NumPy-S3_A sample solve in Linear Algebra using NumPy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A sample solve in Linear Algebra using NumPy\n",
    "\n",
    "#### https://github.com/SelcukDE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 3*x + y = 9\n",
    "# x + 2*y = 8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.array([[3, 1], [1, 2]])\n",
    "b = np.array([9, 8])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3, 1],\n",
       "       [1, 2]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([9, 8])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2., 3.])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.solve(a, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Using array-generating functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For larger arrays it is inpractical to initialize the data manually, using explicit python lists. Instead we can use one of the many functions in `numpy` that generate arrays of different forms. Some of the more common are:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### arange"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# create a range\n",
    "# arguments: start, stop, step\n",
    "x = np.arange(0, 10, 1)\n",
    "\n",
    "x\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.00000000e+00, -8.00000000e-01, -6.00000000e-01, -4.00000000e-01,\n",
       "       -2.00000000e-01, -2.22044605e-16,  2.00000000e-01,  4.00000000e-01,\n",
       "        6.00000000e-01,  8.00000000e-01])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = np.arange(-1, 1, 0.2)\n",
    "y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### linspace and logspace"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.        ,  0.41666667,  0.83333333,  1.25      ,  1.66666667,\n",
       "        2.08333333,  2.5       ,  2.91666667,  3.33333333,  3.75      ,\n",
       "        4.16666667,  4.58333333,  5.        ,  5.41666667,  5.83333333,\n",
       "        6.25      ,  6.66666667,  7.08333333,  7.5       ,  7.91666667,\n",
       "        8.33333333,  8.75      ,  9.16666667,  9.58333333, 10.        ])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# using linspace, both end points ARE included\n",
    "np.linspace(0, 10, 25)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.00000000e+00, 3.03773178e+00, 9.22781435e+00, 2.80316249e+01,\n",
       "       8.51525577e+01, 2.58670631e+02, 7.85771994e+02, 2.38696456e+03,\n",
       "       7.25095809e+03, 2.20264658e+04])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.logspace(0, 10, 10, base=np.e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### random data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.5488135 , 0.71518937, 0.60276338, 0.54488318, 0.4236548 ],\n",
       "       [0.64589411, 0.43758721, 0.891773  , 0.96366276, 0.38344152],\n",
       "       [0.79172504, 0.52889492, 0.56804456, 0.92559664, 0.07103606],\n",
       "       [0.0871293 , 0.0202184 , 0.83261985, 0.77815675, 0.87001215],\n",
       "       [0.97861834, 0.79915856, 0.46147936, 0.78052918, 0.11827443]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# uniform random numbers in [0,1]\n",
    "# seed() must be underlined to the student. what is it used for?\n",
    "np.random.seed(0)\n",
    "np.random.rand(5,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.5488135 , 0.71518937, 0.60276338],\n",
       "       [0.54488318, 0.4236548 , 0.64589411],\n",
       "       [0.43758721, 0.891773  , 0.96366276]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "np.random.rand(3,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.38344152, 0.79172504, 0.52889492],\n",
       "       [0.56804456, 0.92559664, 0.07103606],\n",
       "       [0.0871293 , 0.0202184 , 0.83261985]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.rand(3,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.5488135 , 0.71518937, 0.60276338],\n",
       "       [0.54488318, 0.4236548 , 0.64589411],\n",
       "       [0.43758721, 0.891773  , 0.96366276]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.seed(0)\n",
    "np.random.rand(3,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.38344152, 0.79172504, 0.52889492],\n",
       "       [0.56804456, 0.92559664, 0.07103606],\n",
       "       [0.0871293 , 0.0202184 , 0.83261985]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.rand(3,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.57019677, 0.43860151, 0.98837384, 0.10204481, 0.20887676],\n",
       "       [0.16130952, 0.65310833, 0.2532916 , 0.46631077, 0.24442559],\n",
       "       [0.15896958, 0.11037514, 0.65632959, 0.13818295, 0.19658236],\n",
       "       [0.36872517, 0.82099323, 0.09710128, 0.83794491, 0.09609841],\n",
       "       [0.97645947, 0.4686512 , 0.97676109, 0.60484552, 0.73926358]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.rand(5,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.36433431, -0.67877739, -0.35362786],\n",
       "       [-0.74074747, -0.67502183, -0.13278426],\n",
       "       [ 0.61980106,  1.79116846,  0.17100044]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# standard normal distributed random numbers\n",
    "np.random.randn(3,3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### diag"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0, 0, 0],\n",
       "       [0, 2, 0, 0],\n",
       "       [0, 0, 3, 0],\n",
       "       [0, 0, 0, 4]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# a diagonal matrix\n",
    "np.diag([1,2,3,4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 0, 0, 0, 0],\n",
       "       [1, 0, 0, 0, 0],\n",
       "       [0, 2, 0, 0, 0],\n",
       "       [0, 0, 3, 0, 0],\n",
       "       [0, 0, 0, 4, 0]])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# diagonal with offset from the main diagonal\n",
