{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(section:newtons-method)=\n",
    "# Newton's Method for Solving Equations\n",
    "\n",
    "Last revised on August 7, 2024"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**References:**\n",
    "\n",
    "- Sections 1.2 *Fixed-Point Iteration* and 1.4 *Newton's Method* of {cite}`Sauer`\n",
    "- Sections 2.2 *Fixed-Point Iteration* and 2.3 *Newton's Method and Its Extensions* of {cite}`Burden-Faires`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction\n",
    "\n",
    "Newton's method for solving equations has a number of advantages over the bisection method:\n",
    "- It is usually faster (but not always, and it can even fail completely!)\n",
    "- It can also compute complex roots, such as the non-real roots of polynomial equations.\n",
    "- It can even be adapted to solving systems of non-linear equations; that topic wil be visited later."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# We will often need resources from the modules numpy and pyplot:\n",
    "import numpy as np\n",
    "from numpy import sin, cos\n",
    "from matplotlib.pyplot import figure, plot, title, legend, grid"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "## Derivation as a contraction mapping with \"very small contraction coefficient $C$\"\n",
    "\n",
    "You might have previously seen Newton's method derived using tangent line approximations.\n",
    "That derivation is presented below,\n",
    "but first we approach it another way: as a particularly nice contraction mapping.\n",
    "\n",
    "To compute a root $r$ of a differentiable function $f$, we design a contraction mapping for which the contraction constant $C$ becomes arbitrarily small when we restrict to iterations in a sufficiently small interval around the root: $|x - r| \\leq R$.\n",
    "\n",
    "That is, the error ratio $|E_{k+1}|/|E_k|$ becomes ever smaller as the iterations get closer to the exact solution;\n",
    "the error is thus reducing ever faster than the geometric rate $C^k$ seen in the section on\n",
    "[fixed point iteration](section:fixed-point-iteration).\n",
    "\n",
    "This effect is in turn achieved by getting $|g'(x)|$ arbitrarily small for $|x - r| \\leq R$ with $R$ small enough,\n",
    "and then using the connection between $g'(x)$ and $C$ given in {prf:ref}`a-derivative-based-fixed-point-theorem`.\n",
    "This can be achieved by ensuring that $g'(r) = 0$ at a root $r$ of $f$ — so long as thr root $r$ is *simple*:\n",
    "$f'(r) \\neq 0$ (which is generically true, but not always).\n",
    "\n",
    "To do so, seek $g$ in the above form $g(x) = x - w(x)f(x)$, and choose $w(x)$ appropriately.\n",
    "At the root $r$,\n",
    "\n",
    "$$g'(r) = 1 - w'(r)f(r) - w(r)f'(r) = 1 - w(r)f'(r) \\quad\\text{(using } f(r) = 0,)$$\n",
    "\n",
    "so we ensure $g'(r) = 0$ by requiring $w(r) = 1/f'(r)$ (hence the problem if $f'(r) = 0$).\n",
    "\n",
    "We do not know $r$, but that does not matter!\n",
    "We can just choose $w(x) = 1/f'(x)$ for all $x$ values.\n",
    "That gives\n",
    "\n",
    "$$g(x) = x - {f(x)}/{f'(x)}$$\n",
    "\n",
    "and thus the iteration formula\n",
    "\n",
    "$$x_{k+1} = x_k - {f(x_k)}/{f'(x_k)}$$\n",
    "\n",
    "(That is, $g(x) = x - {f(x)}/{f'(x)}$.)\n",
    "\n",
    "You might recognize this as the formula for Newton's method."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To explore some examples of this, here is a Python function implementing this method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def newtonMethod(f, Df, x0, errorTolerance, maxIterations=20, demoMode=False):\n",
    "    \"\"\"Basic usage is:\n",
    "    (rootApproximation, errorEstimate, iterations) = newtonMethod(f, Df, x0, errorTolerance)\n",
    "    There is an optional input parameter \"demoMode\" which controls whether to\n",
    "    - print intermediate results (for \"study\" purposes), or to\n",
    "    - work silently (for \"production\" use).\n",
    "    The default is silence.\n",
    "    \"\"\"\n",
    "    if demoMode: print(\"Solving by Newton's Method.\")\n",
    "    x = x0\n",
    "    for k in range(maxIterations):\n",
