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Merged
merged 2 commits into from
Jan 17, 2023
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Update Gaussian elimination scripts #185

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240b20d
Update gaussian_elimination scripts
chvmvd Jan 17, 2023
5cdee65
Update gassian_elimination_revised script
chvmvd Jan 17, 2023
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2 changes: 2 additions & 0 deletions docs/02algorithms/10gaussian-elimination/index.mdx
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Expand Up @@ -469,6 +469,8 @@ $x_n = d_n$ になります。求まった $x_n$ をそれよりも上の式す

<ViewSource path="/gaussian-elimination/gaussian_elimination.ipynb" />

冒頭の `x = [0 for _ in range(len(a[0]) - 1)]` は、リストの内包表記と呼ばれ、配列 `x` を `0` で埋める操作です。この操作で `x` は `[0, 0, ..., 0]` となっています。

`reversed` 関数は配列の要素を反転します。

## 部分ピボット選択
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2 changes: 1 addition & 1 deletion docs/02algorithms/11dp/index.mdx
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Expand Up @@ -69,7 +69,7 @@ DP には大きく分けて、二種類あります。トップダウン方式

<ViewSource path="/dp/fib_memoization.ipynb" />

冒頭の `memo = [-1 for _ in range(10000)]` は、リストの内包表記と呼ばれ、配列 `memo` を `-1` で埋める操作です。この操作で `memo` は `[-1, -1, ..., -1]` となっています。
冒頭の `memo = [-1 for _ in range(10000)]` は、配列 `memo` を `-1` で埋めています。

