{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solving Equations by Fixed Point Iteration (of Contraction Mappings)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**References:**\n",
    "\n",
    "- Section 1.2 of [Sauer](../references.html#sauer)\n",
    "- Section 2.2 of [Burden&Faires](../references.html#Burden-Faires)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction\n",
    "\n",
    "in the next section we will meet [Newton's Method](newtons-method-python.ipynb)\n",
    "for root-finding, which you might have seen in a calculus course.\n",
    "This is one very important example of a more genetal strategy of **fixed-point iteration**, so we start with that."
   ]
  },
  {
   "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 matplotlib import pyplot as plt\n",
    "\n",
    "# We can also import items from a module individually, so they can be used by \"first name only\".\n",
    "# Here this is done for mathematical functions; in some later sections it will be done for all imports.\n",
    "from numpy import cos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fixed-point equations\n",
    "\n",
    "A variant of stating equations as *root-finding* ($f(x) = 0$) is *fixed-point* form:\n",
    "given a function $g:\\mathbb{R} \\to \\mathbb{R}$ or $g:\\mathbb{C} \\to \\mathbb{C}$\n",
    "(or even $g:\\mathbb{R}^n \\to \\mathbb{R}^n$; a later topic),\n",
    "find a *fixed point* of $g$.\n",
    "That is, a value $p$ for its argument such that\n",
    "\n",
    "$$g(p) = p$$\n",
    "\n",
    "Such problems are interchangeable with root-finding.\n",
    "One way to convert from $f(x) = 0$ to $g(x) = x$ is defining\n",
    "\n",
    "$$g(x) := x - w(x) f(x)$$\n",
    "\n",
    "for any \"weight function\" $w(x)$.\n",
    "\n",
    "One can convert the other way too, for example defining $f(x) := g(x) - x$.\n",
    "We have already seen this when we converted the equation $x = \\cos x$ to $f(x) = x - \\cos x = 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compare the two setups graphically: in each case, the $x$ value at the intersection of the two curves is the solution we seek."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f_1(x): return x - cos(x)\n",
    "def g_1(x): return cos(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = f_1\n",
    "g = g_1\n",
    "a = -1\n",
    "b = 1\n",
    "\n",
    "x = np.linspace(a, b)\n",
    "\n",
    "plt.figure(figsize=(12,6))\n",
    "plt.title('$y = f(x) = x - \\cos(x)$ and $y=0$')\n",
    "plt.plot(x, f(x))\n",
    "plt.plot([a, b], [0, 0])\n",
    "plt.grid(True)\n",
    "\n",
    "plt.figure(figsize=(12,6))\n",
    "plt.title('$y = g(x) = \\cos (x)$ and $y=x$')\n",
    "plt.plot(x, g(x))\n",
    "plt.plot(x, x)\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The fixed point form can be convenient partly because we almost always have to solve by successive approximations,\n",
    "or *iteration*, and fixed point form suggests _one_ choice of iterative procedure:\n",
    "start with any first approximation $x_0$, and iterate with\n",
    "\n",
    "$$\n",
    "x_1 = g(x_0), \\, x_2 = g(x_1), \\dots, x_{k+1} = g(x_k), \\dots\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proposition 1.**\n",
    "If $g$ is continuous, and **if** the above sequence $\\{x_0, x_1, \\dots \\}$ converges to a limit $p$, then that limit is a fixed point of function $g$: $g(p) = p$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proof:**\n",
    "*From $\\displaystyle \\lim_{k \\to \\infty} x_k = p$, continuity gives*\n",
    "\n",
    "$$\\lim_{k \\to \\infty} g(x_k) = g(p).$$\n",
    "\n",
    "*On the other hand, $g(x_k) = x_{k+1}$, so*\n",
    "\n",
    "$$\\lim_{k \\to \\infty} g(x_k) = \\lim_{k \\to \\infty} x_{k+1} = p.$$\n",
    "\n",
    "*Comparing gives $g(p) = p$.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That second \"if\" is a big one.\n",