    "        fx = f(x)\n",
    "        Dfx = Df(x)\n",
    "        # Note: a careful, robust code would check for the possibility of division by zero here,\n",
    "        # but for now I just want a simple presentation of the basic mathematical idea.\n",
    "        dx = fx/Dfx\n",
    "        x -= dx  # Aside: this is shorthand for \"x = x - dx\"\n",
    "        errorEstimate = abs(dx)\n",
    "        if demoMode:\n",
    "            print(f\"At iteration {k+1} x = {x} with estimated error {errorEstimate:0.3}, backward error {abs(f(x)):0.3}\")\n",
    "        if errorEstimate <= errorTolerance:\n",
    "            iterations = k\n",
    "            return (x, errorEstimate, iterations)\n",
    "    # If we get here, it did not achieve the accuracy target:\n",
    "    iterations = k\n",
    "    return (x, errorEstimate, iterations)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:remark} A Python module\n",
    ":label: module-numerical-methods\n",
    "\n",
    "From now on, all functions like this that implement numerical methods are also collected in the module file `numericalMethods.py`\n",
    "\n",
    "Thus, you could omit the above `def` and instead import `newtonMethod` with\n",
    "\n",
    "    from numericalMethods import newtonMethod\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:example}\n",
    ":label: example-newton-x-cosx\n",
    "Let's start with our favorite equation, $x = \\cos x$.\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:remark} On Python style\n",
    ":label: remark-python-style\n",
    "\n",
    "Since function names in Python (and most programming languages) must be alpha-numeric\n",
    "(with the underscore `_` as a \"special guest letter\"),\n",
    "I will avoid primes in notation for derivatives as much as possible:\n",
    "from now on, the derivative of $f$ is most often denoted as $Df$ rather than $f'$.\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f_1(x): return x - cos(x)\n",
    "def Df_1(x): return 1. + sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solving by Newton's Method.\n",
      "At iteration 1 x = 1.0 with estimated error 1.0, backward error 0.46\n",
      "At iteration 2 x = 0.7503638678402439 with estimated error 0.25, backward error 0.0189\n",
      "At iteration 3 x = 0.7391128909113617 with estimated error 0.0113, backward error 4.65e-05\n",
      "At iteration 4 x = 0.739085133385284 with estimated error 2.78e-05, backward error 2.85e-10\n",
      "At iteration 5 x = 0.7390851332151607 with estimated error 1.7e-10, backward error 0.0\n",
      "\n",
      "The root is approximately 0.7390851332151607\n",
      "The estimated absolute error is 1.7e-10\n",
      "The backward error is 0.0\n",
      "This required 4 iterations\n"
     ]
    }
   ],
   "source": [
    "(root, errorEstimate, iterations) = newtonMethod(f_1, Df_1, x0=0., errorTolerance=1e-8, demoMode=True)\n",
    "print()\n",
    "print(f\"The root is approximately {root}\")\n",
    "print(f\"The estimated absolute error is {errorEstimate:0.3}\")\n",
    "print(f\"The backward error is {abs(f_1(root)):0.3}\")\n",
    "print(f\"This required {iterations} iterations\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we have introduced another way of talking about errors and accuracy, which is further discussed in the section on\n",
    "[measures of error and order of convergence](section:error-measures-convergence-rates)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:definition} Backward Error\n",
    ":label: backward-error\n",
    "- The **backward error** in $\\tilde x$ as an approximation to a root of a function $f$ is $f(\\tilde x)$.\n",
    "\n",
    "- The **absolute backward** error is its absolute value, $|f(\\tilde x)|$.\n",
    "However sometimes the latter is simply called the backward error — as the above code does.\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This has the advantage that we can actually compute it without knowing the exact solution!\n",
    "\n",
    "The backward error also has a useful geometrical meaning:\n",
    "if the function $f$ were changed by this much to a nearbly function $\\tilde f$\n",
