Polynomial_Approximation_Problem_Solution_2_2.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "Polynomial Approximation Problem Solution 2/2",
      "provenance": [],
      "collapsed_sections": [],
      "authorship_tag": "ABX9TyPGEyCnfOBRpk5ffAJY8+Pw",
      "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/cakshay2013/Convex-Optimization-Problems-and-Solutions/blob/main/Polynomial_Approximation_Problem_Solution_2_2.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 486
        },
        "id": "2QwoUpT1nQZx",
        "outputId": "8d1f5551-5694-4e96-ec23-fc339d1817cf"
      },
      "source": [
        "import numpy as np\n",
        "import math as m\n",
        "import matplotlib.pyplot as plt\n",
        "import cvxpy as cp\n",
        "N=10\n",
        " \n",
        "#least squares objective fcn\n",
        "n1=5\n",
        "\n",
        "a1 = cp.Variable(n1)\n",
        "x = np.zeros(2*N+1)\n",
        "x[0] = -1\n",
        "for i in range(0, 2*N-1):\n",
        "  x[i+1] = x[i] + 1/N\n",
        "\n",
        "X1 = np.zeros([n1,2*N+1])\n",
        "for i in range(0, n1-1):\n",
        "  for j in range(0, 2*N):\n",
        "    X1[i,j] = np.power(x[j], i)\n",
        "\n",
        "# Objective fcn\n",
        "#c1=cp.square(cp.norm((np.power(x, n1))- a1@X1))\n",
        "c1=cp.norm((np.power(x, n1))- a1@X1,1)\n",
        "objective = cp.Minimize(c1)\n",
        " \n",
        "#Constraints\n",
        "constraints = []\n",
        "prob = cp.Problem(objective, constraints)\n",
        "prob.solve()\n",
        " \n",
        "#Answers\n",
        "#print(\"x:\", x)\n",
        "#print(\"X1:\", X1)\n",
        "print(\"Optimal polynomial coefficients a1:\", a1.value)\n",
        "print(\"Optimum cost function value c1:\",c1.value)\n",
        " \n",
        "n2=10\n",
        "\n",
        "a2 = cp.Variable(n2)\n",
        "X2 = np.empty([n2,2*N+1])\n",
        "for i in range(0, n2-1):\n",
        "  for j in range(0, 2*N):\n",
        "    X2[i,j] = np.power(x[j], i)\n",
        " \n",
        "#Objective fcn\n",
        "#c2=cp.square(cp.norm((np.power(x, n2))- a2@X2))\n",
        "c2=cp.norm((np.power(x, n2))- a2@X2,1)\n",
        "objective = cp.Minimize(c2)\n",
        " \n",
        "#Constraints\n",
        "constraints = []\n",
        "prob = cp.Problem(objective, constraints)\n",
        "prob.solve()\n",
        " \n",
        "#Answers\n",
        "#print(\"X2:\", X2)\n",
        "print(\"Optimal polynomial coefficients a2:\", a2.value)\n",
        "print(\"Optimum cost function value c2:\",c2.value)\n",
        " \n",
        "n3=20\n",
        "\n",
        "a3 = cp.Variable(n3)\n",
        "X3 = np.empty([n3,2*N+1])\n",
        "for i in range(0, n3-1):\n",
        "  for j in range(0, 2*N):\n",
        "    X3[i,j] = np.power(x[j], i)\n",
        " \n",
        "#Objective fcn\n",
        "#c3=cp.square(cp.norm((np.power(x, n3))- a3@X3))\n",
        "c3=cp.norm((np.power(x, n3))- a3@X3,1)\n",
        "objective = cp.Minimize(c3)\n",
        " \n",
        "#Constraints\n",
        "constraints = []\n",
        "prob = cp.Problem(objective, constraints)\n",
        "prob.solve()\n",
        " \n",
        "#Answers\n",
        "#print(\"X3:\", X3)\n",
        "print(\"Optimal polynomial coefficients a3:\", a3.value)\n",
        "print(\"Optimum cost function value c3:\",c3.value)\n",
        " \n",
        "n = [n1,n2,n3]\n",
        "c = [c1.value,c2.value,c3.value]\n",
        "#c = np.reshape(c, (3,1))\n",
        "\n",
        "plt.plot(n,c)\n",
        "plt.xlabel('x - axis: n') \n",
        "plt.ylabel('y - axis: optimal cost function values') \n",
        "plt.title('optimal cost function values vs n') \n",
        "plt.show()"
      ],
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Optimal polynomial coefficients a1: [ 0.018  -0.1825 -0.072   0.98    0.    ]\n",
            "Optimum cost function value c1: 0.6277800000062301\n",
            "Optimal polynomial coefficients a2: [-0.00627968 -0.01247122  0.07459316  0.07868951 -0.12856083 -0.0956841\n",
            " -0.58539364  0.00493601  1.62146539  0.01311829]\n",
            "Optimum cost function value c2: 0.04417411513961348\n",
            "Optimal polynomial coefficients a3: [-1.99620058e-08 -6.64213910e-08  8.18326082e-06  1.77273520e-05\n",
            " -5.39794074e-04 -4.52398069e-04  1.20097221e-02  4.67065758e-03\n",
            " -1.71136961e+00 -2.48354487e-02  7.30912472e-01  7.36724462e-02\n",
            " -2.47204901e+00 -1.22543470e-01  5.00194913e+00  1.06373708e-01\n",
            " -5.93056779e+00 -3.73541402e-02  3.78441280e+00  1.58478293e+00]\n",
            "Optimum cost function value c3: 1.9614235432829888e-07\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    }
  ]
}