    "Fortunately, it can often be resolved using the idea of a *contraction mapping*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Definiton 1: Mapping.**\n",
    "A function $g(x)$ defined on a closed interval $D = [a, b]$ which sends values back into that interval,\n",
    "$g: D \\to D$, is sometimes called a *map* or *mapping*.\n",
    "\n",
    "(*Aside:* The same applies for a function $g: D \\to D$ where $D$ is a subset of the complex numbers,\n",
    "or even of vectors $\\mathbb{R}^n$ or $\\mathbb{C}^n$.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A mapping is sometimes thought of as moving a region $S$ within its domain $D$ to another such region, by moving each point\n",
    "$x \\in S \\subset D$\n",
    "to its image\n",
    "$g(x) \\in g(S) \\subset D$.\n",
    "\n",
    "A very important case is mappings that shrink the region, by reducing the distance between points:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proposition 2:**\n",
    "Any continuous mapping on a closed interval $[a, b]$ has at least one fixed point."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proof:**\n",
    "*Consider the \"root-finding cousin\", $f(x) = x - g(x)$.*\n",
    "\n",
    "*First, $f(a) = a - g(a) \\leq 0$, since $g(a) \\geq a$ so as to be in the domain $[a,b]$ — similarly, $f(b) = b - g(b) \\geq 0$.*\n",
    "\n",
    "*From the Intermediate Value Theorem, $f$ has a zero $p$, where $f(p) = p - g(p) = 0$.*\n",
    "\n",
    "*In other words, the graph of $y=g(x)$ goes from being above the line $y=x$ at $x=a$ to below it at $x=b$,\n",
    "so at some point $x=p$, the curves meet: $y = x = p$ and $y = g(p)$, so $p = g(p)$.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 1\n",
    "\n",
    "Let us illustrate this with the mapping $g_4(x) := 4 \\cos x$,\n",
    "for which the fact that $|g_4(x)| \\leq 4$ ensures that this is a map of the domain $D = [-4, 4]$ into itself:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def g_4(x): return 4*cos(x)\n",
    "a = -4\n",
    "b = 4\n",
    "x = np.linspace(a, b)\n",
    "\n",
    "plt.figure(figsize=(12,6))\n",
    "plt.title('Fixed points of the map $g_4(x) = 4 \\cos(x)$')\n",
    "plt.plot(x, g_4(x), label='$y=g_4(x)$')\n",
    "plt.plot(x, x, label='$y=x$')\n",
    "ignore_me = plt.legend()  # Note: the bogus output value \"ignore_me\" supresses some useless output"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This example has multiple fixed points (three of them).\n",
    "To ensure both the existence of a **unique** solution, and covergence of the iteration to that solution, we need an extra condition."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Definition 2: Contraction Mapping**.\n",
    "A mapping $g:D \\to D$, is called a *contraction* or *contraction mapping* if there is a constant $C < 1$ such that\n",
    "\n",
    "$$|g(x) - g(y)| \\leq C |x - y|$$\n",
    "\n",
    "for any $x$ and $y$ in $D$.\n",
    "We then call $C$ a *contraction constant*.\n",
    "\n",
    "(*Aside:* The same applies for a domain in $\\mathbb{R}^n$: just replace the absolute value $| \\dots |$ by the vector norm\n",
    "$\\| \\dots \\|$.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Note:** it is not enough to have $| g(x) - g(y) | < | x - y |$ or $C = 1$!\n",
    "We need the ratio\n",
    "$\\displaystyle \\frac{|g(x) - g(y)|}{|x - y|}$\n",
    "to be *uniformly* less than one for all possible values of $x$ and $y$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name=\"CMT\"></a>\n",
    "**Theorem 1 (A Contraction Mapping Theorem).**\n",
    "Any contraction mapping $g$ on a closed, bounded interval $D = [a, b]$ has exactly one fixed point $p$ in $D$.\n",
    "Further, this can be calculated as the limit $\\displaystyle p = \\lim_{k \\to \\infty} x_k$ of the iteration sequence given by $x_{k+1} = g(x_{k})$ for *any* choice of the starting point $x_{0} \\in D$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proof:**\n",