#### ボトムアップ方式

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85 changes: 42 additions & 43 deletions static/gaussian-elimination/gaussian_elimination.ipynb
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@@ -1,47 +1,46 @@
{
"nbformat": 4,
"nbformat_minor": 2,
"metadata": {},
"cells": [
{
"metadata": {},
"source": [
"def gaussian_elimination(a):\n",
" # 前進消去\n",
" for i in range(len(a)):\n",
" # pivot倍で行を割る\n",
" pivot = a[i][i]\n",
" for j in range(i, len(a[i])):\n",
" a[i][j] /= pivot\n",
"\n",
" # i+1行目以降を掃き出す\n",
" for j in range(i + 1, len(a)):\n",
" factor = a[j][i]\n",
" for k in range(i, len(a[i])):\n",
" a[j][k] -= factor * a[i][k]\n",
" # 後退代入\n",
" x = []\n",
" for i in reversed(range(len(a))):\n",
" tmp = a[i][len(a[i]) - 1] / a[i][i]\n",
" for j in range(i):\n",
" a[j][len(a[i]) - 1] -= a[j][i] * tmp\n",
" x = [tmp] + x\n",
" return x\n",
"\n",
"\n",
"print(gaussian_elimination([[1, -2, 3, 3], [-1, 3, -2, 1], [1, -1, 6, 11]]))"
],
"cell_type": "code",
"outputs": [
"nbformat": 4,
"nbformat_minor": 2,
"metadata": {},
"cells": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"[1.0, 2.0, 2.0]\n"
]
"metadata": {},
"source": [
"def gaussian_elimination(a):\n",
" # \u524d\u9032\u6d88\u53bb\n",
" for i in range(len(a)):\n",
" # pivot\u500d\u3067\u884c\u3092\u5272\u308b\n",
" pivot = a[i][i]\n",
" for j in range(i, len(a[i])):\n",
" a[i][j] /= pivot\n",
"\n",
" # i+1\u884c\u76ee\u4ee5\u964d\u3092\u6383\u304d\u51fa\u3059\n",
" for j in range(i + 1, len(a)):\n",
" factor = a[j][i]\n",
" for k in range(i, len(a[i])):\n",
" a[j][k] -= factor * a[i][k]\n",
" # \u5f8c\u9000\u4ee3\u5165\n",
" x = [0 for _ in range(len(a[0]) - 1)]\n",
" for i in reversed(range(len(a))):\n",
" x[i] = a[i][len(a[i]) - 1]\n",
" for j in range(i):\n",
" a[j][len(a[i]) - 1] -= a[j][i] * x[i]\n",
" return x\n",
"\n",
"\n",
"print(gaussian_elimination([[1, -2, 3, 3], [-1, 3, -2, 1], [1, -1, 6, 11]]))"
],
"cell_type": "code",
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"[1.0, 2.0, 2.0]\n"
]
}
],
"execution_count": null
}
],
"execution_count": null
}
]
]
}
95 changes: 47 additions & 48 deletions static/gaussian-elimination/gaussian_elimination_error.ipynb
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Original file line number Diff line number Diff line change
@@ -1,52 +1,51 @@
{
"nbformat": 4,
"nbformat_minor": 2,
"metadata": {},
"cells": [
{
"metadata": {},
"source": [
"def gaussian_elimination(a):\n",
" # 前進消去\n",
" for i in range(len(a)):\n",
" # pivot倍で行を割る\n",
" pivot = a[i][i]\n",
" for j in range(i, len(a[i])):\n",
" a[i][j] /= pivot\n",
"\n",
" # i+1行目以降を掃き出す\n",
" for j in range(i + 1, len(a)):\n",
" factor = a[j][i]\n",
" for k in range(i, len(a[i])):\n",
" a[j][k] -= factor * a[i][k]\n",
" # 後退代入\n",
" x = []\n",
" for i in reversed(range(len(a))):\n",
" tmp = a[i][len(a[i]) - 1] / a[i][i]\n",
" for j in range(i):\n",
" a[j][len(a[i]) - 1] -= a[j][i] * tmp\n",
" x = [tmp] + x\n",
" return x\n",
"\n",
"\n",
"print(gaussian_elimination([[0, -2, 3, 2], [-1, 3, -2, 1], [1, -1, 6, 11]]))"
],
"cell_type": "code",
"outputs": [
"nbformat": 4,
"nbformat_minor": 2,
"metadata": {},
"cells": [
{
"output_type": "error",
"ename": "ZeroDivisionError",
"evalue": "division by zero",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mZeroDivisionError\u001b[0m Traceback (most recent call last)",
"Cell \u001b[0;32mIn [1], line 23\u001b[0m\n\u001b[1;32m 20\u001b[0m x\u001b[39m=\u001b[39m[tmp]\u001b[39m+\u001b[39mx\n\u001b[1;32m 21\u001b[0m \u001b[39mreturn\u001b[39;00m x\n\u001b[0;32m---> 23\u001b[0m \u001b[39mprint\u001b[39m(gaussian_elimination([[\u001b[39m0\u001b[39m,\u001b[39m-\u001b[39m\u001b[39m2\u001b[39m,\u001b[39m3\u001b[39m,\u001b[39m2\u001b[39m],[\u001b[39m-\u001b[39m\u001b[39m1\u001b[39m,\u001b[39m3\u001b[39m,\u001b[39m-\u001b[39m\u001b[39m2\u001b[39m,\u001b[39m1\u001b[39m],[\u001b[39m1\u001b[39m,\u001b[39m-\u001b[39m\u001b[39m1\u001b[39m,\u001b[39m6\u001b[39m,\u001b[39m11\u001b[39m]]))\n",
