Project_4_Case_0_Solving_for_a_known_polynomial_function_using_Singular_Values_or_Best_Fit.ipynb

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  "metadata": {
    "colab": {
      "name": "Copy of Project 4:  Case 0 - Solving for a known polynomial function using Singular Values or Best Fit.ipynb",
      "provenance": [],
      "collapsed_sections": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/KapilKhanal/AdvLinearAlgebra/blob/master/Project_4_Case_0_Solving_for_a_known_polynomial_function_using_Singular_Values_or_Best_Fit.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "9_AimxJu0fZa",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "#  Libraries that we'll need:\n",
        "import numpy as np\n",
        "import sympy as sym\n",
        "#from scipy import misc\n",
        "from scipy import linalg\n",
        "#from PIL import Image\n",
        "#import imageio\n",
        "import matplotlib.pyplot as plt\n",
        "# importing \"random\" for random operations \n",
        "import random "
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ah9MqhC_0nc3",
        "colab_type": "text"
      },
      "source": [
        "#  Singular Value Decomposition and Best Fit\n",
        "\n",
        "We look at some data generated here from three sources:\n",
        "\n",
        "*   Case 0:  A known polynomial function $f(x) = 4 - 3x^2 + 5x^6$\n",
        "*   Case 1:  A known trigonometric function $f(x) = 4 - 2*cos(x) + cos^2 (x)$\n",
        "*   Case 2:  A matrix of data $M$ (specified below) with unknown source function\n",
        "*   Case 3:  An interesting case."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "u294N29608yk",
        "colab_type": "text"
      },
      "source": [
        "#  Overview for Case 0: Fit for a known polynomial function $f(x) = 4 - 3x^2 + 5x^6$\n",
        "We generate data for the polynomial function \n",
        "$$f(x) = 4 -3x^2 +5x^6$$ \n",
        "We know two things: The input (x) and the output (f(x)) for that input.  Pretend we know the (x, f(x)) pairs for **a lot** of different points, say (cough... cough...) $3$ of them.\n",
        "\n",
        "We want to find out what combination of polynomials $1, x, x^2, x^3, x^4, x^5,$ and $x^6$ best matches the data.  For each input/output pair $(x, f(x))$, we generate one row of data in the system $Ac = b$:\n",
        "\n",
        "$$ [  c_0 * 1 + c_1 * x + c_2 * x^2 + c_3 * x^3 + ... c_6 x^6 = f(x) ] $$\n",
        "\n",
        "or\n",
        "\n",
        "$$ \\left[ \\begin{array}{ccccccc} 1 & x & x^2 & x^3 & x^4 & x^5 & x^6 \\end{array} \\right] \\left( \\begin{array}{c} c_0 \\\\ c_1 \\\\ c_2 \\\\ c_3 \\\\ c_4 \\\\ c_5 \\\\c_6 \\\\ \\end{array} \\right) = \\left[ f(x) \\right] $$\n",
        "\n",
        "\n",
        "Be careful -- the unknowns are the coefficients in front - the $c_0, c_1, c_2, ..., c_6$ values.  Since we know $x$, then we know $x, x^2, x^3, x^4, ... x^6$. And we know the output $f(x)$ for that row.\n",
        "\n",
        "So, if we know $x=2$ and $f(2) = 312$, then we populate the row as:\n",
        "\n",
        "$$ [  c_0*1  + c_1*2 + c_2*4 + c_3*8 + ... + c_6*64 = 312 ] $$\n",
        "\n",
        "or\n",
        "$$ \\left[ \\begin{array}{ccccccc} 1 & 2 & 4 & 8 & 16 & x32 & 64 \\end{array} \\right] \\left( \\begin{array}{c} c_0 \\\\ c_1 \\\\ c_2 \\\\ c_3 \\\\ c_4 \\\\ c_5 \\\\c_6 \\\\ \\end{array} \\right) = \\left[ 312 \\right] $$\n",
        "\n",
        "where the vector $\\vec{c}$ contains the unknowns $c_0, c_1, c_2, \\ldots c_6$\n",
        "\n",
        "So... let's generate the matrix:\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tWC4d8yQZGrx",
        "colab_type": "text"
      },
      "source": [
        "# 0.1  Setting up the system:\n",
        "\n",
        "We'll generate some data points for our system using the function \n",
        "$$ f(x) = 4 - 3x^2 + 5x^6$$\n",
        "\n",
        "We'll store the input data in **actual_data_x** and the output in **actual_data_y**.\n",
        "\n",
        "We'll also use the data to set up a system\n",
        "$$M \\vec{x} = \\vec{b}$$\n",