    "then $\\tilde x$ could be an exact root of $\\tilde f$.\n",
    "Hence, if we only know the values of $f$ to within this backward error\n",
    "(for example due to rounding error in evaluating the function)\n",
    "then $\\tilde x$ could well be an exact root,\n",
    "so there is no point in striving for greater accuracy in the approximate root.\n",
    "\n",
    "We will see this in the next example."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Graphing Newton's method iterations as a fixed point iteration\n",
    "\n",
    "Since this is a fixed point iteration with $g(x) = x - (x - \\cos(x)/(1 + \\sin(x))$,\n",
    "let us compare its graph to the ones seen in the section on [fixed point iteration](section:fixed-point-iteration).\n",
    "Now $g$ is neither increasing nor decreasing at the fixed point, so the graph has an unusual form."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def g(x):\n",
    "    return x - (x - cos(x))/(1 + sin(x))\n",
    "a = 0\n",
    "b = 1\n",
    "\n",
    "# An array of x values for graphing\n",
    "x = np.linspace(a, b)\n",
    "\n",
    "iterations = 4  # Not so many are needed now!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Starting near the left end of the domain\n",
      "x_0 = 0.1\n",
      "x_1 = 0.9137633861014282\n",
      "x_2 = 0.7446642419816996\n",
      "x_3 = 0.7390919659607759\n",
      "x_4 = 0.7390851332254692\n"
     ]
    },
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 800x800 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Start at left\n",
    "description = \"Starting near the left end of the domain\"\n",
    "print(description)\n",
    "x_k = 0.1\n",
    "print(f\"x_0 = {x_k}\")\n",
    "figure(figsize=(8,8))\n",
    "title(description)\n",
    "grid(True)\n",
    "plot(x, x, \"g\")\n",
    "plot(x, g(x), \"r\")\n",
    "for k in range(iterations):\n",
    "    g_x_k = g(x_k)\n",
    "    # Graph evalation of g(x_k) from x_k:\n",
    "    plot([x_k, x_k], [x_k, g(x_k)], \"b\")\n",
    "    x_k_plus_1 = g(x_k)\n",
    "    #Connect to the new x_k on the line y = x:\n",
    "    plot([x_k, g(x_k)], [x_k_plus_1, x_k_plus_1], \"b\")\n",
    "    # Update names: the old x_k+1 is the new x_k\n",
    "    x_k = x_k_plus_1\n",
    "    print(f\"x_{k+1} = {x_k}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Starting near the right end of the domain\n",
      "x_0 = 0.9\n",
      "x_1 = 0.7438928778417367\n",
      "x_2 = 0.7390902113045812\n",
      "x_3 = 0.7390851332208545\n",
      "x_4 = 0.7390851332151607\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 800x800 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Start at right\n",
    "description = \"Starting near the right end of the domain\"\n",
    "print(description)\n",
    "x_k = 0.9\n",
    "print(f\"x_0 = {x_k}\")\n",
    "figure(figsize=(8,8))\n",
    "title(description)\n",
    "grid(True)\n",
    "plot(x, x, \"g\")\n",
    "plot(x, g(x), \"r\")\n",
    "for k in range(iterations):\n",
    "    g_x_k = g(x_k)\n",
    "    # Graph evalation of g(x_k) from x_k:\n",
    "    plot([x_k, x_k], [x_k, g(x_k)], \"b\")\n",
    "    x_k_plus_1 = g(x_k)\n",
    "    #Connect to the new x_k on the line y = x:\n",
    "    plot([x_k, g(x_k)], [x_k_plus_1, x_k_plus_1], \"b\")\n",
    "    # Update names: the old x_k+1 is the new x_k\n",
    "    x_k = x_k_plus_1\n",
    "    print(f\"x_{k+1} = {x_k}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In fact, wherever you start, all iterations take you to the right of the root, and then approach the fixed point monotonically — and very fast.\n",
    "We will see an explanation for this in the section on [the convergence rate of Newton's method](section:newtons-method-convergence-rate)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:example} Pushing to the limits of standard 64-bit computer arithmetic\n",
    ":label: example-2\n",
    "\n",
    "Next, demand more accuracy; this time silently.\n",
    "As we will see in a later section, $10^{-16}$ is about the limit of the precision of standard (IEE64) computer arithmetic with 64-bit numbers.\n",
    "\n",
