{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Calculation of polynomials, finding their roots" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Preliminaries" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "tags": [ "hide_code" ] }, "outputs": [], "source": [ "import numpy as np\n", "\n", "import matplotlib.pyplot as plt\n", "# Default style parameters (feel free to modify as you see fit)\n", "params = {'legend.fontsize': 'large',\n", " 'axes.labelsize': 'x-large',\n", " 'axes.titlesize':'x-large',\n", " 'xtick.labelsize':'x-large',\n", " 'ytick.labelsize':'x-large',\n", " 'xtick.direction':'in',\n", " 'ytick.direction':'in',\n", " }\n", "plt.rcParams.update(params)\n", "\n", "accuracy = 1.e-12" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "last_bisection_iterations = 0 # Count how many interactions it took\n", "bisection_verbose = False\n", "\n", "def bisection_method(\n", " f, # The function whose root we are trying to find\n", " a, # The left boundary\n", " b, # The right boundary\n", " tolerance = 1.e-10, # The desired accuracy of the solution\n", " ):\n", " fa = f(a) # The value of the function at the left boundary\n", " fb = f(b) # The value of the function at the right boundary\n", " if (fa * fb > 0.):\n", " return None # Bisection method is not applicable\n", " \n", " global last_bisection_iterations\n", " last_bisection_iterations = 0\n", " \n", " \n", " while ((b-a) > tolerance):\n", " last_bisection_iterations += 1\n", " c = (a + b) / 2. # Take the midpoint\n", " fc = f(c) # Calculate the function at midpoint\n", " \n", " \n", " if bisection_verbose:\n", " print(\"Iteration: {0:5}, c = {1:20.15f}, f(c) = {2:10.15f}\".format(last_bisection_iterations, c, fc))\n", " \n", " if (fc * fa < 0.): \n", " b = c # The midpoint is the new right boundary\n", " fb = fc\n", " else: \n", " a = c # The midpoint is the new left boundary\n", " fa = fc\n", "\n", " return (a+b) / 2. " ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "last_newton_iterations = 0\n", "newton_verbose = True\n", "\n", "def newton_method(\n", " f, # The function whose root we are trying to find\n", " df, # The derivative of the function\n", " x0, # The initial guess\n", " tolerance = 1.e-10, # The desired accuracy of the solution\n", " max_iterations = 100 # Maximum number of iterations\n", " ):\n", " \n", " xprev = xnew = x0\n", " \n", " global last_newton_iterations\n", " last_newton_iterations = 0\n", " diff = 0.\n", " \n", " if newton_verbose:\n", " print(\"Iteration: {0:5}, x = {1:20.15f}, f(x) = {2:10.15f}\".format(last_newton_iterations, x0, f(x0)))\n", " \n", " for i in range(max_iterations):\n", " last_newton_iterations += 1\n", " \n", " xprev = xnew\n", " fval = f(xprev) # The current function value\n", " dfval = df(xprev) # The current function derivative value\n", " \n", " if dfval == 0.:\n", " print(\"Newton's method: the derivative is zero!\")\n", " return None\n", " \n", " xnew = xprev - fval / dfval # The next iteration\n", " \n", " if newton_verbose:\n", " print(\"Iteration: {0:5}, x = {1:20.15f}, f(x) = {2:10.15f}\".format(last_newton_iterations, xnew, f(xnew)))\n", "\n", "\n", " if (abs(xnew-xprev) < tolerance):\n", " return xnew\n", " \n", " \n", " print(\"Newton-Raphson method failed to converge to a required precision in \" + str(max_iterations) + \" iterations\")\n", " # print(\"The error estimate is \", abs(xnew-xprev))\n", " \n", " return None " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Polynomial\n", "\n", "Polynomial $P(x)$ of order $n$ has a general expression\n", "$$\n", "P(x) = \\sum_{j=0}^n a_j \\, x^j\n", "$$\n", "\n", "For calculations we write it in a form\n", "$$\n", "P(x) = a_0 + x (a_1 + x ( \\ldots ))\n", "$$\n", "\n", "This allows on to define iterative procedure for calcualtion of $P(x)$ as well as its derivative $P'(x)$" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "# Evaluate the polynomial with coefficients a\n", "# at a point x\n", "def Poly(x,a):\n", " ret = a[len(a) - 1]\n", " for j in range(len(a) - 2, -1, -1):\n", " ret = ret * x + a[j]\n", " return ret\n", "\n", "# Evaluate the derivative of a polynomial \n", "# with coefficients a at a point x\n", "def dPoly(x,a):\n", " p = a[len(a) - 1]\n", " dp = 0.