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    "# Homework #3\n",
    "### Assignment\n",
    "1. Define a function <code>euclidean_distance</code> that computes the distance between two 1D vectors of size n according to the formula:\n",
    "\n",
    "$$\n",
    "d(a,b) = \\sqrt{\\sum_{i=1}^n{(a_i - b_i)^2}}\n",
    "$$\n",
    "\n",
    "The function should check that the vector have the same size and implement the simplest iterative version and the two methods proposed within this lecture, selectable by an additional parameter <code>mode</code> of the function.\n",
    "\n",
    "Test the function with pairs of randomly generated vectors of size [5, 10, 20, 50] and measure its performance with the %timeit magic.\n",
    "\n",
    "2. Solve the following linear equations system, in the form $Ax = b$, using numpy functions described in the present lecture:\n",
    "\n",
    "$$\n",
    "4x + 3y + 2z = 25 \\\\\n",
    "-2x + 2y + 3z = -10 \\\\\n",
    "3x -5y + 2z = -4\n",
    "$$\n",
    "\n",
    "### Possible solutions to questions 1 & 2 \n",
    "#### Question 1"
   ]
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     "text": [
      "a = [0.04756666 0.17755706 0.21323304 0.08255607 0.36625129]\n",
      "b = [0.66822995 0.39391188 0.54365618 0.53536603 0.05558075]\n",
      "9.24 µs ± 368 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n",
      "ED LO= 0.918\n",
      "15.6 µs ± 321 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n",
      "ED SS= 0.918\n",
      "9.51 µs ± 101 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n",
      "ED DP= 0.918\n",
      "a = [0.04221634 0.60532675 0.97611896 0.25605099 0.9669407  0.2492855\n",
      " 0.01462257 0.48600282 0.30880954 0.11884866]\n",
      "b = [0.61442178 0.07522763 0.71285844 0.18123605 0.90568225 0.35599843\n",
      " 0.67274454 0.11149599 0.19818347 0.14177104]\n",
      "14.9 µs ± 296 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n",
      "ED LO= 1.133\n",
      "16.3 µs ± 1.62 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n",
      "ED SS= 1.133\n",
      "9.6 µs ± 179 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n",
      "ED DP= 1.133\n",
      "a = [0.36454152 0.35431341 0.2452204  0.10963099 0.91409061 0.00741397\n",
      " 0.06116185 0.43287493 0.77105923 0.69210503 0.97740584 0.9063517\n",
      " 0.42872982 0.7640695  0.83426393 0.48779364 0.91748042 0.64518327\n",
      " 0.33178232 0.91955856]\n",
      "b = [0.70857996 0.34099661 0.35538063 0.32251283 0.06596971 0.1466998\n",
      " 0.60591954 0.38263155 0.2213241  0.11373697 0.45443336 0.08267574\n",
      " 0.350592   0.42075258 0.8478867  0.36166327 0.46768857 0.52536167\n",
      " 0.50157416 0.78699725]\n",
      "25.9 µs ± 1.12 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n",
      "ED LO= 1.790\n",
      "15.4 µs ± 253 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n",
      "ED SS= 1.790\n",
      "9.59 µs ± 171 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n",
      "ED DP= 1.790\n",
      "a = [0.66241041 0.8977363  0.76395084 0.09898128 0.41696298 0.33718484\n",
      " 0.2419236  0.95001454 0.92678579 0.00159685 0.08473459 0.16011744\n",
      " 0.30823732 0.4361975  0.87232125 0.5002936  0.29713966 0.94571833\n",
      " 0.75614151 0.04086167 0.52076853 0.76616898 0.18614206 0.49347664\n",
      " 0.26085435 0.1626869  0.24500608 0.61899173 0.32166108 0.69792644\n",
      " 0.28951983 0.83369022 0.94368824 0.69743006 0.06780959 0.82593332\n",
      " 0.18528358 0.38031009 0.32104502 0.91594209 0.2993625  0.12514716\n",
      " 0.23557262 0.50277038 0.2702443  0.05095381 0.40815823 0.56183583\n",
      " 0.31517506 0.82925131]\n",
      "b = [0.27587699 0.42375746 0.77437957 0.9413573  0.77644966 0.41041505\n",
      " 0.017733   0.319019   0.0186726  0.77862412 0.78194412 0.54206786\n",
      " 0.58276347 0.45608191 0.18558384 0.89378062 0.61439171 0.2278767\n",
      " 0.85073462 0.42662207 0.14000123 0.69409834 0.04850972 0.95771897\n",
      " 0.48034958 0.64479738 0.78451301 0.24552058 0.3019615  0.76305405\n",
      " 0.34267731 0.35379068 0.54755861 0.94291601 0.12346746 0.97399342\n",
      " 0.12388551 0.88549354 0.46390048 0.70671859 0.73449434 0.06224399\n",
      " 0.8435283  0.26863556 0.32315355 0.15195794 0.41800812 0.92919966\n",
      " 0.97428459 0.32225621]\n",
      "64.2 µs ± 1.69 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n",
      "ED LO= 2.926\n",
      "16.5 µs ± 273 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n",
      "ED SS= 2.926\n",
      "9.56 µs ± 147 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n",
      "ED DP= 2.926\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "def euclidean_distance(a,b,mode):\n",
    "    l = len(a)\n",
    "    assert(l == len(b)),\"Arrays must have the same length\"\n",
    "    if mode == 0:\n",
    "        # iterative version\n",
    "        vsum = 0\n",
    "        for i in range(l):\n",
    "            vsum += (a[i]-b[i])**2\n",
    "        ed = np.sqrt(vsum)\n",
    "    elif mode == 1:\n",
    "        # sum of squares version\n",
    "        ed = np.sqrt(np.sum(np.square(a-b)))\n",
    "    elif mode == 2:\n",
    "        # dot product\n",
    "        ed = np.sqrt(np.dot(a-b,a-b))\n",
    "    else:\n",
    "        ed = None\n",
    "    \n",
    "    return ed\n",
    "\n",
    "for dimlist in [5,10,20,50]:\n",
    "    a = np.random.random(dimlist)\n",
    "    print('a = {}'.format(a))\n",
    "    b = np.random.random(dimlist)\n",
    "    print('b = {}'.format(b))\n",
    "\n",
    "    %timeit euclidean_distance(a,b,0) # with the loop\n",
    "    ED = euclidean_distance(a,b,0)\n",
    "    print('ED LO= {:.3f}'.format(ED))\n",
    "\n",
    "    %timeit euclidean_distance(a,b,1) # with the sum of squares\n",
    "    ED = euclidean_distance(a,b,1)\n",
    "    print('ED SS= {:.3f}'.format(ED))\n",
    "\n",
    "    %timeit euclidean_distance(a,b,2) # using dot product\n",
    "    ED = euclidean_distance(a,b,2)\n",
    "    print('ED DP= {:.3f}'.format(ED))\n"
   ]
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   "source": [
    "#### Question 2"
   ]
  },
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   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 5.  3. -2.]\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[4, 3, 2], [-2, 2, 3], [3, -5, 2]])\n",
    "b = np.array([25, -10, -4])\n",
    "\n",
    "x = np.linalg.inv(A).dot(b)\n",
    "print(x)"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
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