    "So let's try to compute the root as accurately as we can within these limits:\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "The root is approximately 0.7390851332151607\n",
      "The estimated absolute error is 0.0\n",
      "The backward error is 0.0\n",
      "This required 5 iterations\n"
     ]
    }
   ],
   "source": [
    "(root, errorEstimate, iterations) = newtonMethod(f_1, Df_1, x0=0, errorTolerance=1e-16)\n",
    "print()\n",
    "print(f\"The root is approximately {root}\")\n",
    "print(f\"The estimated absolute error is {errorEstimate}\")\n",
    "print(f\"The backward error is {abs(f_1(root)):0.4}\")\n",
    "print(f\"This required {iterations} iterations\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Observations:\n",
    "- It only took one more iteration to meet the demand for twice as many decimal places of accuracy.\n",
    "- The result is \"exact\" as fas as the computer arithmeric can tell, as shown by the zero backward error:\n",
    "we have indeed reached the accuracy limits of computer arithmetic."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Newton's method works with complex numbers too\n",
    "\n",
    "We will work almost entirely with real values and vectors in $\\mathbb{R}^n$, but actually,\n",
    "everything above also works for complex numbers.\n",
    "In particular, Newton's method works for finding roots of functions $f:\\mathbb{C} \\to \\mathbb{C}$;\n",
    "for example when seeking all roots of a polynomial."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:remark} Notation for complex number in Python\n",
    ":label: python-complex-numbers\n",
    "\n",
    "Python uses `j` for the square root of -1 (as is also sometimes done in engineering) rather than `i`.\n",
    "\n",
    "In general, the complex number $a + b i$ is expressed as `a+bj` (note: `j` at the end, and no spaces).\n",
    "As you might expect, imaginary numbers can be written without the $a$, as `bj`.\n",
    "\n",
    "However, the coefficient `b` is always needed, even when $b=1$: the square roots of -1 are `1j` and `-1j`,\n",
    "not `j` and `-j`, and the latter pair still refer to a variable `j` and its negation.\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(3+4j)\n",
      "5.0\n"
     ]
    }
   ],
   "source": [
    "z = 3+4j\n",
    "print(z)\n",
    "print(abs(z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1j\n"
     ]
    }
   ],
   "source": [
    "print(1j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(-0-1j)\n"
     ]
    }
   ],
   "source": [
    "print(-1j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "but:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'j' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[12], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[38;5;28mprint\u001b[39m(\u001b[43mj\u001b[49m)\n",
      "\u001b[0;31mNameError\u001b[0m: name 'j' is not defined"
     ]
    }
   ],
   "source": [
    "print(j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Giving `j` a value does not interfere:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "j = 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1j\n"
     ]
    }
   ],
   "source": [
    "print(1j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100\n"
     ]
    }
   ],
   "source": [
    "print(j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```{prf:example} All roots of a cubic\n",
    ":label: example-3\n",
    "\n",
    "As an example, let us seek all three cube roots of 8,\n",
    "by solving $x^3 - 8 = 0$ and trying different initial values $x_0$.\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f_2(x): return x**3 - 8\n",
    "def Df_2(x): return 3*x**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***First, $x_0 = 1$***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solving by Newton's Method.\n",
      "At iteration 1 x = 3.3333333333333335 with estimated error 2.33, backward error 29.0\n",
      "At iteration 2 x = 2.462222222222222 with estimated error 0.871, backward error 6.93\n",
      "At iteration 3 x = 2.081341247671579 with estimated error 0.381, backward error 1.02\n",
      "At iteration 4 x = 2.003137499141287 with estimated error 0.0782, backward error 0.0377\n",
