sgm_splines_from_scratch.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Spline interpolation from scratch\n",
    "\n",
    "Example from Chapra & Canale, Ch. 18, Numerical Methods for Engineers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np \n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Input data set\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([3, 4.5, 7, 9])\n",
    "fx = np.array([2.5, 1., 2.5, 0.5])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, fx, 'ro') \n",
    "plt.grid() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Set up the matrix for equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.array([\n",
    "[1, 0, 0, 0, 0, 0, 0, 0, 0],\n",
    "[20.25, 4.5, 1, 0, 0, 0, 0, 0, 0],\n",
    "[0, 0, 0, 20.25, 4.5, 1, 0, 0, 0],\n",
    "[0, 0, 0, 49, 7, 1, 0, 0, 0],\n",
    "[0, 0, 0, 0, 0, 0, 49, 7, 1],\n",
    "[0, 0, 0, 14, 1, 0, -14, -1, 0],\n",
    "[0, 0, 0, 0, 0, 0, 81, 9, 1],\n",
    "[9, 1, 0, -9, -1, 0, 0, 0, 0],\n",
    "[9, 3, 1, 0, 0, 0, 0, 0, 0]\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "# b-vector\n",
    "b = np.array([0, 1, 1, 2.5, 2.5, 0, 0.5, 0, 2.5])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 7.19588997e-15, -1.00000000e+00,  5.50000000e+00,  6.40000000e-01,\n",
       "       -6.76000000e+00,  1.84600000e+01, -1.60000000e+00,  2.46000000e+01,\n",
       "       -9.13000000e+01])"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(np.linalg.inv(A), b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "a1, b1, c1, a2, b2, c2, a3, b3, c3 = np.linalg.solve(A, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f1(x): \n",
    "    return a1*x**2 + b1*x + c1\n",
    "\n",
    "def f2(x): \n",
    "    return a2*x**2 + b2*x + c2\n",
    "\n",
    "def f3(x): \n",
    "    return a3*x**2 + b3*x + c3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.5000000000000133"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xvals = np.arange(0, 12, 0.1)\n",
    "\n",
    "plt.plot(xvals, f1(xvals))\n",
    "plt.plot(xvals, f2(xvals))\n",
    "plt.plot(xvals, f3(xvals))\n",
    "\n",
    "plt.plot(x, fx, 'ro')\n",
    "\n",
    "plt.ylim([-1,6])\n",
    "\n",
    "plt.grid() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Combine functions for correct intervals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3. , 4.5, 7. , 9. ])"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Combined function\n",
    "def combined_function(x, intervals):\n",
    "    if x <= intervals[1]:\n",
    "        return f1(x)\n",
    "    elif intervals[1] < x <= intervals[2]:\n",
    "        return f2(x)\n",
    "    else:\n",
    "        return f3(x)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [],
   "source": [
    "fx_combined = np.array([combined_function(x_val, x) for x_val in xvals])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((120,), (120,))"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fx_combined.shape, xvals.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.\n",
      "The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.\n",
      "The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xvals = np.arange(0, 12, 0.1)\n",
    "\n",
    "plt.plot(xvals, f1(xvals), alpha=0.5)\n",
    "plt.plot(xvals, f2(xvals), alpha=0.5)\n",
    "plt.plot(xvals, f3(xvals), alpha=0.5)\n",
    "\n",
    "# combined function, only\n",
    "\n",
    "plt.plot(xvals, fx_combined, 'k--', lw=2)\n",
    "\n",
    "plt.plot(x, fx, 'ro') \n",
    "\n",
    "plt.ylim([-1,6])\n",
    "\n",
    "plt.grid() \n",
    "\n",
    "plt.savefig(\"./local/splines_from_scratch_example.eps\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "base",
   "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": "3.9.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}