Project_4_Case_1_Solving_for_a_known_trigonometric_function_using_Singular_Values_or_Best_Fit.ipynb

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  "metadata": {
    "colab": {
      "name": "Copy of Project 4:  Case 1 - Solving for a known trigonometric 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_1_Solving_for_a_known_trigonometric_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": "RikUpdZwlBC1",
        "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": "hZ4yDsfTlH6l",
        "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 $f(x) = 4 - 2cos(x) + cos^2 (x)$\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": "FNuhqe22lOzW",
        "colab_type": "text"
      },
      "source": [
        "#  Overview for Case 1: A known trigonometric function $f(x) = 4 - 2\\cos(x) + \\cos^2(x)$  \n",
        "We generate data for the polynomial function \n",
        "$$f(x) = 4 -3\\cos(x) + \\cos^2 (x)$$ \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 $3$ of them.\n",
        "\n",
        "We want to find out what combination of polynomials $1, x, x^2, x^3, x^4, x^5, x^6, x^7$ and $x^8$ 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",
        "$$ [  c_0 * 1 + c_1 * x + c_2 * x^2 + c_3 * x^3 + ... c_8 x^8 = f(x) ] $$\n",
        "\n",
        "or\n",
        "\n",
        "$$ \\left[ \\begin{array}{ccccccccc} 1 & x & x^2 & x^3 & x^4 & x^5 & x^6 & x^7 & x^8 \\end{array} \\right] \\left( \\begin{array}{c} c_0 \\\\ c_1 \\\\ c_2 \\\\ c_3 \\\\ c_4 \\\\ c_5 \\\\c_6 \\\\ c7 \\\\ c8 \\\\ \\end{array} \\right) = \\left[ f(x) \\right] $$\n",
        "\n",
        "\n",
        "Be careful -- the unknowns are the coefficients in front - the $c_0, c_1, c_2, ..., c_8$ values.  Since we know $x$, then we know $x, x^2, x^3, x^4, ... x^8$. 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",
        "$$ [  c_0*1  + c_1*2 + c_2*4 + c_3*8 + ... + c_8*256 = 312 ] $$\n",
        "\n",
        "or\n",
        "$$ \\left[ \\begin{array}{ccccccccc} 1 & 2 & 4 & 8 & 16 & x32 & 64 & 128 & 256\\end{array} \\right] \\left( \\begin{array}{c} c_0 \\\\ c_1 \\\\ c_2 \\\\ c_3 \\\\ c_4 \\\\ c_5 \\\\c_6 \\\\ c_6 \\\\ c_8 \\\\ \\end{array} \\right) = \\left[ 312 \\right] $$\n",
        "\n",
        "where the vector $\\vec{c}$ contains the unknowns $c_0, c_1, c_2, \\ldots c_8$\n",
        "\n",
        "So... let's generate the matrix:\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zGqpUiw8D3CP",
        "colab_type": "text"
      },
      "source": [
        "#1.1 Setting up the system\n",
        "\n",
        "We'll generate some data points for our system using the function \n",
        "$$ f(x) = 4 - 2\\cos(x) + \\cos^2(x)$$\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": "v8SZR2faD-1h",
        "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   data points:\n",
        "for i in range(3):\n",
        "    # pick a random number between -2pi and 2pi\n",
        "    x = 4*np.pi*random.random() - 2*np.pi\n",
        "    f = 4 - 2*np.cos(x) + np.cos(x)**2\n",
        "    actual_data_x.append(x)\n",
        "    actual_data_y.append(f)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "LuBxCrVmEQJ6",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "# uncomment code below  to look at the data in actual_data_x or in actual_data_y\n",
        "\n",
        "#actual_data_x\n",
        "#actual_data_y"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "zlBDrsHMER8j",
        "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, x**7, x**8])\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": "GF2h3SjGEZr0",
        "colab_type": "code",
        "outputId": "7ee5509f-f2fc-4413-b259-846357b9a27c",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 68
        }
      },
      "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": 12,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "matrix([[6.60571282],\n",
              "        [4.03353266],\n",
              "        [5.83103737]])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 12
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "e-MMyhuYEsgT",
        "colab_type": "text"
      },
      "source": [
        "#1.2:  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 $x = -2\\pi$ and $x=2\\pi$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "PGd38j_4qfm1",
        "colab_type": "code",
        "outputId": "affc86f0-01b5-4ed0-ccf7-33a0199d2fb4",
        "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",
        "\n",
        "#set up our 100 data points on the interval from (-2pi, 2pi):\n",
        "xdata = np.array(range(100))/(100/(4*np.pi)) - 2*np.pi\n",
        "# find the ouput for our function on each data point:\n",
        "fdata = 4 - 2*np.cos(xdata) + np.cos(xdata)**2\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": 13,
      "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": "QlXzepbiFEiQ",
        "colab_type": "text"
      },
      "source": [
        "#1.3:  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": "Os9TvUT5qiQO",
        "colab_type": "code",
        "outputId": "b643a80e-9b4d-4918-838f-63c14dd95a22",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 170
        }
      },
      "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"
      ],
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "matrix([[ 9.07637935e+00],\n",
              "        [ 1.18776888e-01],\n",
              "        [ 2.34831683e-01],\n",
              "        [ 8.72100134e-02],\n",
              "        [ 2.00741611e-02],\n",
              "        [ 2.51620992e-03],\n",
              "        [-1.21977480e-03],\n",
              "        [-3.95885124e-06],\n",
              "        [ 4.19187820e-05]])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 14
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_intXzv7FS-D",
        "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": "cyHQjzItqlNn",
        "colab_type": "code",