      "At iteration 5 x = 2.000004911675504 with estimated error 0.00313, backward error 5.89e-05\n",
      "At iteration 6 x = 2.0000000000120624 with estimated error 4.91e-06, backward error 1.45e-10\n",
      "At iteration 7 x = 2.0 with estimated error 1.21e-11, backward error 0.0\n",
      "\n",
      "The first root is approximately 2.0\n",
      "The estimated absolute error is 1.2062351117801901e-11\n",
      "The backward error is 0.0\n",
      "This required 6 iterations\n"
     ]
    }
   ],
   "source": [
    "(root1, errorEstimate1, iterations1) = newtonMethod(f_2, Df_2, x0=1., errorTolerance=1e-8, demoMode=True)\n",
    "print()\n",
    "print(f\"The first root is approximately {root1}\")\n",
    "print(f\"The estimated absolute error is {errorEstimate1}\")\n",
    "print(f\"The backward error is {abs(f_2(root1)):0.4}\")\n",
    "print(f\"This required {iterations1} iterations\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***Next, start at $x_0 = i$ (a.k.a. $x_0 = j$):***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solving by Newton's Method.\n",
      "At iteration 1 x = (-2.6666666666666665+0.6666666666666667j) with estimated error 2.69, backward error 27.2\n",
      "At iteration 2 x = (-1.4663590926566705+0.6105344098423684j) with estimated error 1.2, backward error 10.2\n",
      "At iteration 3 x = (-0.23293230984230862+1.157138282313884j) with estimated error 1.35, backward error 7.21\n",
      "At iteration 4 x = (-1.920232195343855+1.5120026439880303j) with estimated error 1.72, backward error 13.4\n",
      "At iteration 5 x = (-1.1754417924325353+1.4419675366055333j) with estimated error 0.748, backward error 3.76\n",
      "At iteration 6 x = (-0.9389355523964147+1.7160019741718067j) with estimated error 0.362, backward error 0.741\n",
      "At iteration 7 x = (-1.0017352527552088+1.7309534907089796j) with estimated error 0.0646, backward error 0.0246\n",
      "At iteration 8 x = (-0.9999988050398477+1.7320490713246675j) with estimated error 0.00205, backward error 2.53e-05\n",
      "At iteration 9 x = (-1.0000000000014002+1.7320508075706016j) with estimated error 2.11e-06, backward error 2.67e-11\n",
      "At iteration 10 x = (-1+1.7320508075688774j) with estimated error 2.22e-12, backward error 1.99e-15\n",
      "\n",
      "The second root is approximately (-1+1.7320508075688774j)\n",
      "The estimated absolute error is 2.22e-12\n",
      "The backward error is 1.99e-15\n",
      "This required 9 iterations\n"
     ]
    }
   ],
   "source": [
    "(root2, errorEstimate2, iterations2) = newtonMethod(f_2, Df_2, x0=1j, errorTolerance=1e-8, demoMode=True)\n",
    "print()\n",
    "print(f\"The second root is approximately {root2}\")\n",
    "print(f\"The estimated absolute error is {errorEstimate2:0.3}\")\n",
    "print(f\"The backward error is {abs(f_2(root2)):0.3}\")\n",
    "print(f\"This required {iterations2} iterations\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This root is in fact $-1 + i \\sqrt{3}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***Finally, $x_0 = 1 - i$***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "The third root is approximately (-1-1.7320508075688772j)\n",
      "The estimated absolute error is 3.625286715636629e-15\n",
      "The backward error is 1.986e-15\n",
      "This required 9 iterations\n"
     ]
    }
   ],
   "source": [
    "(root3, errorEstimate3, iterations3) = newtonMethod(f_2, Df_2, x0=1-1j, errorTolerance=1e-8, demoMode=False)\n",
    "print()\n",
    "print(f\"The third root is approximately {root3}\")\n",
    "print(f\"The estimated absolute error is {errorEstimate3}\")\n",
    "print(f\"The backward error is {abs(f_2(root3)):0.4}\")\n",
    "print(f\"This required {iterations3} iterations\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This root is in fact $-1 - i \\sqrt{3}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Newton's method derived via tangent line approximations: linearization\n",
    "\n",
    "The more traditional derivation of Newton's method is based on the very widely useful idea of *linearization*;\n",
