{ "metadata": { "name": "", "signature": "sha256:a0f4cda82587c5b8e7a4502192d1210abedac9084077604eb60aea15f262a344" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# SymPy: Open Source Symbolic Mathematics\n", "\n", "This notebook uses the [SymPy](http://sympy.org) package to perform symbolic manipulations,\n", "and combined with numpy and matplotlib, also displays numerical visualizations of symbolically\n", "constructed expressions.\n", "\n", "We first load sympy printing extensions, as well as all of sympy:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.display import display\n", "\n", "from sympy.interactive import printing\n", "printing.init_printing(use_latex='mathjax')\n", "\n", "from __future__ import division\n", "import sympy as sym\n", "from sympy import *\n", "x, y, z = symbols(\"x y z\")\n", "k, m, n = symbols(\"k m n\", integer=True)\n", "f, g, h = map(Function, 'fgh')" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "

## Elementary operations

" ] }, { "cell_type": "code", "collapsed": false, "input": [ "Rational(3,2)*pi + exp(I*x) / (x**2 + y)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{3 \\pi}{2} + \\frac{e^{i x}}{x^{2} + y}$$" ], "metadata": {}, "output_type": "pyout", "prompt_number": 2, "text": [ " \u2148\u22c5x \n", "3\u22c5\u03c0 \u212f \n", "\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\n", " 2 2 \n", " x + y" ] } ], "prompt_number": 2 }, { "cell_type": "code", "collapsed": false, "input": [ "exp(I*x).subs(x,pi).evalf()" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$-1.0$$" ], "metadata": {}, "output_type": "pyout", "prompt_number": 3, "text": [ "-1.00000000000000" ] } ], "prompt_number": 3 }, { "cell_type": "code", "collapsed": true, "input": [ "e = x + 2*y" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 4 }, { "cell_type": "code", "collapsed": false, "input": [ "srepr(e)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 5, "text": [ "\"Add(Symbol('x'), Mul(Integer(2), Symbol('y')))\"" ] } ], "prompt_number": 5 }, { "cell_type": "code", "collapsed": false, "input": [ "exp(pi * sqrt(163)).evalf(50)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$262537412640768743.99999999999925007259719818568888$$" ], "metadata": {}, "output_type": "pyout", "prompt_number": 6, "text": [ "262537412640768743.99999999999925007259719818568888" ] } ], "prompt_number": 6 }, { "cell_type": "markdown", "metadata": {}, "source": [ "