        "outputId": "6f8b835f-aea8-460c-ada6-8089e0e4a9f6",
        "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": 15,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "2.6252657431368585"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 15
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SqBJrx91FaGg",
        "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": "prV8OVeoFbvX",
        "colab_type": "code",
        "outputId": "c592c68e-bad0-4c2a-8539-37a312d5a5c5",
        "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 (-2pi, 2pi):\n",
        "xdata = np.array(range(100))/(100/(4*np.pi)) - 2*np.pi\n",
        "# find the ouput for our function on each data point:\n",
        "fdata = 4 - 2*np.cos(xdata) + np.cos(xdata)**2\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) + c[7,0]*(xdata**7) + c[8,0]*(xdata**8)\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": 17,
      "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": "UUukKVjEF1ZL",
        "colab_type": "text"
      },
      "source": [
        "#1.4 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": "MCh6XRCXqnu2",
        "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)\n",
        "Vh = Vh.T"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "_Or_krPzqqr7",
        "colab_type": "code",
        "colab": {}
      },
      "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",
        "#S\n",
        "#Vh"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "0Z3xxDuhGCHc",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "#  The 2nd singular value is:\n",
        "S[1]"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "7XoANhuXqzbi",
        "colab_type": "code",
        "outputId": "a5c3f419-be26-40cf-91d2-ae69b6485013",
        "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",
        "a1 = np.dot(U, np.dot(sigma, Vh))\n",
        "#does a1 approximate M?\n",
        "np.allclose(M, a1)"
      ],
      "execution_count": 19,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "False"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 19
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "GEiJnb8ZHxe0",
        "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": "MD8yFnR-q1U8",
        "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 = 3\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": "k5xb86Grq3V9",
        "colab_type": "code",
        "colab": {}
      },
      "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",
        "#  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",
        "#sigma_use.shape\n",
        "#Vh_use.shape\n",
        "#U_d.shape\n",
        "#sigma_d.shape\n",
        "#Vh_d.shape"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "dYi5SpNJq5WP",
        "colab_type": "code",
        "outputId": "9b49c727-50cc-426f-f1f2-2d63dd469854",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 170
        }
      },
      "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": 21,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "matrix([[ 0.00024403],\n",
              "        [-0.00072442],\n",
              "        [ 0.00204904],\n",
              "        [-0.005403  ],\n",
              "        [ 0.01275597],\n",
              "        [-0.02465425],\n",
              "        [ 0.02910395],\n",
              "        [ 0.0162938 ],\n",
              "        [ 0.00190014]])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 21
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2HbYoWz0IF4f",
        "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": "0gfbfu6cIKb9",
        "colab_type": "code",
        "outputId": "f417a0c2-d115-477c-efbf-d48f5cd3cca5",
        "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": 22,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "3.1359742465361133e-13"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 22
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "LX49XLRwq9Ys",
        "colab_type": "code",
        "outputId": "3b728808-8d93-487d-ba74-6403c669223d",
        "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 (-2pi, 2pi):\n",
        "xdata = np.array(range(100))/(100/(4*np.pi)) - 2*np.pi\n",
        "\n",
        "# find the ouput for our function on each data point:\n",
        "fdata = 4 - 2*np.cos(xdata) + np.cos(xdata)**2\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) + svdsol[7,0]*(xdata**7)+svdsol[8,0]*(xdata**8)\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); 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='--')\n"
      ],
      "execution_count": 24,
      "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": "LcE53P4SI8S8",
        "colab_type": "text"
      },
      "source": [
        "#1.5  Finding a better solution\n",
        "\n",
        "\n",
        "Both the above solutions give really bad approximations to the polynomial  \n",
        "$$f(x) = 4 - 2\\cos(x) + \\cos^2(x)$$\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": "znsFEjFNcCgI",
        "colab_type": "code",
        "outputId": "5e1ca530-0be4-44f4-c757-623fde56bef1",
        "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 (-2pi, 2pi):\n",
        "xdata = np.array(range(100))/(100/(4*np.pi)) - 2*np.pi\n",
        "\n",
        "# find the ouput for our function on each data point:\n",
        "fdata = 4 - 2*np.cos(xdata) + np.cos(xdata)**2\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) + svdsol[7,0]*(xdata**7)+svdsol[8,0]*(xdata**8)\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) + c[7,0]*(xdata**7) + c[8,0]*(xdata**8)\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",
        "#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); 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": 25,
      "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": "KXZs2FIBJGy-",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        ""
      ],
      "execution_count": 0,
      "outputs": []
    }
  ]
}