        "\n",
        "where **actual_data_x** is used to populate matrix $M$ and **actual_data_y** is used to populate $\\vec{b}$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3xcazoedaEdj",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "#We'll generate random points of data for our system.\n",
        "#The actual points for the system will be stored in containers actual_data_x and actual_data_y\n",
        "\n",
        "actual_data_x = []\n",
        "actual_data_y = []\n",
        "\n",
        "#We'll use (cough... cough...) three(after trying with three now trying with 20)   data points:\n",
        "for i in range(4):\n",
        "    # pick a random number between -5 and 5\n",
        "    x = 6*random.random() - 3\n",
        "    f = 4 - 3*x**2 + 5*x**6\n",
        "    actual_data_x.append(x)\n",
        "    actual_data_y.append(f)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "qBDgSCalahsU",
        "colab_type": "code",
        "outputId": "214d79d7-bbb0-4cdb-f413-8714d5664b93",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 51
        }
      },
      "source": [
        "# uncomment code below  to look at the data in actual_data_x or in actual_data_y\n",
        "\n",
        "print(actual_data_x)\n",
        "\n",
        "actual_data_y"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[2.4286772330985755, 1.3353239164877433, 1.8645115677885897, -1.0774861650489957]\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[1012.4025175124129, 26.99659872102921, 203.63907841102377, 8.34127566830912]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 66
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "XV2S8QHhatXJ",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "#Create the matrix M from the data in actual_data_x\n",
        "M_entries = []\n",
        "\n",
        "for x in actual_data_x:\n",
        "  M_entries.append([1, x, x**2, x**3, x**4, x**5, x**6])\n",
        "\n",
        "M = []\n",
        "M = np.matrix(M_entries)\n",
        "\n",
        "\n",
        "#and create the vector b from the data in actual_data_y:\n",
        "b = []\n",
        "b = (np.matrix(actual_data_y)).transpose()"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "RsAZQ_8TbUrQ",
        "colab_type": "code",
        "outputId": "f01f96aa-1201-47da-bcde-7e99d1ee7458",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 85
        }
      },
      "source": [
        "# Uncomment either line below to show the matrix M\n",
        "# or the first five entries of b\n",
        "\n",
        "M\n",
        "b[:5]"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "matrix([[1012.40251751],\n",
              "        [  26.99659872],\n",
              "        [ 203.63907841],\n",
              "        [   8.34127567]])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 68
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zHF_2CDqcde5",
        "colab_type": "text"
      },
      "source": [
        "#0.2:  Plot the generated system:\n",
        "\n",
        "We'll create a plot showing the function (in red) and the data points sampled for the function (in blue).\n",
        "\n",
        "To generate the graph of the function $f(x)$, we first use $100$ data points from $0$ to $99$, then compress them into an interval of width $6$, and shift it left so that the 100 data points are equally spaced between $x = -3$ and $x=3$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "XViuaKVK3aU3",
        "colab_type": "code",
        "outputId": "e48ce5db-a0a2-450a-e3ed-df80e07c7057",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 295
        }
      },
      "source": [
        "# We'll specify that x is a variable:\n",
        "x = sym.symbols('x')\n",
        "\n",
        "xdata = []\n",
        "fdata = []\n",
        "#set up our 100 data points on the interval from (-3, 3):\n",
        "xdata = np.array(range(100))/(100/6) - 3\n",
        "# find the ouput for our function on each data point:\n",
        "fdata = 4 - 3*xdata**2 + 5*xdata**6\n",
        "\n",
        "\n",
        "# Create the plot in red, and our datapoints in blue:  \n",