    "*The main idea of the proof can be shown with the help of a few pictures.\n",
    "<br>\n",
    "First, uniqeness:\n",
    "between any two of the multiple fixed points above — call them $p_0$ and $p_1$ — the graph of $g(x)$ has to rise with secant slope 1: $(g(p_1) - g(p_0)/(p_1 - p_0) = (p_1 - p_0)/(p_1 - p_0) = 1$, and this violates the contraction property.*\n",
    "\n",
    "*So instead, for a contraction, the graph of a contraction map looks like the one below for our favorite example,\n",
    "$g(x) = \\cos x$ (which we will soon verify to be a contraction on interval $[-1, 1]$):*\n",
    "\n",
    "The second claim, about **convergence** to the fixed point from any initial approximation $x_0$,\n",
    "will be verified below, once we have seen some ideas about measuring errors."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### An easy way of checking whether a differentiable function is a contraction\n",
    "\n",
    "With differentiable functions, the contraction condition can often be easily verified using derivatives:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Theorem 2 (A derivative-based fixed point theorem).**\n",
    "If a function $g:[a,b] \\to [a,b]$ is differentiable and there is a constant $C < 1$ such that $|g'(x)| \\leq C$ for all $x \\in [a, b]$, then $g$ is a contraction mapping, and so has a unique fixed point in this interval."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proof:**\n",
    "*Using the Mean Value Theorem, $g(x) - g(y) = g'(c)(x - y)$ for some $c$ between $x$ and $y$;\n",
    "<br>\n",
    "then taking absolute values,*\n",
    "\n",
    "$$|g(x) - g(y)| = |g'(c)|\n",
    "\\cdot |(x - y)| \\leq C |(x - y)|.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 2. $g(x) = \\cos x$ is a contraction on domain $[-1, 1]$\n",
    "\n",
    "Our favorite example $g(x) = \\cos(x)$ is a contraction, but we have to be a bit careful about the domain.\n",
    "\n",
    "For all real $x$, $g'(x) = -\\sin x$, so $|g'(x)| \\leq 1$; this is almost but not quite enough.\n",
    "\n",
    "However, we have seen that iteration values will settle in the interval $D = [-1,1]$,\n",
    "and considering $g$ as a mapping of this domain,\n",
    "$|g'(x)| \\leq \\sin(1) = 0.841\\dots < 1$: that is, now we have a contraction, with $C = \\sin(1) \\approx 0.841$.\n",
    "\n",
    "And as seen in the graph above, there is indeed a unique fixed point."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The contraction constant $C$ as a measure of how fast the approximations improve (the smaller the better)\n",
    "\n",
    "It can be shown that if $C$ is small (at least when one looks only at a reduced domain $|x - p| < R$) then the convergence is \"fast\" once $|x_k - p| < R$.\n",
    "\n",
    "To see this, we define some jargon for talking about errors.\n",
    "(For more details on error concepts, see the section on\n",
    "[Measures of Error and Convergence Rates](error-measures-convergence-rates.ipynb))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Definition 3: Error**.\n",
    "The *error* in $\\tilde x$ as an approximation to an exact value $x$ is\n",
    "\n",
    "$$\\text{error} := \\text{(approximation)} - \\text{(exact value)} = \\tilde x - x$$\n",
    "\n",
    "This will often be abbreviated as $E$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Definition 4: Absolute Error.**\n",
    "The *absolute error* in $\\tilde x$ an approximation to an exact value $x$ is the magnitude of the error:\n",
    "the absolute value $|E| = |\\tilde x - x|$.\n",
    "\n",
    "(*Aside:* This will later be extended to $x$ and $\\tilde x$ being vectors,\n",
    "by again using the vector norm in place of the absolute value.\n",
    "In fact, I will sometimes blur the distinction by using the \"single line\" absolute value notation for vector norms too.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the case of $x_k$ as an approximation of $p$, we name the error $E_k := x_k - p$.\n",