"Cell \u001b[0;32mIn [1], line 7\u001b[0m, in \u001b[0;36mgaussian_elimination\u001b[0;34m(a)\u001b[0m\n\u001b[1;32m 5\u001b[0m pivot\u001b[39m=\u001b[39ma[i][i]\n\u001b[1;32m 6\u001b[0m \u001b[39mfor\u001b[39;00m j \u001b[39min\u001b[39;00m \u001b[39mrange\u001b[39m(i,\u001b[39mlen\u001b[39m(a[i])):\n\u001b[0;32m----> 7\u001b[0m a[i][j]\u001b[39m/\u001b[39m\u001b[39m=\u001b[39mpivot\n\u001b[1;32m 9\u001b[0m \u001b[39m# i+1行目以降を掃き出す\u001b[39;00m\n\u001b[1;32m 10\u001b[0m \u001b[39mfor\u001b[39;00m j \u001b[39min\u001b[39;00m \u001b[39mrange\u001b[39m(i\u001b[39m+\u001b[39m\u001b[39m1\u001b[39m,\u001b[39mlen\u001b[39m(a)):\n",
"\u001b[0;31mZeroDivisionError\u001b[0m: division by zero"
]
"metadata": {},
"source": [
"def gaussian_elimination(a):\n",
" # \u524d\u9032\u6d88\u53bb\n",
" for i in range(len(a)):\n",
" # pivot\u500d\u3067\u884c\u3092\u5272\u308b\n",
" pivot = a[i][i]\n",
" for j in range(i, len(a[i])):\n",
" a[i][j] /= pivot\n",
"\n",
" # i+1\u884c\u76ee\u4ee5\u964d\u3092\u6383\u304d\u51fa\u3059\n",
" for j in range(i + 1, len(a)):\n",
" factor = a[j][i]\n",
" for k in range(i, len(a[i])):\n",
" a[j][k] -= factor * a[i][k]\n",
" # \u5f8c\u9000\u4ee3\u5165\n",
" x = [0 for _ in range(len(a[0]) - 1)]\n",
" for i in reversed(range(len(a))):\n",
" x[i] = a[i][len(a[i]) - 1]\n",
" for j in range(i):\n",
" a[j][len(a[i]) - 1] -= a[j][i] * x[i]\n",
" return x\n",
"\n",
"\n",
"print(gaussian_elimination([[0, -2, 3, 2], [-1, 3, -2, 1], [1, -1, 6, 11]]))"
],
"cell_type": "code",
"outputs": [
{
"output_type": "error",
"ename": "ZeroDivisionError",
"evalue": "division by zero",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mZeroDivisionError\u001b[0m Traceback (most recent call last)",
"Cell \u001b[0;32mIn [1], line 23\u001b[0m\n\u001b[1;32m 19\u001b[0m a[j][\u001b[39mlen\u001b[39m(a[i]) \u001b[39m-\u001b[39m \u001b[39m1\u001b[39m] \u001b[39m-\u001b[39m\u001b[39m=\u001b[39m a[j][i] \u001b[39m*\u001b[39m x[i]\n\u001b[1;32m 20\u001b[0m \u001b[39mreturn\u001b[39;00m x\n\u001b[0;32m---> 23\u001b[0m \u001b[39mprint\u001b[39m(gaussian_elimination([[\u001b[39m0\u001b[39m, \u001b[39m-\u001b[39m\u001b[39m2\u001b[39m, \u001b[39m3\u001b[39m, \u001b[39m2\u001b[39m], [\u001b[39m-\u001b[39m\u001b[39m1\u001b[39m, \u001b[39m3\u001b[39m, \u001b[39m-\u001b[39m\u001b[39m2\u001b[39m, \u001b[39m1\u001b[39m], [\u001b[39m1\u001b[39m, \u001b[39m-\u001b[39m\u001b[39m1\u001b[39m, \u001b[39m6\u001b[39m, \u001b[39m11\u001b[39m]]))\n",
"Cell \u001b[0;32mIn [1], line 7\u001b[0m, in \u001b[0;36mgaussian_elimination\u001b[0;34m(a)\u001b[0m\n\u001b[1;32m 5\u001b[0m pivot \u001b[39m=\u001b[39m a[i][i]\n\u001b[1;32m 6\u001b[0m \u001b[39mfor\u001b[39;00m j \u001b[39min\u001b[39;00m \u001b[39mrange\u001b[39m(i, \u001b[39mlen\u001b[39m(a[i])):\n\u001b[0;32m----> 7\u001b[0m a[i][j] \u001b[39m/\u001b[39m\u001b[39m=\u001b[39m pivot\n\u001b[1;32m 9\u001b[0m \u001b[39m# i+1\u884c\u76ee\u4ee5\u964d\u3092\u6383\u304d\u51fa\u3059\u001b[39;00m\n\u001b[1;32m 10\u001b[0m \u001b[39mfor\u001b[39;00m j \u001b[39min\u001b[39;00m \u001b[39mrange\u001b[39m(i \u001b[39m+\u001b[39m \u001b[39m1\u001b[39m, \u001b[39mlen\u001b[39m(a)):\n",
"\u001b[0;31mZeroDivisionError\u001b[0m: division by zero"
]
}
],
"execution_count": null
}
],
"execution_count": null
}
]
]
}
11 changes: 4 additions & 7 deletions static/gaussian-elimination/gaussian_elimination_revised.ipynb
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -12,9 +12,7 @@
" # 部分ピボット選択\n",
" for j in range(i + 1, len(a)):\n",
" if abs(a[i][i]) < abs(a[j][i]):\n",
" tmp = a[i]\n",
" a[i] = a[j]\n",
" a[j] = tmp\n",
" a[i], a[j] = a[j], a[i]\n",
" # pivot倍で行を割る\n",
" pivot = a[i][i]\n",
" for j in range(i, len(a[i])):\n",
Expand All @@ -26,12 +24,11 @@
" for k in range(i, len(a[i])):\n",
" a[j][k] -= factor * a[i][k]\n",
" # 後退代入\n",
" x = []\n",
" x = [0 for _ in range(len(a[0]) - 1)]\n",
" for i in reversed(range(len(a))):\n",
" tmp = a[i][len(a[i]) - 1] / a[i][i]\n",
" x[i] = a[i][len(a[i]) - 1]\n",
" for j in range(i):\n",
" a[j][len(a[i]) - 1] -= a[j][i] * tmp\n",
" x = [tmp] + x\n",
" a[j][len(a[i]) - 1] -= a[j][i] * x[i]\n",
" return x\n",
"\n",
"\n",
Expand Down