        "plt.plot(xdata,fdata, color='red') + plt.plot(actual_data_x, actual_data_y, 'bo')\n",
        "\n",
        "# Add a title\n",
        "plt.title('Our test data (in blue) compared to the actual data (in red)')\n",
        "\n",
        "# Add X and y Label\n",
        "plt.xlabel('x axis')\n",
        "plt.ylabel('output')\n",
        "\n",
        "# Add a grid\n",
        "plt.grid(alpha=.4,linestyle='--')\n"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uWyKTBIaeI11",
        "colab_type": "text"
      },
      "source": [
        "#0.3:  Solving the system using the method of Best-Fit:\n",
        "\n",
        "Given a system \n",
        "$$ M \\vec{x} = \\vec{b}$$\n",
        "\n",
        "where matrix $M$ is **not necessarily square**, we can multiply by $M^T$ on the left to form the system\n",
        "\n",
        "$$ \\left( M^T M \\right) \\vec{x} = M^T \\vec{b} $$\n",
        "\n",
        "Now, $M^T M$ is a square matrix, so we can use it's inverse to solve for $\\vec{x}$:\n",
        "\n",
        "$$ \\vec{x} = \\left( M^T M \\right)^{-1} M^T \\vec{b}$$\n",
        "\n",
        "This solution method is called the method of Best-Fit.\n",
        "\n",
        "**A word of caution**:  We're initially trying to fit a 6th degree polynomial with only 3 data points.  That shouldn't have a good solution, and it could have no solution.  If you get no solution, start at the top and re-run the code to generate a new set of random data points for our system."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Lt_-LiCm4mX5",
        "colab_type": "code",
        "outputId": "c1e774d6-9a31-4cc0-8dd1-83b71fc79aab",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 136
        }
      },
      "source": [
        "#  Find the solution to Mx = b using Best Fit:\n",
        "#  (We've already used 'x' as a variable, so we'll call the solution vector c.)\n",
        "c = linalg.inv(M.transpose() * M) * M.transpose() * b\n",
        "c\n"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "matrix([[-78.30487224],\n",
              "        [100.25394994],\n",
              "        [-35.69095946],\n",
              "        [-61.07590829],\n",
              "        [150.14085129],\n",
              "        [ -8.19343953],\n",
              "        [  1.02747379]])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 70
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "32HO69A9f9WO",
        "colab_type": "text"
      },
      "source": [
        "We can test our solution against the output of our original system.\n",
        "\n",
        "The original output in our system is represented by $\\vec{b}$.\n",
        "\n",
        "Our suggested solution to $M\\vec{x} = \\vec{b}$ is $\\vec{c}$.  That is, we think\n",
        "\n",
        "$$ M \\vec{c} \\approx \\vec{b}$$\n",
        "\n",
        "If we have a good fit, then \n",
        "\n",
        "$$\\vec{b} - M \\vec{c} \\approx 0$$\n",
        "\n",
        "To test our solution, we find the) magnitude of $\\vec{b} - M\\vec{c}$:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "alHjgx4L4ydh",
        "colab_type": "code",
        "outputId": "6dc6c3b3-5366-40cd-c82a-1e107b97e790",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        }
      },
      "source": [
        "#  Let's see how close this actually gets us by comparing the original b vector to the result of Mc:\n",
        "error = ((b-M*c).transpose() * (b-M*c))[0,0]\n",
        "error**0.5"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "3014.621194590967"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 71
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rO3fm0sVhI_M",
        "colab_type": "text"
      },
      "source": [
        "We can plot the actual solution (in red) with our best-fit solution (in green), using the test data shown as blue points.  It may be helpful to adjust the limits on the y-axis to zoom in or out on the output.\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "bygM7WnqhIHw",
        "colab_type": "code",
