M104A HW6.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.0, 1.513705285705932, 11.660133509251635, 0.0]\n",
      "[0.0, 1.513705285705932, 11.660133509251635, 0.0]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "'''\n",
    "First of all, I'd like to acknowledge Amir O. for helping me debug my index issues.\n",
    "\n",
    "Name: Daniel Arredondo\n",
    "Last Edited: 07/15/2020\n",
    "\n",
    "Triadiagonal Matrix Solver\n",
    "'''\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as mat\n",
    "import math as m\n",
    "%matplotlib inline\n",
    "\n",
    "coordinates = [(0.0,0.0),(1.0,1.0),(2.0,16.0)] #list of vectors, x_j, y_j\n",
    "test = [0,1,2,3]\n",
    "testf = []\n",
    "for x in test:\n",
    "    testf.append((x,m.exp(x)))\n",
    "    \n",
    "def z_node_function(f_array): \n",
    "    #input: list of 'coordinates', list whose elements are two values.\n",
    "    # First we calculate little a, b, c, d\n",
    "    n = len(f_array)\n",
    "    a = []\n",
    "    b = []\n",
    "    c = []\n",
    "    d = []\n",
    "    for i in range(1,n-1):        \n",
    "        a.append(2*( f_array[i+1][0]-f_array[i-1][0] ))\n",
    "        d1 = (f_array[i+1][1]-f_array[i][1])/(f_array[i+1][0]-f_array[i][0]) \n",
    "        d1 = d1 - (f_array[i][1]-f_array[i-1][1])/(f_array[i][0]-f_array[i-1][0])\n",
    "        d.append(6*d1)\n",
    "    for ii in range(1, n-2):\n",
    "        b.append(f_array[ii+1][0]-f_array[ii][0])\n",
    "        c.append(f_array[ii][0]-f_array[ii-1][0])\n",
    "        \n",
    "    # Next, we calculate our z nodes, substituting our value n\n",
    "    # for the length of a, such as to avoid calculating the endpoints\n",
    "    # z_0 and z_n\n",
    "    \n",
    "    n = len(a)\n",
    "    #print(n)\n",
    "    m = [a[0]]\n",
    "    y = [d[0]]\n",
    "    l = []\n",
    "    \n",
    "    for j in range(n-1):\n",
    "        l.append(c[j]/m[j])\n",
    "        m.append(a[j+1] - l[j]*b[j])\n",
    "        \n",
    "    for k in range(1, n):\n",
    "        y.append(d[k] - l[k-1] * y[k-1])\n",
    "    \n",
    "    z = [0]*(n)\n",
    "    z[-1] = y[-1]/m[-1]\n",
    "    \n",
    "    for iii in range(n-2, -1, -1):\n",
    "        z[iii] = ((y[iii] - b[iii]*z[iii+1])/m[iii] )\n",
    "    \n",
    "    z.insert(0,0.0)\n",
    "    z.append(0.0)\n",
    "    print(z)\n",
    "    return z # output: a list of z nodes\n",
    "\n",
    "\n",
    "\n",
    "def CapitalCoefficients(z_nodes, f_array):\n",
    "    #inputs: our z nodes and our previously aformentioned 'coordinates'\n",
    "    #outputs: A list whose elements are lists of A_j, B_j, C_j, D_j for each spline\n",
    "    n = len(z_nodes)\n",
    "    A=[]\n",
    "    B=[]\n",
    "    C=[]\n",
    "    D=[]\n",
    "    for i in range(n-1): #we evaluate our spline coefficients with our given z_j, and each for-loop\n",
    "                        # represents a different set of coefficients for a different interval on the spline\n",
    "        A.append((z_nodes[i+1]-z_nodes[i])/(6*(f_array[i+1][0]-f_array[i][0])))\n",
    "        B.append((0.5)*z_nodes[i])\n",
    "        C.append((1/(f_array[i+1][0]-f_array[i][0]))*(f_array[i+1][1]-f_array[i][1]) - \\\n",
    "            (1/6)*(f_array[i+1][0]-f_array[i][0])*(z_nodes[i+1]+2*z_nodes[i]))\n",
    "        D.append(f_array[i][1])\n",
    "    Cof = [A,B,C,D] #list of lists\n",
    "    return Cof\n",
    "\n",
    "# z = z_node_function(coordinates)\n",
    "# c = CapitalCoefficients(z, coordinates)\n",
    "\n",
    "tz = z_node_function(testf)\n",
    "ta = CapitalCoefficients(tz, testf)\n",
    "# print(tz, ta[0])\n",
    "\n",
    "\n",
    "\n",
    "def s_j(X, f_array):\n",
    "    #inputs: X is a list of points where we evalute our spline at, f_array is our 'cooridinate' nodes\n",
