07-newton.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# \n",
        "\n",
        "# Newton’s method\n",
        "\n",
        "We get started by loading our package that brings in plotting and other\n",
        "features, including those provided by the `Roots` package:"
      ],
      "id": "b8cbbdd1-de64-4b59-aba0-1d32bee0b835"
    },
    {
      "cell_type": "code",
      "execution_count": 0,
      "metadata": {},
      "outputs": [],
      "source": [
        "using MTH229\n",
        "using Plots\n",
        "plotly()"
      ],
      "id": "2"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "------------------------------------------------------------------------\n",
        "\n",
        "### Quick background\n",
        "\n",
        "Read about this material here: [Newton’s\n",
        "Method](http://mth229.github.io/newton.html).\n",
        "\n",
        "Symbolic math - as is done behind the scenes at the Wolfram alpha web\n",
        "site - is pretty nice. For so many problems it can easily do what is\n",
        "tedious work. However, for some questions, only numeric solutions are\n",
        "possible. For example, there is no general formula to solve a fifth\n",
        "order polynomial the way there is a quadratic formula for solving\n",
        "quadratic polynomials. Even an innocuous polynomial like\n",
        "$f(x) = x^5 - x - 1$ has no easy algebraic solution for is one root.\n",
        "\n",
        "A graph shows what looks like just one answer between $1$ and $2$,\n",
        "closer to $1$"
      ],
      "id": "81a37c8c-7213-40f1-8caa-27248cf27243"
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/html": [
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\" />"
            ]
          }
        }
      ],
      "source": [
        "f(x) = x^5 - x - 1\n",
        "plot(f, -2, 2)\n",
        "plot!(zero, -2, 2)"
      ],
      "id": "6"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "From this, we can call the bisection method to identify the value:"
      ],
      "id": "c5d1bcc3-34eb-4b24-a0ce-5bee8988d05e"
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "1.1673039782614187"
            ]
          }
        }
      ],
      "source": [
        "fzero(f, -2, 2)"
      ],
      "id": "8"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "We’ve seen the bisection method previously to find a root, but this is\n",
        "somewhat cumbersome to use as it needs a\n",
        "[bracketing](https://en.wikipedia.org/wiki/Bisection_method#The_method)\n",
        "interval to begin. Moreover, bisection can be computationally slow (by\n",
        "comparison).\n",
        "\n",
        "Here we discuss Newton’s method. Like the bisection method it is an\n",
        "*iterative algorithm*. However instead of identifying a bracketing\n",
        "interval, we only need to identify a **nearby** *initial* guess, $x_0$.\n",
        "\n",
        "Starting with $x_0$ the algorithm to produce $x_1$ is easy to describe:\n",
        "\n",
        "-   form the tangent line at $(x_0, f(x_0))$.\n",
        "\n",
        "-   let $x_1$ be the intersection point of this tangent line with the\n",
        "    $x$ axis.\n",
        "\n",
        "If we can go from $x_0$ to $x_1$ we can repeat the update step to get\n",
        "$x_2$ and then $x_3$, …\n",
        "\n",
        "Graphically, this figure illustrates the process until the guesses get\n",
        "so close they crowd the figure. (Ignore the code to produce the figure.)"
      ],
      "id": "9ff9a5a2-a7a9-4e30-94bc-40771140ded1"
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/html": [
              "<img 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\" />"
            ]
          }
        }
      ],
      "source": [
        "f(x) = x^5 - x - 1\n",
        "plot(f, 0.9, 1.3; legend=false, linewidth=2)\n",
        "plot!(zero)\n",
        "newton_plot!(f, 1.0; annotate_steps=3)"
      ],
      "id": "10"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "In the figure, the sequence of guesses can be seen, basically $x_0=1$,\n",
        "$x_1=1.25$, $x_2=1.178\\dots$, …, $1.16730\\dots$.\n",
        "\n",
        "To find these values numerically, we first need an algebraic\n",
        "representation. For this problem, we can describe the tangent line’s\n",
        "slope by *either* $f'(x_0)$ *or* by using “rise over run”:\n",
        "\n",
        "$$\n",
        "f'(x_0) = \\frac{f(x_0) - 0}{x_0 - x_1}.\n",
        "$$\n",
        "\n",
        "Solving for $x_0$, this yields the update formula:\n",
        "\n",
        "$$\n",
        "x_1 = x_0 - f(x_0) / f'(x_0).\n",
        "$$\n",
        "\n",
        "That is, the new guess shifts the old guess by an increment\n",
        "$f(x_0)/f'(x_0)$.\n",
        "\n",
        "In `Julia`, we can do one step with:"
      ],
      "id": "5d51810b-7bf1-4c05-ad69-8b33cf403e95"
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "1.25"
            ]
          }
        }
      ],
      "source": [
        "f(x) = x^5 - x - 1\n",
        "x = 1\n",
        "x = x - f(x) / f'(x)"
      ],
      "id": "12"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "(We don’t use indexing, but rather update our binding for the `x`\n",
        "variable.)\n",
        "\n",
        "Is `x` close to being the zero? We don’t know the actual zero - we are\n",
        "trying to approximate it - but we do know the function’s value at the\n",
        "actual zero. For this new guess the function value is"
      ],
      "id": "ab096154-8613-4b34-b1ee-7f5b80f79456"
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "0.8017578125"
            ]
          }
        }
      ],
      "source": [
        "f(x)"
      ],
      "id": "14"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "This is much closer to $0$ than $f(1)$, the value at our initial guess,\n",
        "but not nearly as close as we can get using Newton’s method. We just\n",
        "need to **iterate** - run a few more steps.\n",
        "\n",
        "We do another step just by running the last line. For example, we run 5\n",
        "more steps by copying and pasting the same expression:"
      ],