        "outputId": "9c3e3add-c83b-4fcf-b305-4355d9b6f6a4",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 295
        }
      },
      "source": [
        "#Show a plot of the solutions using the best-fit method, and the SVD method:\n",
        "\n",
        "# We'll specify that x is a variable:\n",
        "x = sym.symbols('x')\n",
        "\n",
        "xdata = []\n",
        "fdata = []\n",
        "#set up our 100 data points on the interval from (-3, 3):\n",
        "xdata = np.array(range(100))/(100/6) - 3\n",
        "# find the ouput for our function on each data point:\n",
        "fdata = 4 - 3*xdata**2 + 5*xdata**6\n",
        "\n",
        "best_fit = c[0,0] + c[1,0]*xdata + c[2,0]*(xdata**2) + c[3,0]*(xdata**3) + c[4,0]*(xdata**4) + c[5,0]*(xdata**5) + c[6,0]*(xdata**6)\n",
        "# Create the actual plot in red, and our datapoints in blue,\n",
        "# and the best-fit method in green:  \n",
        "plt.plot(xdata,fdata, color='red', linewidth=2 ) + plt.plot(actual_data_x, actual_data_y, 'bo', alpha=0.3)+plt.plot(xdata,best_fit, color='green')\n",
        "\n",
        "#Restrict the output values to this range:\n",
        "plt.ylim(-30,30)\n",
        "\n",
        "# Add a title\n",
        "plt.title('Our test data (blue) for the original function (red) and our best-fit solution (green)')\n",
        "\n",
        "# Add X and y Label\n",
        "plt.xlabel('x axis')\n",
        "plt.ylabel('output')\n",
        "\n",
        "# Add a grid\n",
        "plt.grid(alpha=.4,linestyle='--')"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "6JCKDSbQx20d",
        "colab_type": "text"
      },
      "source": [
        "#0.4 Solving the system using Singular Value Decomposition\n",
        "\n",
        "Singular Value Decomposition decomposes a matrix M into three matrices\n",
        "\n",
        "$$ M = U \\Sigma V^\\dagger$$\n",
        "\n",
        "where $U$ and $V$ are square matrices and $\\Sigma$ is a rectangular matrix with singular values on the main diagonal."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "PFNKQo23KFEv",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "#  For this method, we use SVD on the data matrix M:\n",
        "U, S, Vh = linalg.svd(M, full_matrices=True)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "zReOuVkOKoQ_",
        "colab_type": "code",
        "outputId": "17bca7d6-5cb9-42c0-90df-18dffd368645",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 391
        }
      },
      "source": [
        "#  Uncomment the lines below to show the matrix U, S, and Vh:\n",
        "#  Notice that in S only the singular values are stored, not the entire matrix.\n",
        "U\n",
        "print(S)\n",
        "Vh = Vh.T"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[230.62319532   8.68544769   3.13479129   1.01445599]\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([[-5.31915281e-03, -1.21748309e-02, -2.84628521e-02,\n",
              "        -6.69594890e-02, -1.58958123e-01, -3.79169879e-01,\n",
              "        -9.08566063e-01],\n",
              "       [-1.15463511e-01, -2.09816603e-01, -2.85851013e-01,\n",
              "        -4.47177771e-01, -5.36771194e-01, -5.01360437e-01,\n",
              "         3.48540970e-01],\n",
              "       [-3.68888536e-01,  2.67757552e-01, -4.58457181e-01,\n",
              "         2.85136911e-01, -5.23171126e-01,  4.63603219e-01,\n",
              "        -1.10023290e-01],\n",
              "       [ 3.84582460e-01,  6.30968833e-01,  2.89970763e-01,\n",
              "         3.34648363e-01, -3.48994275e-01, -3.34269850e-01,\n",
              "         1.56104954e-01],\n",
              "       [-3.43029050e-01, -4.37070778e-01,  8.01953507e-02,\n",
              "         7.34300723e-01,  8.36776130e-04, -3.67138554e-01,\n",
              "         1.04306962e-01],\n",
              "       [-1.80999017e-01, -2.40942932e-01,  7.59386127e-01,\n",
              "        -1.51134130e-01, -4.63517736e-01,  3.03205317e-01,\n",
              "        -5.38041713e-02],\n",
              "       [-7.43120374e-01,  4.86753151e-01,  2.00285923e-01,\n",
