lab0.ipynb

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        "<a href=\"https://colab.research.google.com/github/yajuna/tmath308/blob/master/lab0.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
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      "cell_type": "markdown",
      "metadata": {
        "id": "DUOmkHmbXeRJ"
      },
      "source": [
        "# Lab 0: Introduction to Python\n",
        "\n",
        "Welcome to Python lab! In these exercises, we will learn some basic commands in Python. A couple of these exercises are mentioned in class."
      ]
    },
    {
      "cell_type": "code",
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      "metadata": {
        "id": "kByj4SCMXeRK"
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      "source": [
        "# import numpy and scipy for numerical and scientific computations; import visualization tool\n",
        "import numpy as np\n",
        "import scipy as sp\n",
        "\n",
        "# visualization tools\n",
        "import plotly.graph_objects as go\n",
        "from plotly.subplots import make_subplots\n",
        "from typing import Tuple, Iterable"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bL8oryUhXeRL"
      },
      "source": [
        "# We have looked at this system in class\n",
        "\n",
        "\\begin{align}\n",
        "3x_1+2x_2+x_3=1\\\\\n",
        "-x_1+x_2+5x_3=2\\\\\n",
        "2x_1-3x_2=3\\\\\n",
        "x_2+x_3=4\\end{align}\n",
        "and know that it has no **exact** solutions. In this part of the lab, we study the least square solution to it."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "VwxrEoaSXeRL",
        "outputId": "ff33c87f-6b0a-4e94-fdda-03bfa01d9ddb",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
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      "outputs": [
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          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[[ 3  2  1]\n",
            " [-1  1  5]\n",
            " [ 2 -3  0]\n",
            " [ 0  1  1]] [[1]\n",
            " [2]\n",
            " [3]\n",
            " [4]]\n"
          ]
        }
      ],
      "source": [
        "# define coefficient matrix and right hand side\n",
        "\n",
        "B = np.array([[3,2,1],[-1,1,5],[2,-3,0],[0,1,1]])\n",
        "b = np.array([[1],[2],[3],[4]])\n",
        "\n",
        "# type print(B,b) to print the above variables\n",
        "print(B,b)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "V4W6s7uQXeRL"
      },
      "source": [
        "The python command x = np.linalg.lstsq(B,b) computes the least square solution of the system $Bx=b$. Bye that we mean the $x$ such that the difference $Bx-b$ is the smallest in some sense (the 2-norm)."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "R36OvxVcXeRL",
        "outputId": "ad2c3238-2b69-42bb-d659-db65cd8e2f60",
        "colab": {
          "base_uri": "https://localhost:8080/"
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      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "(array([[ 0.57342657],\n",
            "       [-0.41258741],\n",
            "       [ 0.72027972]]), array([14.76923077]), 3, array([5.59367481, 3.70281003, 3.31662479]))\n"
          ]
        }
      ],
      "source": [
        "# type in the command x = np.linalg.lstsq(B,b, rcond = None) and then type print(x) to see the output\n",
        "\n",
        "x = np.linalg.lstsq(B,b, rcond = None)\n",
        "print(x)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "UCLPdnXCXeRL"
      },
      "source": [
        "You probably have noticed that the x has several components, not all are of them are $x$ as mentioned above. Below, type x[0], x[1], x[2], and x[3] and see what you get."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "RZnFTyrhXeRM",
        "outputId": "f2ef09ad-2faf-4cb0-a261-261aa0b37a6a",
        "colab": {
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        }
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      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "3\n"
          ]
        }
      ],
      "source": [
        "# type x[0]-x[3], then type in r = np.dot(B,x[0])-b. Type r and see what you get;\n",
        "# then type in rankB = np.linalg.matrix_rank(B) and see what you get\n",
        "rankB = np.linalg.matrix_rank(B)\n",
        "print(rankB)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rgsV7BxZXeRM"
      },
      "source": [
        "# In class, we have discussed two systems being equivalent. This part solve the two systems and show you get the same solutions.\n",