    "using the fact that a differentiable function can be approximated over a small part of its domain by a straight line — its tangent line — and it is easy to compute the root of this linear function.\n",
    "\n",
    "So start with a first approximation $x_0$ to a solution $r$ of $f(x) = 0$.\n",
    "\n",
    "### Step 1: Linearize at $x_0$.\n",
    "\n",
    "The tangent line to the graph of this function wih center $x_0$,\n",
    "also know as the *linearization of $f$ at $x_0$*, is\n",
    "\n",
    "$$L_0(x) = f(x_0) + f'(x_0) (x - x_0).$$\n",
    "\n",
    "(Note that $L_0(x_0) = f(x_0)$ and $L_0'(x_0) = f'(x_0)$.)\n",
    "\n",
    "### Step 2: Find the zero of this linearization\n",
    "\n",
    "Hopefully, the two functions $f$ and $L_0$ are close, so that the root of $L_0$ is close to a root of $f$;\n",
    "close enough to be a better approximation of the root $r$ than $x_0$ is.\n",
    "\n",
    "Give the name $x_1$ to this root of $L_0$:\n",
    "it solves $L_0(x_1) = f(x_0) + f'(x_0) (x_1 - x_0) = 0$, so\n",
    "\n",
    "$$x_1 = x_0 - {f(x_0)}/{f'(x_0)}$$\n",
    "\n",
    "### Step 3: Iterate\n",
    "\n",
    "We can then use this new value $x_1$ as the center for a new linearization\n",
    "$L_1(x) = f(x_1) + f'(x_1)(x - x_1)$,\n",
    "and repeat to get a hopefully even better approximate root,\n",
    "\n",
    "$$x_2 = x_1 - f(x_1)/f'(x_1)$$\n",
    "\n",
    "And so on: at each step, we get from approximation $x_k$ to a new one $x_{k+1}$ with\n",
    "\n",
    "$$x_{k+1} = x_k - {f(x_k)}/{f'(x_k)}$$\n",
    "\n",
    "And indeed this is the same formula seen above for Newton's method."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Illustration: a few steps of Newton's method for $x - \\cos(x) = 0$.**\n",
    "\n",
    "This approach to Newton's method via linearization and tangent lines suggests another graphical presentation;\n",
    "again we use the example of $f(x) = x - \\cos (x)$.\n",
    "This has $Df(x) = 1 + \\sin(x)$, so the linearization at center $a$ is\n",
    "\n",
    "$$L(x) = (a - \\cos(a)) + (1 + \\sin(a))(x-a)$$\n",
    "\n",
    "For Newton's method starting at $x_0 = 0$, this gives\n",
    "\n",
    "$$L_0(x) = -1 + x$$\n",
    "\n",
    "and its root — the next iterate in Newton's method — is $x_1 = 1$\n",
    "\n",
    "Then the linearization at center $x_1$ is\n",
    "\n",
    "$$\n",
    "L_1(x) = (1 - \\cos(1) + (1 + \\sin(1))(x-1),\n",
    "\\approx 0.4596 + 1.8415(x-1)\n",
    "$$\n",
    "giving $x_2 \\approx 1 - 0.4596/1.8415 \\approx 0.7504$.\n",
    "\n",
    "Let's graph a few steps."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "def L_0(x): return -1 + x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x_1=1.0\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1200x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=(12,6))\n",
    "title(\"First iteration, from $x_0 = 0$\")\n",
    "left = -0.1\n",
    "right = 1.1\n",
    "x = np.linspace(left, right)\n",
    "plot(x, f_1(x), label=\"$x - \\cos(x)$\")\n",
    "plot([left, right], [0, 0], \"k\", label=\"$x=0$\")  # The x-axis, in black\n",
    "x_0 = 0\n",
    "plot([x_0], [f_1(x_0)], \"g*\")\n",
    "plot(x, L_0(x), \"y\", label=\"$L_0(x)$\")\n",
    "plot([x_0], [f_1(x_0)], \"g*\")\n",
    "x_1 = x_0 - f_1(x_0)/Df_1(x_0)\n",
    "print(f\"{x_1=}\")\n",
    "plot([x_1], [0], \"r*\")\n",
    "legend()\n",
    "grid(True);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x_2=0.7503638678402439\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1200x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def L_1(x): return (x_1 - cos(x_1)) +  (1 + sin(x_1))*(x - x_1)\n",
    "\n",
    "figure(figsize=(12,6))\n",
    "title(\"Second iteration, from $x_1 = 1$\")\n",
    "# Shrink the domain\n",
    "left = 0.7\n",
    "right = 1.05\n",
    "x = np.linspace(left, right)\n",
    "\n",
    "plot(x, f_1(x), label=\"$x - \\cos(x)$\")\n",
    "plot([left, right], [0, 0], \"k\", label=\"$x=0$\")  # The x-axis, in black\n",
    "plot([x_1], [f_1(x_1)], \"g*\")\n",
    "plot(x, L_1(x), \"y\", label=\"$L_1(x)$\")\n",
    "x_2 = x_1 - f_1(x_1)/Df_1(x_1)\n",
    "print(f\"{x_2=}\")\n",
    "plot([x_2], [0], \"r*\")\n",
    "legend()\n",