              "        -2.00541284e-01,  2.76142314e-01, -2.26890846e-01,\n",
              "         5.27085256e-02]])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 74
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "swO65JoHLbgn",
        "colab_type": "code",
        "outputId": "aa702ff8-ccb2-48e0-895e-19dffee1a024",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        }
      },
      "source": [
        "#  The 2nd singular value is:\n",
        "S[1]"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "8.685447692333632"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 75
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "zs4i_zcQMg1d",
        "colab_type": "code",
        "outputId": "d21e8910-d910-43d5-8d5d-07f1ddaf3062",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        }
      },
      "source": [
        "# We can compare the original matrix M to the matrix reconstructed (called a1) from the SVD:\n",
        "# If they are close, then allclose(M, a1) should be True.\n",
        "\n",
        "#First, recast S as a matrix instead of a vector:\n",
        "sigma = np.zeros((U.shape[1], Vh.shape[0]))\n",
        "for i in range(min(sigma.shape)):\n",
        "     sigma[i, i] = S[i]\n",
        "#Reconstruct the matrix:\n",
        "a1 = np.dot(U, np.dot(sigma, Vh))\n",
        "\n",
        "#does a1 approximate M?\n",
        "np.allclose(M, a1)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "True"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 76
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "UsQKVQln09qn",
        "colab_type": "text"
      },
      "source": [
        "The SVD decomposition of matrix $M$ may have non-zero singular values in $\\Sigma.  To solve\n",
        "$$ M \\vec{x} = \\vec{b}$$\n",
        "\n",
        "we can replace $M$ with $U \\Sigma V^\\dagger$:\n",
        "$$ U \\, \\Sigma \\, V^\\dagger \\, \\vec{x} = \\vec{b}$$\n",
        "\n",
        "The inverse of matrices $U$ and $V$ are $U^\\dagger$ and $V^\\dagger$.  If we decompose $\\Sigma$ into a block matrix composed of the non-zero singular values, then we can restrict our singular value decomposition into invertible and non-invertible parts:\n",
        "\n",
        "$$ U \\, \\Sigma \\, V^\\dagger = \\left( \\begin{array}{c|c} U_{use} & U_{discard} \\\\ \\end{array} \\right) \\left( \\begin{array}{c|c} \\Sigma_{use} & 0 \\\\ \\hline 0 & \\Sigma_{discard} \\\\ \\end{array} \\right)  \\left( \\begin{array}{c|c} V_{use} & V_{discard} \\\\ \\end{array} \\right)^\\dagger$$\n",
        "\n",
        "Effectively, this allows us to replace $U \\Sigma V^\\dagger \\vec{x} = \\vec{b}$ with\n",
        "\n",
        "$$ U_{use} \\, \\Sigma_{use} \\, V_{use}^\\dagger \\, \\vec{x} = \\vec{b}$$\n",
        "\n",
        "The solution to this system is then\n",
        "\n",
        "$$\\vec{x} = V_{use} \\, \\Sigma_{use}^{-1} \\, U_{use}^\\dagger \\, \\vec{b}$$\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "OWnxUptVMreR",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "#  Based on the number of non-zero singular values in S, divide each of U, sigma, and Vh into \n",
        "#  U_use, Sigma_use, and Vh_use.\n",
        "#  and  discard U_d, Sigma_d, Vh_d\n",
        "useSigs = 4\n",
        "\n",
        "#Use these parts of U, S, and Vh\n",
        "(Vrows, Vcols) = Vh.shape\n",
        "(Urows, Ucols) = U.shape\n",
        "U_use = U[:Urows, :useSigs]\n",
        "Vh_use = Vh[:Vrows, :useSigs]\n",
        "sigma_use = sigma[:useSigs, :useSigs]\n",
        "\n",
        "#discard these parts of U, S, and Vh:\n",
        "U_d = U[:Urows, useSigs:]\n",
        "Vh_d = Vh[:Vrows, useSigs:]\n",
        "sigma_d = sigma[useSigs:, useSigs:]"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "FJvxlnbLM0X8",
        "colab_type": "code",
        "outputId": "2833f56d-80f9-4150-9452-3dd8a72e1ef6",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 51
        }
      },
      "source": [
        "#  We should be able to figure out the size of U and Vh that we'll use:\n",
        "#\n",
