Merge pull request rasbt#28 from rasbt/matmul

The 5fth way to multiply 2 matrices
youyinnn · Oct 22, 2022 · e5458d2 · e5458d2
2 parents 7deca80 + 0503ac6
commit e5458d2
Showing 1 changed file with 163 additions and 2 deletions.
diff --git a/math/Four-matrix-multiplications.ipynb b/math/Four-matrix-multiplications.ipynb
@@ -220,7 +220,7 @@
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@@ -232,7 +232,7 @@
        "       [61.82, 62.96, 64.1 ]])"
       ]
      },
-     "execution_count": 11,
+     "execution_count": 6,
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@@ -283,6 +283,167 @@
     "    \n",
     "AB"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "75c9ea85-8a99-4442-a097-f2d83cad8cbb",
+   "metadata": {},
+   "source": [
+    "## Bonus: The 5th way -- Block multiplication\n",
+    "\n",
+    "In block multiplication, we can divide up a large matrix into smaller blocks and then multiply them separately.\n",
+    "\n",
+    "Suppose we have the following matrix multiplication\n",
+    "\n",
+    "$$\n",
+    "A=\\begin{bmatrix}\n",
+    "1.1 & 1.2 & 1.3 & 1.4\\\\\n",
+    "2.1 & 2.2 & 2.3 & 2.4\\\\\n",
+    "3.1 & 3.2 & 3.3 & 3.4\\\\\n",
+    "4.1 & 4.2 & 4.3 & 4.4\n",
+    "\\end{bmatrix}\n",
+    "B=\\begin{bmatrix}\n",
+    "5.1 & 5.2 & 5.3 & 5.4\\\\\n",
+    "6.1 & 6.2 & 6.3 & 6.4\\\\\n",
+    "7.1 & 7.2 & 7.3 & 7.4\\\\\n",
+    "8.1 & 8.2 & 8.3 & 8.4\n",
+    "\\end{bmatrix}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "AB=\\begin{bmatrix}\n",
+    "33.5 & 34.0 & 34.5 & 35.0\\\\\n",
+    "59.9 & 60.8 & 61.7 & 62.6\\\\\n",
+    "86.3 & 87.6 & 88.9 & 90.2\\\\\n",
+    "112.7 & 114.4 & 116.1 & 117.8\n",
+    "\\end{bmatrix}\n",
+    "$$\n",
+    "\n",
+    "The multiplication of the blocks looks as follows:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2840e3f0-a99d-4676-a74d-34c62199d526",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "A = \\left[\\begin{array}{ll}\n",
+    "A_1 & A_2 \\\\\n",
+    "A_3 & A_4\n",
+    "\\end{array}\\right]\n",
+    "B = \\left[\\begin{array}{ll}\n",
+    "B_1 & B_2 \\\\\n",
+    "B_3 & B_4\n",
+    "\\end{array}\\right] \\\\ \n",
+    "AB =\n",
+    "\\left[\\begin{array}{ll} A_1 B_1+A_2 B_3 & A_1 B_2+A_2 B_4 \\\\\n",
+    "A_3 B_1+A_4 B_3 & A_3 B_2+A_4 B_4\n",
+    "\\end{array}\\right]\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "d4f153c6-c698-44af-8cb3-6c177315173b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "A = np.array([[1.1, 1.2, 1.3, 1.4],\n",
+    "              [2.1, 2.2, 2.3, 2.4],\n",
+    "              [3.1, 3.2, 3.3, 3.4],\n",
+    "              [4.1, 4.2, 4.3, 4.4]])\n",
+    "\n",
+    "B = np.array([[5.1, 5.2, 5.3, 5.4],\n",
+    "              [6.1, 6.2, 6.3, 6.4],\n",
+    "              [7.1, 7.2, 7.3, 7.4],\n",
+    "              [8.1, 8.2, 8.3, 8.4]])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "57540937-2e52-4a8a-bb87-42c9ed55966e",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([[ 33.5,  34. ,  34.5,  35. ],\n",
+       "       [ 59.9,  60.8,  61.7,  62.6],\n",
+       "       [ 86.3,  87.6,  88.9,  90.2],\n",
+       "       [112.7, 114.4, 116.1, 117.8]])"
+      ]
+     },
+     "execution_count": 9,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "A @ B"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "a217d3bf-62a6-43c8-a71d-6544e1e79ea1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "A1, A2 = A[:2, :2], A[:2, 2:]\n",
+    "A3, A4 = A[2:, :2], A[2:, 2:]\n",
+    "\n",
+    "B1, B2 = B[:2, :2], B[:2, 2:]\n",
+    "B3, B4 = B[2:, :2], B[2:, 2:]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "1b888d6b-5ede-4dd2-a619-298fdcbcc41c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "C1, C2 = A1@B1 + A2@B3, A1@B2 + A2@B4\n",
+    "C3, C4 = A3@B1 + A4@B3, A3@B2 + A4@B4"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "id": "947f1463-c956-4098-882a-9ea22a51e7ef",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([[ 33.5,  34. ,  34.5,  35. ],\n",
+       "       [ 59.9,  60.8,  61.7,  62.6],\n",
+       "       [ 86.3,  87.6,  88.9,  90.2],\n",
+       "       [112.7, 114.4, 116.1, 117.8]])"
+      ]
+     },
+     "execution_count": 12,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "AB = np.vstack((np.hstack((C1, C2)),\n",
+    "                np.hstack((C3, C4))))\n",
+    "\n",
+    "AB"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "15be5d4a-4c1f-4c2c-ba27-7eed5875711c",
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
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