\n", " for j in range(len(a) - 2, -1, -1):\n", " dp = dp * x + p\n", " p = p * x + a[j]\n", " return dp" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us multiply $P(x)$ by $(x-c)$. The result is a polynomial $\\tilde{P}(x)$ of order $n+1$:\n", "$$\n", "\\tilde{P}(x) = (x-c) \\, P(x) = \\sum_{j=0}^{n+1} \\tilde{a}_j x^j.\n", "$$\n", "\n", "The coefficients $\\tilde{a}_j(x)$ can be readily expressed in terms of $a_j(x)$:\n", "$$\n", "\\tilde{a}_j = a_{j-1} - c \\, a_j, \\qquad j = 1,\\ldots,n+1,\n", "$$\n", "and\n", "$$\n", "\\tilde{a}_0 = -c a_0 .\n", "$$\n", "\n", "These relations can also be inverted to express coefficients $a_j$ of $P(x) = \\tilde{P(x)} / (x-c)$ in terms of $\\tilde{a}_j$ defining the division of a polynomial (deflation):\n", "$$\n", "a_j = \\tilde{a}_{j+1} + c \\, a_{j+1}, \\quad j = 0,\\ldots,n\n", "$$\n", "with $a_{n+1} = 0$." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "# Multiply polynomial by (x - c)\n", "def PolyMult(a,c):\n", " n = len(a)\n", " ret = a[:]\n", " ret.append(ret[-1])\n", " for j in range(n-1,0,-1):\n", " ret[j] = ret[j-1] - c * ret[j]\n", " ret[0] = -c * ret[0]\n", " return ret\n", "\n", "# Divide the polynomial by (x - c), \n", "# assuming x = c is one of the roots\n", "def PolyDiv(a,c):\n", " n = len(a) - 1\n", " ret = a[:]\n", " ret[-1] = 0.\n", " for j in range(n-1,-1,-1):\n", " ret[j] = a[j+1] + c * ret[j+1]\n", " ret.pop()\n", " return ret" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us now consider a specific example of a polynomial, namely the 6th order Legendre polynomial $P_6(x)$:\n", "$$\n", "P_6(x) = \\frac{1}{16} \\left( 231 x^6 - 315 x^4 + 105 x^2 - 5 \\right).\n", "$$" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "coefficients_Legendre6 = [-5./16., 0., \n", " 105./16., 0., \n", " -315./16., 0., \n", " 231./16.]\n", "\n", "def fP6(x):\n", " return Poly(x,coefficients_Legendre6)\n", "\n", "def dfP6(x):\n", " return dPoly(x,coefficients_Legendre6)\n", "\n", "xplot = np.linspace(-1,1,100)\n", "\n", "plt.xlabel(\"${x}$\")\n", "plt.ylabel('${P_6(x)}$')\n", "plt.plot(xplot,fP6(xplot),label = '${y_1}')\n", "plt.axhline(y = 0., color = 'black', linestyle = '--')\n", "#plt.plot([1.841406], [0], 'ro')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us use $P_6(x)$ to test polynomial multiplication and deflation" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Coefficients before multiplication: [-0.3125, 0.0, 6.5625, 0.0, -19.6875, 0.0, 14.4375]\n", " Coefficients after multiplication: [0.05252187499999998, -0.37515624999999997, -0.031084374999999775, 6.347031249999998, -18.106746875, 8.208906250000002, 42.132864375, -72.57403125, 19.38562500000001, 51.05624999999999, -50.53125, 14.4375]\n", " Coefficients after division: [-0.312499999999972, 2.6645352591003757e-14, 6.562500000000022, 1.4210854715202004e-14, -19.687499999999996, 0.0, 14.4375]\n" ] } ], "source": [ "coef1 = coefficients_Legendre6\n", "iters = 5\n", "xc = 0.7\n", "\n", "print(\"Coefficients before multiplication:\", coef1)\n", "for i in range(0,iters):\n", " coef1 = PolyMult(coef1,xc)\n", " \n", "print(\" Coefficients after multiplication:\", coef1)\n", "for i in range(0,iters):\n", " coef1 = PolyDiv(coef1,xc)\n", "\n", "print(\" Coefficients after division:\", coef1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We recover the coefficients after multiplying and dividing the polynomial by same factor several times in a row, although this procedure introduces visible round-off error. One thus has to be mindful of the error accumulation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Roots of the Legendre polynomials play an important role, for instance, when applied to numerical integration (such as Gauss-Legendre quadratures).