    "Then $C$ measures a worst case for how fast the error decreases as $k$ increases, and this is \"exponentially fast\":"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proposition 3.**\n",
    "$|E_{k+1}| \\leq C |E_{k}|$, or $|E_{k+1}|/|E_{k}|\\leq C$,\n",
    "and so\n",
    "\n",
    "$$|E_k| \\leq C^k |x_0 - p|$$\n",
    "\n",
    "That is, the error decreases at worst in a geometric sequence,\n",
    "which is exponential decrease with respect to the variable $k$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Proof.**\n",
    "\n",
    "*$E_{k+1} = x_{k+1} - p = g(x_{k}) - g(p)$, using $g(p) = p$.\n",
    "Thus the contraction property gives*\n",
    "\n",
    "$$|E_{k+1}| = |g(x_k) - g(p)| \\leq C |x_k - p| = C |E_k|$$\n",
    "\n",
    "*Applying this again,*\n",
    "\n",
    "$$|E_k| \\leq C |E_{k-1}| \\leq C \\cdot C |E_{k-2}| = C^2 |E_{k-2}|$$\n",
    "\n",
    "*and repeating $k-2$ more times,*\n",
    "\n",
    "$$|E_k|\\leq  C^k |E_0| = C^k |x_0 - p|.$$\n",
    "\n",
    "**Aside:** We will often use this \"recursive\" strategy of relating the error in one iterate to that in the previous iterate."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**The rest of the proof of the Contraction Mapping Theorem (Theorem 5): guaranteed convergence.**\n",
    "\n",
    "*This now follows from the above proposition:\n",
    "<br>\n",
    "for* **any** *initial approximation $x_0$, we know that\n",
    "$|E_k|\\leq C^k |x_0 - p|$, and with $C < 1$, this can be made as small as we want by choosing a large enough value of $k$.\n",
    "Thus*\n",
    "\n",
    "$$\\lim_{k \\to \\infty} |E_k| = \\lim_{k \\to \\infty} |x_k - p| = 0,$$\n",
    "*which is another way of saying that $\\displaystyle \\lim_{k \\to \\infty} x_k = p$, or $x_k \\to p$, as claimed.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3. Solving $x = \\cos x$ with a naive fixed point iteration\n",
    "\n",
    "We have seen that _one_ way to convert the example\n",
    "$f(x) = x - \\cos x = 0$ to a fixed point iteration is $g(x) = \\cos x$,\n",
    "and that this is a contraction on $D = [-1, 1]$\n",
    "\n",
    "Here is what this iteration looks like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solving x = cos(x) starting to the left, at x_0 = 0\n",
      "x_1 = 1.0\n",
      "x_2 = 0.5403023058681398\n",
      "x_3 = 0.8575532158463934\n",
      "x_4 = 0.6542897904977791\n",
      "x_5 = 0.7934803587425656\n",
      "x_6 = 0.7013687736227565\n",
      "x_7 = 0.7639596829006542\n",
      "x_8 = 0.7221024250267079\n",
      "x_9 = 0.7504177617637604\n",
      "x_10 = 0.7314040424225099\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solving x = cos(x) starting to the right, at x_0 = 1\n",
      "x_1 = 0.5403023058681398\n",
      "x_2 = 0.8575532158463934\n",
      "x_3 = 0.6542897904977791\n",
      "x_4 = 0.7934803587425656\n",
      "x_5 = 0.7013687736227565\n",
      "x_6 = 0.7639596829006542\n",
      "x_7 = 0.7221024250267079\n",
      "x_8 = 0.7504177617637604\n",
      "x_9 = 0.7314040424225099\n",
      "x_10 = 0.7442373549005569\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "g = g_1\n",
    "a = 0\n",
    "b = 1\n",
    "x = np.linspace(a, b)\n",
    "iterations = 10\n",
    "\n",
    "# Start at left\n",
    "print(f\"Solving x = cos(x) starting to the left, at x_0 = {a}\")\n",
    "x_k = a\n",
    "plt.figure(figsize=(8,8))\n",
    "plt.title(f\"Solving $x = \\cos x$ starting to the left, at $x_0$ = {a}\")\n",
    "plt.plot(x, x, 'g')\n",
    "plt.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",
    "    plt.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",
    "    plt.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_plus_1}\")\n",
    "\n",
    "plt.show()\n",
    "\n",
    "# Start at right\n",
    "print(f\"Solving x = cos(x) starting to the right, at x_0 = {b}\")\n",
    "x_k = b\n",