    "    #outpus: a list of size X, which allows us to graph along with X in mat.plot\n",
    "    z = z_node_function(f_array)\n",
    "    CapCof = CapitalCoefficients(z, f_array)\n",
    "    y = []\n",
    "    for x in X: # we begin the process of evaluating our input value\n",
    "        que = 0 #this controls a lower bounds in a for loop later for checking\n",
    "                # where the x value lies between which node\n",
    "        if (x < f_array[0][0] or x > f_array[len(f_array)-1][0]):\n",
    "            pass # A quality check. Any input values outside of outside of our nodes are thrown away and no evaluated\n",
    "        else:\n",
    "            for k in range(que, len(f_array)): #que changes through false if-statements, resets on each new x input\n",
    "                if abs(x - f_array[k][0]) < 0.000001: #This is how we check if our input value is actually an\n",
    "                                                        # interpolation node\n",
    "                    y.append(f_array[k][1]) # we append the interpolation node\n",
    "                if x > f_array[k][0] and x < f_array[k+1][0]:# Here we are evaluting our spline with out input x\n",
    "                    y.append(CapCof[0][k]*(x-f_array[k][0])**3 +\\\n",
    "                        CapCof[1][k]*(x-f_array[k][0])**2 +\\\n",
    "                        CapCof[2][k]*(x-f_array[k][0]) +\\\n",
    "                        CapCof[3][k])\n",
    "                else: que += 1; #If our x is not within the bounds of x(j) and x(j+1) nodes, we increase the lower\n",
    "                            # bounds to reevalute the input x at the next set of nodes x(j+1) and x(j+2)\n",
    "    return y\n",
    "\n",
    "\n",
    "\n",
    "prec_x = np.arange(0,3,0.03)\n",
    "prec_f = []\n",
    "approx_f = s_j(prec_x, testf)\n",
    "for jk in prec_x:\n",
    "    prec_f.append(m.exp(jk))\n",
    "\n",
    "mat.plot(prec_x, prec_f, '--', label='Exact')\n",
    "mat.plot(prec_x, approx_f, '', label='Approxiamtion')\n",
    "mat.legend(loc='upper left')\n",
    "mat.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem 2\n",
    "\n",
    "Spline Graphing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.0, 0.039320428702891375, 0.23179835421574355, 2.0972572196142436, -1.8071900908005378, 6.728424761331322, -1.0190512391169801, 0.2476361703205055, 0.0]\n",
      "[0.0, 1.0743448551050194, -4.96085513931473, 0.04199952355248365, -0.6511897020468728, 2.8576565438941572, 1.1550304915607377, -0.5211854292129106, 0.0]\n",
      "[0.0, 0.039320428702891375, 0.23179835421574355, 2.0972572196142436, -1.8071900908005378, 6.728424761331322, -1.0190512391169801, 0.2476361703205055, 0.0]\n",
      "[0.0, 1.0743448551050194, -4.96085513931473, 0.04199952355248365, -0.6511897020468728, 2.8576565438941572, 1.1550304915607377, -0.5211854292129106, 0.0]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_j = [(0,1.5) ,(0.618,0.9),(0.935,0.6),(1.255,0.35),(1.636,0.2) ,(1.905,0.1),(2.317,0.5),(2.827,1.0),(3.330,1.5)]\n",
    "y_j = [(0,0.75),(0.618,0.9),(0.935,1.0),(1.255,0.8) ,(1.636,0.45),(1.905,0.2),(2.317,0.1),(2.827,0.2),(3.330,0.25)]\n",
    "\n",
    "t_j_x = np.arange(0,3.33,2*0.00333)\n",
    "\n",
    "x_z = z_node_function(x_j)\n",
    "y_z = z_node_function(y_j)\n",
    "\n",
    "#print(len(x_z),len(y_z))\n",
    "\n",
    "x_cof = CapitalCoefficients(x_z, x_j)\n",
    "y_cof = CapitalCoefficients(y_z, y_j)\n",
    "\n",
    "#print(len(x_cof[0]), len(y_cof[0]))\n",
    "\n",
    "x___s = s_j(t_j_x, x_j)\n",
    "y___s = s_j(t_j_x, y_j)\n",
    "\n",
    "#print(len(t_j_x),len(x___s), len(y___s))\n",
    "\n",
    "mat.plot(x___s, y___s, 'blue')\n",
    "mat.show()"
   ]
  }
 ],
 "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": "3.7.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}