parallel\n", | |
| "# do imports and push our lambda function over\n", | |
| "dview.execute('import numpy')\n", | |
| "dview.execute('from numpy import matrix as ImmutableMatrix')\n", | |
| "dview.push(dict(lstr=lstr))\n", | |
| "p_result = dview.map_sync(lambda y: numpy.linalg.det(eval(lstr)(3,y)), range(30))\n", | |
| "p_result" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$\\begin{bmatrix}0.0, & 9.9284346199e-51, & 1.17528889921e-49, & 0.0, & 0.0, & 0.0, & 0.0, & 0.0, & 0.0, & 1.32486805867e-62, & -1.46681392565e-30, & 0.0, & 0.0, & -7.24249763767e-49, & 0.0, & 2.48513164374e-45, & 0.0, & 0.0, & 6.83960894448e-29, & 0.0, & 0.0, & 2.98238783422e-29, & -2.62584900178e-45, & 0.0, & 0.0, & 0.0, & 0.0, & 1.22565792285e-25, & 0.0, & 0.0\\end{bmatrix}$$" | |
| ], | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "prompt_number": 9, | |
| "text": [ | |
| "[0.0, 9.9284346199e-51, 1.17528889921e-49, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.324\n", | |
| "86805867e-62, -1.46681392565e-30, 0.0, 0.0, -7.24249763767e-49, 0.0, 2.4851316\n", | |
| "4374e-45, 0.0, 0.0, 6.83960894448e-29, 0.0, 0.0, 2.98238783422e-29, -2.6258490\n", | |
| "0178e-45, 0.0, 0.0, 0.0, 0.0, 1.22565792285e-25, 0.0, 0.0]" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 9 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "p_result == s_result" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "prompt_number": 10, | |
| "text": [ | |
| "True" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 10 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "So, I kind of got things working, but it doesn't feel like the right way to do it, probably due to the way `lambdify` works. Is there a way to generate more parallelizable code from Sympy?" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "##Versions" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "import sys\n", | |
| "sys.version" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "prompt_number": 11, | |
| "text": [ | |
| "'2.7.3 (default, Nov 29 2012, 17:18:24) \\n[GCC 4.2.1 Compatible Apple Clang 4.1 ((tags/Apple/clang-421.11.66))]'" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 11 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "import IPython\n", | |
| "IPython.version_info" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "prompt_number": 12, | |
| "text": [ | |
| "(1, 1, 0, '')" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 12 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "s.__version__ # sympy" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "prompt_number": 13, | |
| "text": [ | |
| "'0.7.4.1'" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 13 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "numpy.__version__" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "metadata": {}, | |
| "output_type": "pyout", | |
| "prompt_number": 14, | |
| "text": [ | |
| "'1.8.0'" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 14 | |
| } | |
| ], | |
| "metadata": {} | |
| } | |
| ] | |
| } |
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment