{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "using ForwardDiff\n",
    "using Plots\n",
    "using Calculus\n",
    "plotly()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#number of degrees of freedom\n",
    "n = 10\n",
    "\n",
    "left = 0 #x_L default. never changed\n",
    "right = 11 #x_R default. to be changed later\n",
    "#number of springs\n",
    "num_springs = n+1\n",
    "#lengths of each spring\n",
    "L = ones(num_springs,1);\n",
    "#spring constants:\n",
    "k_spring = ones(num_springs,1)\n",
    "for i in 1:num_springs\n",
    "    if(i %2 == 0)\n",
    "        k_spring[i] = 0.5\n",
    "    end\n",
    "end\n",
    "#Construct spring system matrix: \n",
    "cols = n+2\n",
    "rows = num_springs\n",
    "A = eye(rows,cols)\n",
    "for i = 1:rows\n",
    "    A[i,i+1] = -1\n",
    "end\n",
    "#define PE\n",
    "PE(x) = sum(1/2*k_spring.*(abs.(A*[left;x;right])-L).^2)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "K = 10000 #maximum number of iterations\n",
    "epsilon = 1e-8 \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#f is a function (PE or rosenbrock). x0 is the initial condition.\n",
    "function myGradientDescent(f, x0)\n",
    "    #YOUR CODE HERE\n",
    "    #Return optimal x vector\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#f is a function (PE or rosenbrock). x0 is the initial condition.\n",
    "function myNewtonMethod(f, x0)\n",
    "    #YOUR CODE HERE\n",
    "    #Return optimal x vector\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#rosenbrock function and initial condition.\n",
    "rosenbrock(x) =  (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2\n",
    "rosenbrockx0 = [0; 0];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Part 1\n",
    "println(\"Gradient Descent\")\n",
    "res = myGradientDescent(rosenbrock, [0;0])\n",
    "println(res)\n",
    "println(\"Newton's Method\")\n",
    "res = myNewtonMethod(rosenbrock, [0,0])\n",
    "println(res)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Problem 2, try to solve on large boundary.\n",
    "left = 0\n",
    "right = 11\n",
    "x0 = linspace(1, right-1, n)*0.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Part 3, try to solve on smaller boundary. \n",
    "left = 0\n",
    "right = 5\n",
    "x0 = linspace(1, right-1, n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Part 4, Compress the system to get to a smaller boundary.\n",
    "left = 0\n",
    "right = 11\n",
    "x0 = linspace(1, right-1, n)\n",
    "while right > 1\n",
    "    right = right-0.1\n",
    "    res = myNewtonMethod(PE,x0)\n",
    "    dres = diff([left;res;right])\n",
    "    #println(dres)\n",
    "    if(any(x->x<0, dres))\n",
    "        println(\"FAILURE when right= \", right)\n",
    "        println(res)\n",
    "        break;\n",
    "    end\n",
    "    x0 = res\n",
    "end\n",
    "println(left, \" \", right, \" \", x0)"
   ]
  }
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