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    "### Machine Learning  (Background): Gram-Schmidt process\n",
    "$\\mathbf{u_k}\\leftarrow \\mathbf{v_k}-\\sum_{j=1}^{k-1} proj_{\\mathbf{u}_j}(\\mathbf{v_k})$\n",
    "<br>$, proj_{\\mathbf{u}_j}(\\mathbf{v_k})=\\frac{< \\mathbf{v}_k,\\mathbf{u}_j>}\n",
    "{< \\mathbf{u}_j,\\mathbf{u}_j>}\\mathbf{u}_j$\n",
    "###### by Hamed Shah-Hosseini\n",
    "Explanation at: https://www.pinterest.com/HamedShahHosseini/Machine-Learning/Background-Knowledge\n",
    "<br>Explanation in Persian: https://www.instagram.com/words.persian\n",
    "<br>Code that: https://github.com/ostad-ai/Machine-Learning"
   ]
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  {
   "cell_type": "code",
   "execution_count": 51,
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   "source": [
    "# importing the required module\n",
    "# درون‌بَری سنجانه نیازداشته\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "3d6ffd39",
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    "# this modified method can have smaller rounding errors\n",
    "# این روش سنجیده‌سازی شده، میتواند دارای ایرَنگ‌های گِرد کردن کوچکتری باشد\n",
    "def modified_gram_schmidt(A,normalize=False): # row vectors\n",
    "    A=A.astype('float32')\n",
    "    n=A.shape[0] # no. of rows\n",
    "    for j in range(n):\n",
    "        for k in range(j):\n",
    "            A[j]-=np.dot(A[j],A[k])/np.dot(A[k],A[k])*A[k]\n",
    "    if normalize:\n",
    "        for j in range(n):\n",
    "            A[j]/=np.linalg.norm(A[j])\n",
    "    return A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "78a08b83",
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   "source": [
    "# The classical method which may have large rounding errors\n",
    "# روش سَررَده‌ای که میتواند دارای ایرَنگ‌های گِرد کردن بزرگ باشد\n",
    "def gram_schmidt(A,normalize=False):# row vectors\n",
    "    A=A.astype('float32')\n",
    "    n=A.shape[0] # no. of rows\n",
    "    temprow=np.zeros(A.shape[1])\n",
    "    for j in range(n):\n",
    "        temprow.fill(0)\n",
    "        for k in range(j):\n",
    "            temprow+=np.dot(A[j],A[k])/np.dot(A[k],A[k])*A[k]\n",
    "        A[j]-=temprow\n",
    "    if normalize:\n",
    "        for j in range(n):\n",
    "            A[j]/=np.linalg.norm(A[j])\n",
    "    return A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "c5f5d909",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Classical method:\n",
      " [[ 1.  0.  1.  0.]\n",
      " [ 0.  1.  0.  1.]\n",
      " [-1.  0.  1.  0.]]\n",
      "Modified method:\n",
      " [[ 1.  0.  1.  0.]\n",
      " [ 0.  1.  0.  1.]\n",
      " [-1.  0.  1.  0.]]\n"
     ]
    }
   ],
   "source": [
    "# example, rows of matrix are vectors\n",
    "# نمونه، رجهای ماتکدان، بُردارها هستند\n",
    "A=np.array([[1,0,1,0],[1,1,1,1],[0,1,2,1]]) \n",
    "print('Classical method:\\n',gram_schmidt(A))\n",
    "print('Modified method:\\n',modified_gram_schmidt(A))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "111a3338",
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   "source": [
    "Hint: You can check that the processed vectors are orthogonal.<br>\n",
    "نکته: میتوانید چک کنید که بُردارهای پردازش شده، فرکنجی هستند"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "55608cb7",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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