        "\n",
        "In class, we have discussed the two systems\n",
        "\\begin{align}\n",
        "5x_1+10x_2-5x_3=5\\\\\n",
        "2x_1-x_2+x_3=3\\\\\n",
        "-x_1+2x_2+3x_3=7\\end{align}\n",
        "\n",
        "and\n",
        "\n",
        "\\begin{align}\n",
        "x_1+2x_2-x_3=1\\\\\n",
        "4x_2+2x_3=8\\\\\n",
        "-5x_2+3x_3=1\\end{align}\n",
        "\n",
        "Now we will solve them with np.linalg.solve\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "S7UgTV1EXeRM",
        "outputId": "a2cc21a3-2754-4ec8-a3c5-d683b1ef2c58",
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          "base_uri": "https://localhost:8080/"
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      "outputs": [
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          "name": "stdout",
          "text": [
            "[[1.]\n",
            " [1.]\n",
            " [2.]] [[1.]\n",
            " [1.]\n",
            " [2.]]\n"
          ]
        }
      ],
      "source": [
        "# run the following code and read the output. Explain the meaning of C, c, D, and d.\n",
        "\n",
        "C = np.array([[5,-10,5],[2,-1,1],[-1,2,3]])\n",
        "c = np.array([[5],[3],[7]])\n",
        "solc = np.linalg.solve(C,c)\n",
        "\n",
        "D = np.array([[1,2,-1],[0,4,2],[0,-5,3]])\n",
        "d = np.array([[1],[8],[1]])\n",
        "sold = np.linalg.solve(D,d)\n",
        "\n",
        "print(solc, sold)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Hlc3TttXXeRM"
      },
      "source": [
        "# In class, we discussed the geometric meaning of $2\\times 2$ systems. In this part, we will look at the geometric meaning of $3\\times 3$ systems.\n",
        "\n",
        "Consider the two linear equation $x_1- 2x_2 + x_3 = 0$ and $2x_2 - 8x_3 = 8$. We know they represent planes in 3D space. Run the following code."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "y_Hm-oX4XeRM",
        "outputId": "26fb9249-51e1-460a-bf4a-8aacadeb50bd",
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          "metadata": {}
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      ],
      "source": [
        "# run the following code, and make a comment to explain what this code means. Comments are made following a '#' sign.\n",
        "# Update: if you have not taken tmath126 yet, you do not need to worry about this portion\n",
        "\n",
        "n1 = (1,-2,1)\n",
        "n2 = (0,2,-8)\n",
        "\n",
        "d1 = 0\n",
        "d2 = -8\n",
        "\n",
        "def plotPlane(fig: go.Figure,\n",
        "              normal: Tuple[int, int, int],\n",
        "              d: int,\n",
        "              values: Iterable,\n",
        "              colorScaleName: str) -> None:\n",
        "    \"\"\"\n",
        "        :param fig: figure to plot on\n",
        "        :param colorScaleName: choose from <https://plotly.com/javascript/colorscales/>\n",
        "    \"\"\"\n",
        "    # x, y, z\n",
        "    x, y = np.meshgrid(values, values)\n",
        "    z = (-normal[0] * x - normal[1] * y - d) * 1. / normal[2]\n",
        "\n",
        "    # draw plane\n",
        "    surface = go.Surface(x=x, y=y, z=z, colorscale=colorScaleName, showscale=False)\n",
        "    fig.add_trace(surface, row=1, col=1)\n",
        "\n",
        "\n",
        "# create figure\n",
        "fig = make_subplots(rows=1, cols=1, specs=[[{'type': 'surface'}]])\n",
        "# plot two intersectioned surfaces\n",
        "values = range(-10, 11)\n",
        "plotPlane(fig, n1, d1, values, \"Hot\")\n",
        "plotPlane(fig, n2, d2, values, \"phase\")\n",
        "fig.show()\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zKvTkO0NXeRN"
      },
      "source": [
        "# Generate a random matrix $A$ of $4\\times 4$ and a vector $y$ of size $4\\times 1$. Solve the system $(A|\\mathbf{y})$ (or $Az=y$)."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "0kj_I2AuXeRN"
      },
      "outputs": [],
      "source": [
        "# type in your codes here\n",
        "\n",
        "A = np.random.randint(0, 59, (4,4))\n",
        "y = np.random.randint(0, 6, (4,1))\n",
        "soln = np.linalg.solve(A,y)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "NNMOJvnZXeRN"
      },
      "outputs": [],
      "source": [
        "\n",
        "\n",
        "\n"
      ]
    }
  ],
  "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": "3.7.4"
    },
    "colab": {
      "provenance": [],
      "include_colab_link": true
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}