## Calculus

" ] }, { "cell_type": "code", "collapsed": false, "input": [ "limit((sin(x)-x)/x**3, x, 0)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{1}{6}$$" ], "metadata": {}, "output_type": "pyout", "prompt_number": 14, "text": [ "-1/6" ] } ], "prompt_number": 14 }, { "cell_type": "code", "collapsed": false, "input": [ "(1/cos(x)).series(x, 0, 6)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$1 + \\frac{x^{2}}{2} + \\frac{5 x^{4}}{24} + \\mathcal{O}\\left(x^{6}\\right)$$" ], "metadata": {}, "output_type": "pyout", "prompt_number": 15, "text": [ " 2 4 \n", " x 5\u22c5x \u239b 6\u239e\n", "1 + \u2500\u2500 + \u2500\u2500\u2500\u2500 + O\u239dx \u23a0\n", " 2 24 " ] } ], "prompt_number": 15 }, { "cell_type": "code", "collapsed": false, "input": [ "diff(cos(x**2)**2 / (1+x), x)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{4 x \\cos{\\left (x^{2} \\right )}}{x + 1} \\sin{\\left (x^{2} \\right )} - \\frac{\\cos^{2}{\\left (x^{2} \\right )}}{\\left(x + 1\\right)^{2}}$$" ], "metadata": {}, "output_type": "pyout", "prompt_number": 16, "text": [ " \u239b 2\u239e \u239b 2\u239e 2\u239b 2\u239e\n", " 4\u22c5x\u22c5sin\u239dx \u23a0\u22c5cos\u239dx \u23a0 cos \u239dx \u23a0\n", "- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", " x + 1 2\n", " (x + 1) " ] } ], "prompt_number": 16 }, { "cell_type": "code", "collapsed": false, "input": [ "integrate(x**2 * cos(x), (x, 0, pi/2))" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$-2 + \\frac{\\pi^{2}}{4}$$" ], "metadata": {}, "output_type": "pyout", "prompt_number": 17, "text": [ " 2\n", " \u03c0 \n", "-2 + \u2500\u2500\n", " 4 " ] } ], "prompt_number": 17 }, { "cell_type": "code", "collapsed": false, "input": [ "eqn = Eq(Derivative(f(x),x,x) + 9*f(x), 1)\n", "display(eqn)\n", "dsolve(eqn, f(x))" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$9 f{\\left (x \\right )} + \\frac{d^{2}}{d x^{2}} f{\\left (x \\right )} = 1$$" ], "metadata": {}, "output_type": "display_data", "text": [ " 2 \n", " d \n", "9\u22c5f(x) + \u2500\u2500\u2500(f(x)) = 1\n", " 2 \n", " dx " ] }, { "latex": [ "$$f{\\left (x \\right )} = C_{1} \\sin{\\left (3 x \\right )} + C_{2} \\cos{\\left (3 x \\right )} + \\frac{1}{9}$$" ], "metadata": {}, "output_type": "pyout", "prompt_number": 18, "text": [ "f(x) = C\u2081\u22c5sin(3\u22c5x) + C\u2082\u22c5cos(3\u22c5x) + 1/9" ] } ], "prompt_number": 18 }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Illustrating Taylor series\n", "\n", "We will define a function to compute the Taylor series expansions of a symbolically defined expression at\n", "various orders and visualize all the approximations together with the original function" ] }, { "cell_type": "code", "collapsed": false, "input": [ "%matplotlib inline\n", "import numpy as np\n", "import matplotlib.pyplot as plt" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 19 }, { "cell_type": "code", "collapsed": true, "input": [ "# You can change the default figure size to be a bit larger if you want,\n", "# uncomment the next line for that:\n", "#plt.rc('figure', figsize=(10, 6))" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 20 }, { "cell_type": "code", "collapsed": true, "input": [ "def plot_taylor_approximations(func, x0=None, orders=(2, 4), xrange=(0,1), yrange=None, npts=200):\n", " \"\"\"Plot the Taylor series approximations to a function at various orders.\n", "\n", " Parameters\n", " ----------\n", " func : a sympy function\n", " x0 : float\n", " Origin of the Taylor series expansion. If not given, x0=xrange[0].\n", " orders : list\n", " List of integers with the orders of Taylor series to show. Default is (2, 4).\n", " xrange : 2-tuple or array.\n", " Either an (xmin, xmax) tuple indicating the x range for the plot (default is (0, 1)),\n", " or the actual array of values to use.\n", " yrange : 2-tuple\n", " (ymin, ymax) tuple indicating the y range for the plot. If not given,\n", " the full range of values will be automatically used. \n", " npts : int\n", " Number of points to sample the x range with. Default is 200.\n", " \"\"\"\n", " if not callable(func):\n", " raise ValueError('func must be callable')\n", " if isinstance(xrange, (list, tuple)):\n", " x = np.linspace(float(xrange[0]), float(xrange[1]), npts)\n", " else:\n", " x = xrange\n", " if x0 is None: x0 = x[0]\n", " xs = sym.Symbol('x')\n", " # Make a numpy-callable form of the original function for plotting\n", " fx = func(xs)\n", " f = sym.lambdify(xs, fx, modules=['numpy'])\n", " # We could use latex(fx) instead of str(), but matploblib gets confused\n", " # with some of the (valid) latex constructs sympy emits. So we play it safe.\n", " plt.plot(x, f(x), label=str(fx), lw=2)\n", " # Build the Taylor approximations, plotting as we go\n", " apps = {}\n", " for order in orders:\n", " app = fx.series(xs, x0, n=order).removeO()\n", " apps[order] = app\n", " # Must be careful here: if the approximation is a constant, we can't\n", " # blindly use lambdify as it won't do the right thing. In that case, \n", " # evaluate the number as a float and fill the y array with that value.\n", " if isinstance(app, sym.numbers.Number):\n", " y = np.zeros_like(x)\n", " y.fill(app.evalf())\n", " else:\n", " fa = sym.lambdify(xs, app, modules=['numpy'])\n", " y = fa(x)\n", " tex = sym.latex(app).replace('$', '')\n", " plt.plot(x, y, label=r'$n=%s:\\, %s\$' % (order, tex) )\n", " \n", " # Plot refinements\n", " if yrange is not None:\n", " plt.ylim(*yrange)\n", " plt.grid()\n", " plt.legend(loc='best').get_frame().set_alpha(0.8)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 21 }, { "cell_type": "markdown", "metadata": {}, "source": [ "With this function defined, we can now use it for any sympy function or expression" ] }, { "cell_type": "code", "collapsed": false, "input": [ "plot_taylor_approximations(sin, 0, [2, 4, 6], (0, 2*pi), (-2,2))" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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xR1rVbMXHHT/WWo7CxixJSmLR+fNENGtm1ueVj16RH5qmKVYUjlDZKIsd8xITGVGjhtYy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\nj+zsbFG9enVx5syZAttoOqPX+4aqNm3aULFiRa1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j5C9DlYffn/LOqBo9QLdu3ejWrVuh740cObLQ9rx584w9TcXTvbs84dm33z40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\nzcHdHdavL+s4pabmaw/qz6/WGv09as+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55OfnM3jwYJ555hmlYxlEjWXMy5cv07dvX0Au\nhbz00kt07txZ4VT6++6773jppZfIycnB09OTJUuWKB2pVG7fvs2mTZv0ujciHpgSBEEo59R7u18Q\nBEHQi2joBUEQyjnR0AuCIJRzoqEXBEEo50RDLwiCUM6Jhl4QBKGcEw29IAhCOScaekEQhHLu/wGT\ntYbY1r+2UgAAAABJRU5ErkJggg==\n", "text": [ "" ] } ], "prompt_number": 23 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This shows easily how a Taylor series is useless beyond its convergence radius, illustrated by \n", "a simple function that has singularities on the real axis:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# For an expression made from elementary functions, we must first make it into\n", "# a callable function, the simplest way is to use the Python lambda construct.\n", "plot_taylor_approximations(lambda x: 1/cos(x), 0, [2,4,6], (0, 2*pi), (-5,5))" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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FixfZtWuXYRqOHoVz56BJE6heXf9x6enp+Pn5eU+Yl5FFf3h4uMvHqghFoVAo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\nMXEPYHMgez/kK1ciiIkRvR6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