    "np.diag([1,2,3,4], k=-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 0, 1, 0, 0, 0],\n",
       "       [0, 0, 0, 2, 0, 0],\n",
       "       [0, 0, 0, 0, 3, 0],\n",
       "       [0, 0, 0, 0, 0, 4],\n",
       "       [0, 0, 0, 0, 0, 0],\n",
       "       [0, 0, 0, 0, 0, 0]])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.diag([1,2,3,4], k=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Manipulating arrays"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fancy indexing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also use index masks: If the index mask is an Numpy array of data type `bool`, then an element is selected (True) or not (False) depending on the value of the index mask at the position of each element: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 2, 3, 4])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = np.array([i for i in range(5)])\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 2, 3, 4])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = np.array(range(5))\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 2])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "row_mask = np.array([True, False, True, False, False])\n",
    "B[row_mask]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 2])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "row_mask = np.array([1, 0, 1, 0, 0], dtype = bool)\n",
    "B[row_mask]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This feature is very useful to conditionally select elements from an array, using for example comparison operators:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0. , 0.5, 1. , 1.5, 2. , 2.5, 3. , 3.5, 4. , 4.5, 5. , 5.5, 6. ,\n",
       "       6.5, 7. , 7.5, 8. , 8.5, 9. , 9.5])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.arange(0, 10, 0.5)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([False, False, False, False, False, False, False, False, False,\n",
       "       False, False,  True,  True,  True,  True, False, False, False,\n",
       "       False, False])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mask = (5 < x) & (x < 7.5)\n",
    "mask"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5.5, 6. , 6.5, 7. ])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x[mask]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Functions for extracting data from arrays and creating arrays"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ``where``"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The index mask can be converted to position index using the `where` function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([False, False, False, False, False, False, False, False, False,\n",
       "       False, False,  True,  True,  True,  True, False, False, False,\n",
       "       False, False])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mask"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([11, 12, 13, 14], dtype=int64),)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "indices = np.where(mask)\n",
    "indices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5.5, 6. , 6.5, 7. ])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x[indices]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5.5, 6. , 6.5, 7. ])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x[mask]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ``diag``"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With the diag function we can also extract the diagonal and subdiagonals of an array:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2, 3, 4],\n",
       "       [5, 6, 7, 8],\n",
       "       [1, 2, 3, 4],\n",
       "       [5, 6, 7, 8]])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[1, 2, 3, 4], [5, 6, 7, 8,], [1, 2, 3, 4], [5, 6, 7, 8,]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 6, 3, 8])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.diag(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0, 0, 0],\n",
       "       [0, 6, 0, 0],\n",
       "       [0, 0, 3, 0],\n",
       "       [0, 0, 0, 8]])"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.diag([1, 6, 3, 8])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2, 3, 4],\n",
       "       [5, 6, 7, 8],\n",
       "       [1, 2, 3, 4],\n",
       "       [5, 6, 7, 8]])"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 7, 4])"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.diag(A, k=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4])"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.diag(A, k=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ``take``"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `take` function is similar to fancy indexing described above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-3, -2, -1,  0,  1,  2])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v2 = np.arange(-3, 3)\n",
    "v2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2,  0,  2])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# fancy indexing\n",
    "row_index= [1, 3, 5]\n",
    "v2[row_index]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2,  0,  2])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v2.take(row_index)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But `take` also works on lists and other objects:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ``choose``"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Constructs an array by picking elements from several arrays:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "which = [1, 0, 1, 2]\n",
    "choices = [[-2, -3, -4, -5], [6, 7, 8, 9], [1,2,3,4]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 6, -3,  8,  4])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.choose(which, choices)"
   ]
  }
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