    "grid(True);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1200x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def L_2(x): return (x_2 - cos(x_2)) +  (1 + sin(x_2))*(x - x_2)\n",
    "\n",
    "figure(figsize=(12,6))\n",
    "title(\"Third iteration, from $x_2$\")\n",
    "# Shrink the domain some more\n",
    "left = 0.735\n",
    "right = 0.755\n",
    "x = np.linspace(left, right)\n",
    "\n",
    "plot(x, f_1(x), label=\"$x - \\cos(x)$\")\n",
    "plot([left, right], [0, 0], \"k\", label=\"$x=0$\")  # The x-axis, in black\n",
    "plot([x_2], [f_1(x_2)], \"g*\")\n",
    "plot(x, L_2(x), \"y\", label=\"$L_2(x)$\")\n",
    "x_3 = x_2 - f_1(x_2)/Df_1(x_2)\n",
    "plot([x_3], [0], \"r*\")\n",
    "legend()\n",
    "grid(True);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How accurate and fast is this?\n",
    "\n",
    "For the bisection method, we have seen in the section on [root-finding by interval halving](section:root-finding-by-interval-halving)\n",
    "a fairly simple way to get an upper limit on the absolute error in the approximations.\n",
    "\n",
    "For absolute guarantees of accuracy, things do not go quite as well for Newton's method,\n",
    "but we can at least get a very \"probable\" *estimate* of how large the error can be.\n",
    "This requires some calculus, and more specifically Taylor's theorem,\n",
    "reviewed in the section on [Taylor's theorem](section:taylors-theorem).\n",
    "\n",
    "On the other hand, the example graphs above illustrate that the successive linearizations become ever more accurate as approximations of the function $f$ itself, so that the approximation $x_3$ looks \"perfect\" on the graph —\n",
    "the speed of Newton's method looks far better than for bisection.\n",
    "\n",
    "we will return to the question of both the speed and accuracy of Newton's method in section on\n",
    "[the convergence rate of Newton's method](section:newtons-method-convergence-rate)."
   ]
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    "---\n",
    "\n",
    "## Exercises"
   ]
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   "source": [
    "### Exercise 1\n",
    "\n",
    "a) Show that Newton's method applied to\n",
    "\n",
    "$$ f(x) = x^k - a $$\n",
    "\n",
    "leads to fixed point iteration with function\n",
    "\n",
    "$$ g(x) = \\frac{(k-1) x + \\displaystyle \\frac{a}{x^{k-1}}}{k}. $$\n",
    "\n",
    "b) Then verify mathematically that the iteration\n",
    "$x_{k+1} = g(x_k)$ has super-linear convergence."
   ]
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    "### Exercise 2\n",
    "\n",
    "a) Create a Python function for Newton's method,\n",
    "with usage\n",
    "    \n",
    "    (root, errorEstimate, iterations, functionEvaluations) = newtonMethod(f, Df, x_0, errorTolerance, maxIterations)\n",
    "\n",
    "(The last input parameter `maxIterations` could be optional, with a default like `maxIterations=100`.)\n",
    "\n",
    "b) based on your function `bisection2` create a third (and final!) version with usage\n",
    "\n",
    "    (root, errorBound, iterations, functionEvaluations) = bisection(f, a, b, errorTolerance, maxIterations)\n",
    "\n",
    "c) Use both of these to solve the equation\n",
    "\n",
    "$$ f_1(x) = 10 - 2x + \\sin(x) = 0 $$\n",
    "\n",
    "i) with (estimated) absolute error of no more than $10^{-6}$, and then\n",
    "\n",
    "ii) with (estimated) absolute error of no more than $10^{-15}$.\n",
    "\n",
    "Note in particular how many iterations and how many function evaluations are needed.\n",
    "\n",
    "Graph the function, which will help to find a good starting interval $[a, b]$ and initial approximation $x_0$.\n",
    "\n",
    "d) Repeat, this time finding the unique real root of\n",
    "\n",
    "$$ f_2(x) = x^3 - 3.3 x^2 + 3.63 x - 1.331 = 0 $$\n",
    "\n",
    "Again graph the function, to find a good starting interval $[a, b]$ and initial approximation $x_0$.\n",
    "\n",
    "e) This second case will behave differently than for $f_1$ in part (c): describe the difference.\n",
    "(We will discuss the reasons in class.)"
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