    "plt.figure(figsize=(8,8))\n",
    "plt.title(f\"Solving $x = \\cos(x)$ starting to the right, at $x_0$ = {b}\")\n",
    "plt.plot(x, x, 'g')\n",
    "plt.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",
    "    plt.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",
    "    plt.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_plus_1}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In each case, one gets a \"box spiral\" in to the fixed point.\n",
    "It always looks like this when $g$ is *decreasing* near the fixed point.\n",
    "\n",
    "If instead $g$ is *increasing* near the fixed point, the iterates approach monotonically, either from above or below:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 4. Solving $f(x) = x^2 - 5x + 4 = 0$ in interval $[0, 3]$\n",
    "\n",
    "The roots are 1 and 4; for now we aim at the first of these,\n",
    "so we chose a domain $[0, 3]$ that contains just this root.\n",
    "\n",
    "Let us get a fixed point for by \"partially solving for $x$\": solving for the $x$ in the $5 x$ term:\n",
    "\n",
    "$$x = g(x) = (x^2 + 4)/5$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f_2(x): return x**2 - 5*x + 4\n",
    "def g_2(x): return (x**2 + 4)/5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = f_2\n",
    "g = g_2\n",
    "a = 0\n",
    "b = 3\n",
    "x = np.linspace(a, b)\n",
    "\n",
    "plt.figure(figsize=(12,6))\n",
    "plt.title('$y = f_2(x) = x^2-5x+4$ and $y = 0$')\n",
    "plt.plot(x, f(x))\n",
    "plt.plot([a, b], [0, 0])\n",
    "plt.grid(True)\n",
    "\n",
    "plt.figure(figsize=(12,6))\n",
    "plt.title('$y = g_2(x) = (x^2 + 4)/5$ and $y=x$')\n",
    "plt.plot(x, g(x))\n",
    "plt.plot(x, x)\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Starting to the left, at x_0 = 0\n",
      "x_1 = 0.8\n",
      "x_2 = 0.9280000000000002\n",
      "x_3 = 0.9722368000000001\n",
      "x_4 = 0.9890488790548482\n",
      "x_5 = 0.9956435370319303\n",
      "x_6 = 0.9982612105666906\n",
      "x_7 = 0.9993050889044148\n",
      "x_8 = 0.999722132142052\n",
      "x_9 = 0.9998888682989302\n",
      "x_10 = 0.999955549789623\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Starting to the right, at x_0 = 3\n",
      "x_1 = 2.6\n",
      "x_2 = 2.152\n",
      "x_3 = 1.7262208\n",
      "x_4 = 1.3959676500705283\n",
      "x_5 = 1.1897451360086866\n",
      "x_6 = 1.0830986977312658\n",
      "x_7 = 1.0346205578054328\n",
      "x_8 = 1.014087939726725\n",
      "x_9 = 1.0056748698998388\n",
      "x_10 = 1.0022763887896116\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "iterations = 10\n",
    "g = g_2\n",
    "# Start at left\n",
    "a = 0\n",
    "b = 1.5\n",
    "x = np.linspace(a, b)\n",
    "print(f\"Starting to the left, at x_0 = {a}\")\n",
    "x_k = a\n",
    "plt.figure(figsize=(8,8))\n",
    "plt.title('Starting at left')\n",
    "plt.grid(True)\n",
    "plt.plot(x, x, 'g')\n",
    "plt.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",
    "    plt.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",
    "    plt.plot([x_k, g_2(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_plus_1}\")\n",
    "\n",
    "plt.show()\n",
    "\n",
    "# Start at right\n",
    "a = 0\n",
    "b = 3\n",
    "x = np.linspace(a, b)\n",
    "print(f\"Starting to the right, at x_0 = {b}\")\n",
    "x_k = b\n",
    "plt.figure(figsize=(8,8))\n",
    "plt.title('Starting at left')\n",
    "plt.grid(True)\n",
    "plt.plot(x, x, 'g')\n",
    "plt.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",
    "    plt.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",
    "    plt.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_plus_1}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "This work is licensed under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/)"
   ]
  }
 ],
 "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": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}