\n", "\n", "Let us evaluate the roots using what we learned so far.\n", "\n", "$P_6(x)$ has six roots on the real axis. By plotting $P_6(x)$ we can easily figure out the brackets for each root and use the bisection method to evaluate the roots to desired precision" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "xplot = np.linspace(-1,1,100)\n", "\n", "plt.xlabel(\"${x}$\")\n", "plt.ylabel('${P_6(x)}$')\n", "plt.plot(xplot,fP6(xplot),label = '${P_6(x)}$')\n", "#plt.plot(xroots,[0. for k in range(0,6)], 'ro', label = 'roots')\n", "plt.plot([-1.,-1.],[-0.05,0.05], color='red')\n", "plt.plot([-0.75,-0.75],[-0.05,0.05], color='red')\n", "plt.plot([-0.5,-0.5],[-0.05,0.05], color='red')\n", "plt.plot([0.,0.],[-0.05,0.05], color='red')\n", "plt.plot([0.5,0.5],[-0.05,0.05], color='red')\n", "plt.plot([0.75,0.75],[-0.05,0.05], color='red')\n", "plt.plot([1.,1.],[-0.05,0.05], color='red')\n", "plt.axhline(y = 0., color = 'black', linestyle = '--')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Root 1 between -1.0 and -0.75 is x = -0.9324695142277051\n", "Root 2 between -0.75 and -0.5 is x = -0.6612093864532653\n", "Root 3 between -0.5 and 0.0 is x = -0.23861918607144617\n", "Root 4 between 0.0 and 0.5 is x = 0.23861918607144617\n", "Root 5 between 0.5 and 0.75 is x = 0.6612093864532653\n", "Root 6 between 0.75 and 1.0 is x = 0.9324695142277051\n" ] }, { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "xroots = []\n", "\n", "# Root 1\n", "xleft = -1. \n", "xright = -0.75\n", "xroots.append(bisection_method(fP6,xleft,xright))\n", "print(\"Root 1 between\", xleft, \"and\", xright, \"is x =\",xroots[-1])\n", "xleft = -0.75 \n", "xright = -0.5\n", "xroots.append(bisection_method(fP6,xleft,xright))\n", "print(\"Root 2 between\", xleft, \"and\", xright, \"is x =\",xroots[-1])\n", "xleft = -0.5\n", "xright = 0.\n", "xroots.append(bisection_method(fP6,xleft,xright))\n", "print(\"Root 3 between\", xleft, \"and\", xright, \"is x =\",xroots[-1])\n", "xleft = 0.\n", "xright = 0.5\n", "xroots.append(bisection_method(fP6,xleft,xright))\n", "print(\"Root 4 between\", xleft, \"and\", xright, \"is x =\",xroots[-1])\n", "xleft = 0.5 \n", "xright = 0.75\n", "xroots.append(bisection_method(fP6,xleft,xright))\n", "print(\"Root 5 between\", xleft, \"and\", xright, \"is x =\",xroots[-1])\n", "xleft = 0.75\n", "xright = 1.\n", "xroots.append(bisection_method(fP6,xleft,xright))\n", "print(\"Root 6 between\", xleft, \"and\", xright, \"is x =\",xroots[-1])\n", "\n", "xplot = np.linspace(-1,1,100)\n", "\n", "plt.xlabel(\"${x}$\")\n", "plt.ylabel('${P_6(x)}$')\n", "plt.plot(xplot,fP6(xplot),label = '${P_6(x)}$')\n", "plt.plot(xroots,[0. for k in range(0,6)], 'ro', label = 'roots')\n", "plt.axhline(y = 0., color = 'black', linestyle = '--')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Could we devise a procedure to find the roots without relying on manual inspection of the $P_6(x)$ plot which would be easily generalizable to other polynomials?" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "def PolyRoots(\n", " a, # The coefficients of the polynomial that we are solving\n", " x0 = -1., # The initial guess for the first root\n", " accuracy = 1.e-10, # The desired accuracy of the solution\n", " polishing = True, # Whether to polish the roots further with Newton's method\n", " max_iterations = 100 # Maximum number of iterations in Newton's method\n", "):\n", " ret = []\n", " n = len(a)\n", " apoly = a[:]\n", " current_root = x0\n", " \n", " def f(x):\n", " return Poly(x,apoly)\n", " def df(x):\n", " return dPoly(x,apoly)\n", " \n", " print(\"Searching all the roots using deflation and Newton's method\")\n", " # Loop over all the roots\n", " for k in range(0,n-1,1):\n", " current_root = newton_method(f,df,current_root,accuracy,max_iterations)\n", " if (current_root == None):\n", " print(\"Failed to find the next root!