{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Newton's method with automatic differentiation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the following demo, we illustrate an example application of the `gradpy` package to determine the roots of a vector-valued, multivariate function using Newton's method." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%matplotlib inline\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from gradpy.autodiff import Var, Array" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We begin by initializing a vector-valued, multivariate function\n", "\n", "$$\n", "\\vec{f}(x,y) = \\left[\\begin{array}{ll}\n", "\\left(x^2+y^2-1\\right)^3-x^2y^3 \\\\\n", "\\dfrac{9}{4} x^2-\\left(y-\\dfrac{1}{2}\\right)^2-2\n", "\\end{array}\n", "\\right]\n", "$$\n", "\n", "as an `Array` object, defined in terms of `Var` variables $x$ and $y$. The zero contours of the individual components of $\\vec{f}$ and their intersections, which correspond to the solutions $\\vec{f}(x,y)=\\mathbf{0}$, are plotted for reference. Analytically we can determine the four intersection points to be\n", "\n", "$$\n", "(x,y) = \\left\\{(-1,1), (1,1), (1,0), (-1,0)\\right\\}\n", "$$" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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RDuatnBw83Ny4cuxYo6MIFyfFWji9QC8vLoqP56O8PNq6uoyOIxxEl9nM+7t2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\nvp6Fa9f6jiNhTGUt3tQ3NfHI8uVkDxjA5wYP9h1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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x = Var()\n", "y = Var()\n", "f = Array([(x**2 + y**2 - 1)**3 - x**2 * y**3, # heart-shaped contour\n", " 9./4.*x**2 - (y - 0.5)**2 - 2]) # hyperbola\n", "\n", "def plot_f(ax):\n", " # Plot the components and true zeros of f. Each component of f delineates a contour,\n", " # and their intersections define the zeros of f.\n", " xp, yp = np.meshgrid(np.arange(-3.5,3.51,0.01),np.arange(-3.5,3.51,0.01))\n", " xz = [-1,1,1,-1]\n", " yz = [1,1,0,0]\n", " cs1 = ax.contour(xp,yp,(xp**2+yp**2-1)**3-xp**2*yp**3,levels=[0],colors='hotpink')\n", " cs1.collections[0].set_label('$f$ [0]')\n", " cs2 = ax.contour(xp,yp,9./4.*xp**2-(yp-0.5)**2-2,levels=[0],colors='darkcyan')\n", " cs2.collections[0].set_label('$f$ [1]')\n", " ax.scatter(xz,yz,color='none',edgecolor='k',s=60,zorder=5,label='roots')\n", "\n", " ax.tick_params(labelsize=14)\n", " ax.legend(fontsize=12)\n", " ax.axis('equal')\n", " \n", "fig, ax = plt.subplots(1,1,figsize=(8,8))\n", "plot_f(ax)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we apply Newton's method to numerically determine the roots of $\\vec{f}(x,y)$; that is, the solutions to $\\vec{f}(x,y)=\\mathbf{0}$. Our implementation of a multivariate Newton's method which handles `autodiff` objects can be found in the supplementary module `newton_multivar.py`. A simple example use of `newton_multivar` is:\n", "\n", "``` Python\n", "root, res, Niter, traj = newton_multivar(f, [x,y], [-3.,2.], tol=1e-12, maxiter=100000)\n", "```\n", "\n", "The input arguments are the `autodiff Operation` or `Array` object f whose roots we seek, a list of independent `Var` variables [x,y], an initial guess for each variable (here [-3.,2.]), and optional settings for the desired tolerance on the 2-norm of the residual and maximum number of iterations. Returned are the roots, residual, number of iterations, and the complete trajectory of values taken by each independent variable through the Newton's iterations.\n", "\n", "We test four different initial points to locate each of the four roots, and plot the trajectories taken by successive Newton's method iterations in locating each root." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Initial guess: [-3.0, 2.0]\n", "----------------------\n", "Converged to root: [-1.0, 1.0]\n", "2-norm of residual: 3.025026694190274e-13\n", "Number of iterations: 12\n", "\n", "\n", "Initial guess: [3.0, 2.0]\n", "----------------------\n", "Converged to root: [1.0, 1.0]\n", "2-norm of residual: 3.025026694190274e-13\n", "Number of iterations: 12\n", "\n", "\n", "Initial guess: [3.0, -2.0]\n", "----------------------\n", "Converged to root: [1.00000038, -1.7e-06]\n", "2-norm of residual: 6.397105068111509e-13\n", "Number of iterations: 40\n", "\n", "\n", "Initial guess: [-3.0, -2.0]\n", "----------------------\n", "Converged to root: [-1.00000038, -1.7e-06]\n", "2-norm of residual: 6.397105068111509e-13\n", "Number of iterations: 40\n", "\n", "\n" ] }, { "data": { "image/png": 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a+nruXLqUlKgofjxlitPhqC5Cky3lNfOTk/nK8OE8tWED2adPOx2OUkq53fNb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\n8dpkjjyZFdtbcgurefXjY1TVu4kLN1FV7+bVj4/1aMI1d+5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7309ycjJjxoxBURRuvPHG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\nBOSnbADacPQoL2zZwk0jRnBOS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"text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from newton_multivar import newton_multivar\n", "\n", "fig, ax = plt.subplots(1,1,figsize=(10,10))\n", "# test four different initial points to locate each of 4 roots.\n", "g1 = [-3.,2.]\n", "g2 = [3.,2.]\n", "g3 = [3.,-2.]\n", "g4 = [-3.,-2.]\n", "for i,g in enumerate([g1,g2,g3,g4]):\n", " print(\"Initial guess:\",g)\n", " print(\"----------------------\")\n", " root, res, Niter, traj = newton_multivar(f,[x,y],g)\n", " print(\"Converged to root:\",[float('%.8f'%r) for r in root])\n", " print(\"2-norm of residual:\",res)\n", " print(\"Number of iterations:\",Niter)\n", " print(\"\\n\")\n", " ax.plot(traj[:,0],traj[:,1],'o-',label='trajectory '+str(i+1),alpha=0.6)\n", "\n", "plot_f(ax)\n", "ax.set_title(\"Applying Newton's method to find the roots of\\n\"+\n", " \"$\\\\vec{f}(x,y)=[(x^2+y^2-1)^3-x^2y^3$,\\n\"+\n", " \" $\\\\frac{9}{4}x^2-(y-\\\\frac{1}{2})^2-2]$\",size=14)\n", "plt.show()" ] } ], "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": "3.6.6" } }, "nbformat": 4, "nbformat_minor": 2 }