        "#  U S Vh  =   [         .     ] [             .         ]  [            .      ]\n",
        "#              [ U_use   .     ] [  sigma_use  .         ]  [   Vh_use   .      ]\n",
        "#              [         . U_d ] [.......................]  [            . Vh_d ]\n",
        "#              [         .     ] [             . sigma_d ]  [            .      ]\n",
        "#\n",
        "#\n",
        "#  Uncomment these lines (one at a time) to check that U_use, sigma_use, and Vh_use are the right sizes, as well as U_d, sigma_d, and Vh_d\n",
        "U_use.shape\n",
        "print(sigma_use.shape)\n",
        "Vh_use.shape\n",
        "U_d.shape\n",
        "sigma_d.shape\n",
        "Vh_d.shape"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "(4, 4)\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(7, 3)"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 81
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Mm72yqKVT7jz",
        "colab_type": "code",
        "outputId": "32612432-2a7a-45ec-983f-52600f2be5c7",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 136
        }
      },
      "source": [
        "#  If we use Singular Value Decomposition to break down M as a product of matrices\n",
        "#  M = U sigma Vh.conjugate_transpose()\n",
        "#  Then the part of A which corresponds to non-zero singular values is:\n",
        "#  M_use = U_use sigma_use Vh_use.conjugate_transpose()\n",
        "#\n",
        "#  So, the equation Mx = b becomes:  (U_use sigma_use Vh_use.conjugate_transpose()) x = b\n",
        "#  or\n",
        "#  svd_sol = (U_use sigma_use Vh_use.conjugate_transpose()).inverse() b\n",
        "#  or\n",
        "#  svd_sol = Vh_use * Sigma_use.inverse() * U_use.transpose() * b\n",
        "#  (since Vh_use.conjugate_transpose() = Vh_use.inverse() and U_use.transpose() = U_use.inverse() )\n",
        "svdsol = (Vh_use @ linalg.inv(sigma_use) @ U_use.transpose()) @ b\n",
        "svdsol"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "matrix([[-0.08072279],\n",
              "        [-0.39359024],\n",
              "        [ 2.93600285],\n",
              "        [-0.07685189],\n",
              "        [ 1.61350511],\n",
              "        [-0.41740239],\n",
              "        [ 3.94828318]])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 82
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NUyAAGKI424a",
        "colab_type": "text"
      },
      "source": [
        "We can test our solution against the output of our original system.\n",
        "\n",
        "The original output in our system is represented by $\\vec{b}$.\n",
        "\n",
        "Our suggested solution to $M\\vec{x} = \\vec{b}$ is $\\vec{svdsol}$.  That is, we think\n",
        "\n",
        "$$ M \\vec{svdsol} \\approx \\vec{b}$$\n",
        "\n",
        "If we have a good fit, then \n",
        "\n",
        "$$\\vec{b} - M \\vec{svdsol} \\approx 0$$\n",
        "\n",
        "To test our solution, we find the) magnitude of $\\vec{b} - M\\vec{svdsol}$:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hC0UDpjX43HF",
        "colab_type": "code",
        "outputId": "012b2bba-4863-4d13-fac2-11e0a34bd2b3",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        }
      },
      "source": [
        "#  Let's see how close this actually gets us by comparing the original b vector to the result of Mc:\n",
        "error = ((b-M*svdsol).transpose() * (b-M*svdsol))[0,0]\n",
        "error**0.5"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "167.23715571649322"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 83
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "6F6M7Y3I5bGY",
        "colab_type": "code",
        "outputId": "5ba75aaf-8773-48c0-8e80-c01fa5b6f85d",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 295
        }
      },
      "source": [
        "#Show a plot of the solutions using the best-fit method, and the SVD method:\n",
        "\n",
        "# We'll specify that x is a variable:\n",
        "x = sym.symbols('x')\n",
        "\n",
        "xdata = []\n",
        "fdata = []\n",