\")\n", " break\n", " ret.append(current_root)\n", " print(\"Root \", k+1, \"is x = \",current_root)\n", " # Deflate the polynomial\n", " apoly = PolyDiv(apoly, current_root)\n", " \n", " if polishing:\n", " print(\"Polishing the roots by reapplying Newton's method to the original polynomial\")\n", " apoly = a[:]\n", " for k in range(0,n-1,1):\n", " ret[k] = newton_method(f,df,ret[k],accuracy,max_iterations)\n", " print(\"Root \", k+1, \"is x = \",ret[k])\n", " \n", " return ret" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Searching all the roots using deflation and Newton's method\n", "Root 1 is x = -0.932469514203152\n", "Root 2 is x = -0.6612093864662645\n", "Root 3 is x = -0.23861918608319668\n", "Root 4 is x = 0.23861918608319652\n", "Root 5 is x = 0.6612093864662646\n", "Root 6 is x = 0.9324695142031523\n", "[-0.932469514203152, -0.6612093864662645, -0.23861918608319668, 0.23861918608319652, 0.6612093864662646, 0.9324695142031523]\n" ] }, { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x0 = -1.\n", "newton_verbose = False\n", "polishing = False\n", "xroots = PolyRoots(coefficients_Legendre6, x0, accuracy, polishing)\n", "print(xroots)\n", "xplot = np.linspace(-1.,1.,100)\n", "\n", "plt.xlabel(\"${x}$\")\n", "plt.ylabel('${P_6(x)}$')\n", "plt.plot(xplot,fP6(xplot),label = '${P_6(x)}$')\n", "plt.plot(xroots,[0. for k in range(0,len(xroots))], 'ro', label = 'roots')\n", "plt.axhline(y = 0., color = 'black', linestyle = '--')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can straightforwardly calculate the roots of other polynomials\n", "\n", "For example, $P_6(x)$:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Searching all the roots using deflation and Newton's method\n", "Root 1 is x = -0.9681602395076263\n", "Root 2 is x = -0.8360311073266355\n", "Root 3 is x = -0.6133714327005905\n", "Root 4 is x = -0.3242534234038087\n", "Root 5 is x = -2.9050086835705924e-16\n", "Root 6 is x = 0.32425342340380925\n", "Root 7 is x = 0.6133714327005846\n", "Root 8 is x = 0.8360311073266581\n", "Root 9 is x = 0.968160239507609\n" ] }, { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "coefficients_Legendre9 = [\n", " 0., 315./128., \n", " 0., -4620./128., \n", " 0., 18018./128., \n", " 0., -25740./128., \n", " 0., 12155./128.]\n", "\n", "x0 = -1.\n", "newton_verbose = False\n", "polishing = False\n", "xroots = PolyRoots(coefficients_Legendre9, x0, accuracy, polishing)\n", "xplot = np.linspace(-1.,1.,100)\n", "\n", "plt.xlabel(\"${x}$\")\n", "plt.ylabel('${P_{9}(x)}$')\n", "plt.plot(xplot,Poly(xplot,coefficients_Legendre9),label = '${P_{9}(x)}$')\n", "plt.plot(xroots,[0. for k in range(0,len(xroots))], 'ro', label = 'roots')\n", "plt.axhline(y = 0., color = 'black', linestyle = '--')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Hermite polynomials" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Searching all the roots using deflation and Newton's method\n", "Root 1 is x = -2.0201828704560856\n", "Root 2 is x = -0.9585724646138192\n", "Root 3 is x = 1.0319700683878226e-15\n", "Root 4 is x = 0.9585724646138175\n", "Root 5 is x = 2.0201828704560865\n", "Polishing the roots by reapplying Newton's method to the original polynomial\n", "Root 1 is x = -2.0201828704560856\n", "Root 2 is x = -0.9585724646138186\n", "Root 3 is x = 0.0\n", "Root 4 is x = 0.9585724646138185\n", "Root 5 is x = 2.0201828704560856\n" ] }, { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "coefficients_Hermite5 = [\n", " 0., 120.,\n", " 0., -160.,\n", " 0., 32.]\n", "\n", "x0 = -2.\n", "newton_verbose = False\n", "polishing = True\n", "xroots = PolyRoots(coefficients_Hermite5, x0, accuracy, polishing)\n", "\n", "xplot = np.linspace(-2.5,2.5,100)\n", "\n", "plt.xlabel(\"${x}$\")\n", "plt.ylabel('${H_{5}(x)}$')\n", "plt.plot(xplot,Poly(xplot,coefficients_Hermite5),label = '${H_{5}(x)}$')\n", "plt.plot(xroots,[0. for k in range(0,len(xroots))], 'ro', label = 'roots')\n", "plt.axhline(y = 0., color = 'black', linestyle = '--')\n", "plt.legend()\n", "plt.show()" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.10.16" } }, "nbformat": 4, "nbformat_minor": 4 }