{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(newtons-method)=\n",
    "# Newton's Method for Solving Equations"
   ]
  },
  {
   "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 above geometric rate $C^k$.\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 above connection between $g'(x)$ and $C$.\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 Newton's method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def newton_method(f, Df, x0, errorTolerance, maxIterations=20, demoMode=False):\n",
    "    \"\"\"Basic usage is:\n",
    "    (rootApproximation, errorEstimate, iterations) = newton_method(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 `newton_method` with\n",
    "\n",
    "    from numericalMethods import newton_method\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 = 0.7390851333852839 with estimated error 2.78e-05, backward error 2.85e-10\n",
      "At iteration 5 x = 0.7390851332151606 with estimated error 1.7e-10, backward error 1.11e-16\n",
      "\n",
      "The root is approximately 0.7390851332151606\n",
      "The estimated absolute error is 1.7e-10\n",
      "The backward error is 1.11e-16\n",
      "This required 4 iterations\n"
     ]
    }
   ],
   "source": [
    "(root, errorEstimate, iterations) = newton_method(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 [Measures of Error and Order of Convergence](error-measures-convergence-rates.ipynb)."
   ]
  },
  {
   "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](fixed-point-iteration-python.ipynb).\n",
    "Now $g$ is neither increasing nor decreasing at the fixed point,\n",
    "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"
     ]
    },
    {
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\n",
      "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 = 0.7438928778417369\n",
      "x_2 = 0.7390902113045812\n",
      "x_3 = 0.7390851332208545\n",
      "x_4 = 0.7390851332151607\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "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 {doc}`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 6.633694101535508e-17\n",
      "The backward error is 0.0\n",
      "This required 5 iterations\n"
     ]
    }
   ],
   "source": [
    "(root, errorEstimate, iterations) = newton_method(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)",
      "Input \u001b[0;32mIn [12]\u001b[0m, in \u001b[0;36m<cell line: 1>\u001b[0;34m()\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) = newton_method(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+0.6105344098423685j) 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 = (-0.9389355523964146+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) = newton_method(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 3.629748304956849e-15\n",
      "The backward error is 1.986e-15\n",
      "This required 9 iterations\n"
     ]
    }
   ],
   "source": [
    "(root3, errorEstimate3, iterations3) = newton_method(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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\n",
      "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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\n",
      "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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\n",
      "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 [Root Finding by Interval Halving](root-finding-by-interval-halving-python.ipynb)\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](taylors-theorem.ipynb).\n",
    "\n",
    "So we will return to the question of both the speed and accuracy of Newton's method in {doc}`newtons-method-convergence-rate`.\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",
    "This will also be explained in the section on [The Convergence Rate of Newton's Method](newtons-method-convergence-rate.ipynb)."
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    "---\n",
    "\n",
    "## Exercises"
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    "### 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) = newton_method(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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