        "#set up our 100 data points on the interval from (-3, 3):\n",
        "xdata = np.array(range(100))/(100/6) - 3\n",
        "# find the ouput for our function on each data point:\n",
        "fdata = 4 - 3*xdata**2 + 5*xdata**6\n",
        "\n",
        "#construct the solution from svdsol\n",
        "svd_fit = svdsol[0,0] + svdsol[1,0]*xdata + svdsol[2,0]*(xdata**2) + svdsol[3,0]*(xdata**3) + svdsol[4,0]*(xdata**4) + svdsol[5,0]*(xdata**5) + svdsol[6,0]*(xdata**6)\n",
        "# Create the actual plot in red, and our datapoints in blue,\n",
        "# and the svd solution in aqua:  \n",
        "plt.plot(xdata,fdata, color='red', linewidth=2 ) + plt.plot(actual_data_x, actual_data_y, 'bo', alpha=0.3) + plt.plot(xdata, svd_fit, 'aqua')\n",
        "\n",
        "#Restrict the output values to this range:\n",
        "plt.ylim(-30,30)\n",
        "# Add a title\n",
        "plt.title('Our test data (blue) for the original function (red)  and svd solution (aqua)')\n",
        "\n",
        "# Add X and y Label\n",
        "plt.xlabel('x axis')\n",
        "plt.ylabel('output')\n",
        "\n",
        "# Add a grid\n",
        "plt.grid(alpha=.4,linestyle='--')"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NKBuj3wo5r79",
        "colab_type": "text"
      },
      "source": [
        "#0.5  Finding a better solution\n",
        "\n",
        "\n",
        "Both the above solutions give really bad approximations to the polynomial  \n",
        "$$f(x) = 4 -3x^2 +5x^6$$\n",
        "\n",
        "Why?  We only used three data points for our data.  Go back and modify the code so that you use *MORE DATA*\n",
        "\n",
        "(Hint:  You have to change two of the '3' numbers in the code above:  One to generate more data; the other to use the right number of singular values for your data)\n",
        "\n",
        "The graph below should show both the best-fit and svd solution:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "jfQsu02SYVQ-",
        "colab_type": "code",
        "outputId": "cd8719d2-1380-4717-f9d9-0a2acda9b2fc",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 295
        }
      },
      "source": [
        "#Show a plot of the solutions using the best-fit method, and the SVD method:\n",
        "\n",
        "# We'll specify that x is a variable:\n",
        "x = sym.symbols('x')\n",
        "\n",
        "xdata = []\n",
        "fdata = []\n",
        "#set up our 100 data points on the interval from (-3, 3):\n",
        "xdata = np.array(range(100))/(100/6) - 3\n",
        "# find the ouput for our function on each data point:\n",
        "fdata = 4 - 3*xdata**2 + 5*xdata**6\n",
        "\n",
        "#construct the solution from bestfit:\n",
        "best_fit = c[0,0] + c[1,0]*xdata + c[2,0]*(xdata**2) + c[3,0]*(xdata**3) + c[4,0]*(xdata**4) + c[5,0]*(xdata**5) + c[6,0]*(xdata**6)\n",
        "\n",
        "#construct the solution from svdsol\n",
        "svd_fit = svdsol[0,0] + svdsol[1,0]*xdata + svdsol[2,0]*(xdata**2) + svdsol[3,0]*(xdata**3) + svdsol[4,0]*(xdata**4) + svdsol[5,0]*(xdata**5) + svdsol[6,0]*(xdata**6)\n",
        "\n",
        "# Create the actual plot in red, and our datapoints in blue,\n",
        "# the best-fit method in green, and the svd solution in aqua:  \n",
        "plt.plot(xdata,fdata, color='red', linewidth=2 ) + plt.plot(actual_data_x, actual_data_y, 'bo', alpha=0.3) + plt.plot(xdata,best_fit, color='green') + plt.plot(xdata, svd_fit, 'aqua')\n",
        "\n",
        "\n",
        "#Restrict the output values to this range:\n",
        "plt.ylim(-30,30)\n",
        "# Add a title\n",
        "plt.title('Our test data (blue) for the original function (red); our best-fit solution (green);  and svd solution (aqua)')\n",
        "\n",
        "# Add X and y Label\n",
        "plt.xlabel('x axis')\n",
        "plt.ylabel('output')\n",
        "\n",
        "# Add a grid\n",
        "plt.grid(alpha=.4,linestyle='--')"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "anbAm6nrY_xc",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        ""
      ],
      "execution_count": 0,
      "outputs": []
    }
  ]
}