      "id": "1d881930-fe59-4af3-902f-db26af41d40c"
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "1.1673039782614187"
            ]
          }
        }
      ],
      "source": [
        "x = x - f(x) / f'(x)\n",
        "x = x - f(x) / f'(x)\n",
        "x = x - f(x) / f'(x)\n",
        "x = x - f(x) / f'(x)\n",
        "x = x - f(x) / f'(x)"
      ],
      "id": "16"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "The value of `x` updates. But is it getting closer to a *zero*? If so,\n",
        "then $f(x)$ should be close to zero. We can see both values with:"
      ],
      "id": "db2972ce-5948-473e-8441-5b3bbeb93a09"
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "(1.1673039782614187, 6.661338147750939e-16)"
            ]
          }
        }
      ],
      "source": [
        "x, f(x)"
      ],
      "id": "18"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "This shows $f(x)$ is not exactly $0.0$ but it is as close as we can get.\n",
        "Repeating the algorithm does not change the value of `x`. (On a\n",
        "computer, floating point issues creep in when values are close to 0, and\n",
        "these prevent values being mathematically exact.) As we can’t improve,\n",
        "we stop. Our value of `x` is an *approximate* zero and `f(x)` is within\n",
        "machine tolerance of being `0`.\n",
        "\n",
        "How do we know how to stop? When the algorithm works, we will stop when\n",
        "the `x` value *basically* stops updating, as `f(x)` is basically `0`.\n",
        "However, the algorithm need not work, so any implementation must keep\n",
        "track of how many steps are taken and stop when this gets out of hand.\n",
        "\n",
        "For convenience, the `newton` function in the `MTH229` package (using\n",
        "the `Roots` package) will iterate until convergence. If we pass in the\n",
        "optional argument `verbose=true` we will see the sequence of steps.\n",
        "\n",
        "For example, for $f(x) = x^3 - 2x - 5$, a function that Newton himself\n",
        "considered, a solution near $2$, is found with:"
      ],
      "id": "7b3c216c-6f6c-4a45-a645-7f099610f30c"
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "2.0945514815423265"
            ]
          }
        }
      ],
      "source": [
        "x = 2\n",
        "f(x) = x^3 - 2x -5\n",
        "xstar = newton(f, 2)"
      ],
      "id": "20"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "We can see the approximate zero and the function value, as follows:"
      ],
      "id": "e2465c81-0400-4867-91ef-331d783ef7dd"
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "(2.0945514815423265, -8.881784197001252e-16)"
            ]
          }
        }
      ],
      "source": [
        "xstar, f(xstar)"
      ],
      "id": "22"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "------------------------------------------------------------------------\n",
        "\n",
        "### Using fzero from the Roots package\n",
        "\n",
        "As mentioned, the `newton` function in the `Roots` package implements\n",
        "Newton’s method. The `Roots` package also provides the `fzero` function\n",
        "for finding roots. This function will:\n",
        "\n",
        "-   use bisection when called as `fzero(f, a, b)` with the zero\n",
        "    **between** the bracketing interval `[a,b]`\n",
        "-   use a method like Newton’s method when called as `fzero(f, c)` with\n",
        "    the zero **near** `c`.\n",
        "\n",
        "For example:"
      ],
      "id": "45d701d1-5d33-4423-818b-916d9c699548"
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [
        {
          "output_type": "display_data",
          "metadata": {},
          "data": {
            "text/plain": [
              "3.141592653589793"
            ]
          }
        }
      ],
      "source": [
        "fzero(sin, 3)   # start with initial guess of 3, returns 3.141592653589793"
      ],
      "id": "24"
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "The utility of this function is that it does not require a derivative to\n",
        "be taken and it is a little less sensitive than Newton’s method to the\n",
        "initial guess. The use of `fzero` is recommended: just plot to find a\n",
        "nearby guess and use `fzero(f,c)` to improve this guess.\n",
        "\n",
        "### When Newton’s method fails\n",
        "\n",
        "The error in the $n$th step using Newton’s method at a simple zero\n",
        "follows a formula: $|e_{n+1}| \\leq (1/2) |f''(a)/f'(b)| \\cdot |e_n|^2$,\n",
        "for some $a$ and $b$. Generally this ensures that the error at step\n",
        "$n+1$ is smaller than the error at step $n$ squared. But this can fail\n",
        "due to various cases:\n",
        "\n",
        "-   the initial guess is not *near* the zero,\n",
        "\n",
        "-   the derivative, $|f'(x)|$, is too small, or\n",
        "\n",
        "-   the second derivative, $|f''(x)|$, is too big, or possibly\n",
        "    undefined.\n",
        "\n",
        "### Quadratic convergence\n",
        "\n",
        "When Newton’s method converges to a *simple zero* it is said to have\n",
        "*quadratic convergence*. A simple zero is one with multiplicity 1 and\n",
        "quadratic convergence says basically that the error at the $i+1$st step\n",
        "is like the error for $i$th step squared. In particular, if the error is\n",
        "like $10^{-3}$ on one step, it will be like $10^{-6}$, then $10^{-12}$\n",
        "then $10^{-24}$ on subsequent steps. (Which is typically beyond the\n",
        "limit of a floating point approximation.) This is why one can *usually*\n",
        "take just 5, or so, steps to get to an answer.\n",
        "\n",
        "Not so for multiple roots and some simple roots.\n",
        "\n",
        "------------------------------------------------------------------------"
      ],
      "id": "3eb7ec93-b5f0-4b4d-99aa-3944e8d560b3"
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [],
      "source": [
        "#  Your commands go here\n"
      ],
      "id": "26"
    }
  ],
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