Preclass 1.2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "run_control": {
     "marked": true
    }
   },
   "outputs": [],
   "source": [
    "function coefficients = global_polynomial(x, y)\n",
    "    xT = x.';\n",
    "    point_count = length(x);\n",
    "    degree = point_count - 1;\n",
    "\n",
    "    A = zeros(point_count);\n",
    "    for col_number = 1:point_count\n",
    "        A(:, col_number) = xT;\n",
    "    end\n",
    "\n",
    "    A = A.^(degree:-1:0);\n",
    "\n",
    "    coefficients = (A\\y.');\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Xdata = [0, 1, 4, 5];\n",
    "Ydata = [-1, 3, 1, -3];\n",
    "coeff = global_polynomial(Xdata, Ydata);\n",
    "\n",
    "Xtest = -6:6;\n",
    "Ytest = [];\n",
    "for x_i = Xtest; Ytest = [Ytest, polyval(coeff, x_i)]; end\n",
    "plot(Xtest, Ytest, \"b-\", Xdata, Ydata, \"ro\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have three points:\n",
    "$$(x_0, y_0)$$\n",
    "$$(x_1, y_1)$$\n",
    "$$(x_2, y_2)$$\n",
    "\n",
    "We will have two spline equations:\n",
    "\n",
    "$$f_1(x) = a_1x^3 + b_1x^2 + c_1x + d_1$$\n",
    "$$f_1'(x) = 3a_1x^2 + 2b_1x + c_1$$\n",
    "$$f_1''(x) = 6a_1x + 2b_1$$\n",
    "\n",
    "$$f_2(x) = a_2x^3 + b_2x^2 + c_2x + d_2$$\n",
    "$$f_2'(x) = 3a_2x^2 + 2b_2x + c_2$$\n",
    "$$f_2''(x) = 6a_2x + 2b_2$$\n",
    "\n",
    "\n",
    "The coefficients must be such that we have:\n",
    "\n",
    "- Value match:\n",
    "\n",
    "$$f_1(x_0) = y_0$$\n",
    "$$f_1(x_1) = y_1$$\n",
    "\n",
    "$$f_2(x_1) = y_1$$\n",
    "$$f_2(x_2) = y_2$$\n",
    "\n",
    "- Transition match:\n",
    "\n",
    "$$f_1'(x_1) = f_2'(x_1)$$\n",
    "$$f_1''(x_1) = f_2''(x_1)$$\n",
    "\n",
    "- Natural spline:\n",
    "\n",
    "$$f_1''(x_0) = 0$$\n",
    "$$f_2''(x_2) = 0$$\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [],
   "source": [
    "function spline_coeffs = natural_spline(x, y)\n",
    "    % This could be built from partial matreces but\n",
    "    % this manual specification is the quick and dirty way.\n",
    "    A = [\n",
    "        x(1)^3, x(1)^2, x(1), 1, 0, 0, 0, 0;\n",
    "        x(2)^3, x(2)^2, x(2), 1, 0, 0, 0, 0;\n",
    "        0, 0, 0, 0, x(2)^3, x(2)^2, x(2), 1;\n",
    "        0, 0, 0, 0, x(3)^3, x(3)^2, x(3), 1;\n",
    "        3*x(2)^2, 2*x(2), 1, 0, -3*x(2)^2, -2*x(2), -1, 0;\n",
    "        6*x(2), 2, 0, 0, -6*x(2), -2, 0, 0;\n",
    "        6*x(1), 2, 0, 0, 0, 0, 0, 0;\n",
    "        0, 0, 0, 0, 6*x(3), 2, 0, 0;\n",
    "    ];\n",
    "    \n",
    "    Y = [y(1), y(2), y(2), y(3), 0, 0, 0, 0];\n",
    "    \n",
    "    spline_coeffs = A\\Y.';\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Xdata = [0, 1, 4];\n",
    "Ydata = [-1,3,1];\n",
    "spline_coeffs = natural_spline(Xdata, Ydata);\n",
    "\n",
    "Xtest = 0:0.25:4;\n",
    "Ytest = [];\n",
    "for x_i = Xtest;\n",
    "    if x_i >= Xdata(1) & x_i <= Xdata(2)\n",
    "        Ytest = [Ytest, polyval(spline_coeffs(1:4), x_i)];\n",
    "    else\n",
    "        Ytest = [Ytest, polyval(spline_coeffs(5:8), x_i)];\n",
    "    end\n",
    "end\n",
    "\n",
    "plot(Xtest, Ytest, \"b-\", Xdata, Ydata, \"ro\")"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Octave",
   "language": "octave",
   "name": "octave"
  },
  "language_info": {
   "file_extension": ".m",
   "help_links": [
    {
     "text": "GNU Octave",
     "url": "https://www.gnu.org/software/octave/support.html"
    },
    {
     "text": "Octave Kernel",
     "url": "https://github.com/Calysto/octave_kernel"
    },
    {
     "text": "MetaKernel Magics",
     "url": "https://metakernel.readthedocs.io/en/latest/source/README.html"
    }
   ],
   "mimetype": "text/x-octave",
   "name": "octave",
   "version": "4.2.2"
  },
  "varInspector": {
   "cols": {
    "lenName": 16,
    "lenType": 16,
    "lenVar": 40
   },
   "kernels_config": {
    "python": {
     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
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