{"id":3281,"date":"2026-08-25T17:36:34","date_gmt":"2026-08-25T08:36:34","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=3281"},"modified":"2026-08-25T17:36:34","modified_gmt":"2026-08-25T08:36:34","slug":"5-2_sparse","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/num\/5-2_sparse\/","title":{"rendered":"5.2 \u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f (\u5b9a\u5e38\u53cd\u5fa9\u6cd5)"},"content":{"rendered":"\nn \u5143\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f Ax = b \u306b\u5bfe\u3057\u3066\u6b21\u306e\u3088\u3046\u306a\u5f62\u5f0f\u306e\u53cd\u5fa9\u6cd5\u3092\u5b9a\u5e38\u53cd\u5fa9\u89e3\u6cd5\u3068\u3044\u3046. \u4e0a\u4ed8\u304d\u306e (k) \u306f k \u56de\u76ee\u306e\u53cd\u5fa9\u3092\u8868\u3059.
\n\\[
\nx^{(k+1)} = M x^{(k)} + N b
\n\\]\n\u3053\u3053\u3067, M \u3068 N \u306f\u53cd\u5fa9\u56de\u6570 k \u306b\u4f9d\u5b58\u3057\u306a\u3044\u884c\u5217\u3067\u3042\u308b. M \u306f\u53cd\u5fa9\u884c\u5217\u3068\u547c\u3070\u308c\u308b.<\/p>\n

\u3053\u306e\u53cd\u5fa9\u304c\u53ce\u675f\u3059\u308b\u305f\u3081\u306e\u6761\u4ef6\u306f\u53cd\u5fa9\u884c\u5217 M \u306e\u30b9\u30da\u30af\u30c8\u30eb\u534a\u5f84 (\u884c\u5217\u306e\u56fa\u6709\u5024\u306e\u6700\u5927\u7d76\u5bfe\u5024) \\rho(M) \u304c 1 \u3088\u308a\u5c0f\u3055\u3044\u3053\u3068\u3067\u3042\u308b.
\n\\[
\n\\rho(M) = max|\\lambda_i| < 1 \\space (\\lambda_i \u306f\u884c\u5217 M \u306e\u56fa\u6709\u5024, i = 1, \\dots, n)\n\\]\n\u4ee3\u8868\u7684\u306a\u5b9a\u5e38\u53cd\u5fa9\u89e3\u6cd5\u3068\u3057\u3066\u306f, \u30e4\u30b3\u30d3\u6cd5, \u30ac\u30a6\u30b9\u30fb\u30b6\u30a4\u30c7\u30eb\u6cd5, SOR (\u9010\u6b21\u904e\u5270\u7de9\u548c) \u6cd5\u304c\u3042\u308b. SOR \u6cd5\u306f\u5b9f\u7528\u7684\u306b\u3082\u3088\u304f\u4f7f\u7528\u3055\u308c\u3066\u3044\u308b.\n\n\n\n

5.2.1 \u30e4\u30b3\u30d3\u6cd5<\/h3>\n

n \u5143\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f Ax = b \u306e\u7b2c i \u756a\u76ee\u306e\u5f0f
\n\\[
\n\\sum_{j=1}^n a_{ij} x_j = b_i
\n\\]\n\u3092 \\(x_i\\) \u306b\u3064\u3044\u3066\u89e3\u304d, \u305d\u306e\u5024\u3092\u65b0\u305f\u306a \\(x_i\\) \u3068\u3059\u308b\u53cd\u5fa9\u3092 \\(i = 1, \\dots, n\\) \u306e\u5168\u3066\u306b\u3064\u3044\u3066\u884c\u3046.
\n\\[
\nx_i^{(k+1)} = (b_i – \\sum_{j \\ne i, j=1}^n a_{ij} x_j^{(k)}) \/ a_{ii}
\n\\]\n\u3044\u307e, A \u3092\u5de6\u4e0b\u4e09\u89d2\u884c\u5217 L (\u5bfe\u89d2\u8981\u7d20\u306f 0), \u5bfe\u89d2\u884c\u5217 D, \u53f3\u4e0a\u4e09\u89d2\u884c\u5217 U (\u5bfe\u89d2\u8981\u7d20\u306f 0)\u306b\u5206\u96e2\u3057\u3066, A = L + D + U \u3068\u3059\u308b. \u305d\u3046\u3059\u308b\u3068\u4e0a\u306e\u53cd\u5fa9\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b.
\n\\[
\nx^{(k+1)} = D^{-1}(b – (L + U) x^{(k)}) = -D^{-1}(L + U) x^{(k)} + D^{-1}b
\n\\]\n\u3059\u306a\u308f\u3061, \u53cd\u5fa9\u884c\u5217 M \u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b.
\n\\[
\nM = -D^{-1}(L + U)
\n\\]\n\u6b21\u5f0f\u304c\u5168\u3066\u306e i \u306b\u3064\u3044\u3066\u6210\u308a\u7acb\u3064\u3068\u304d\u306b\u884c\u5217 A \u306f\u72ed\u7fa9\u306e\u5bfe\u89d2\u512a\u4f4d\u3067\u3042\u308b\u3068\u8a00\u3046.
\n\\[
\n |a_{ii}| \\ge \\sum_{j \\ne i, j=1}^n |a_{ij}|
\n\\]\nA \u304c\u72ed\u7fa9\u306e\u5bfe\u89d2\u512a\u4f4d\u3067\u3042\u308b\u3068\u304d, \u30e4\u30b3\u30d3\u6cd5\u306e\u53cd\u5fa9\u884c\u5217 M \u306e\u30b9\u30da\u30af\u30c8\u30eb\u534a\u5f84 \\(\\rho\\)(M) \u306f 1 \u3088\u308a\u5c0f\u3055\u3044. \u3059\u306a\u308f\u3061, \u30e4\u30b3\u30d3\u6cd5\u306f\u53ce\u675f\u3059\u308b.<\/p>\n

5.2.2 \u30ac\u30a6\u30b9\u30fb\u30b6\u30a4\u30c7\u30eb\u6cd5<\/h3>\n

\u30e4\u30b3\u30d3\u6cd5\u3068\u4f3c\u3066\u3044\u308b\u304c, \\(i = 1, \\dots, n\\) \u306e\u5168\u90e8\u306e\u8a08\u7b97\u3092\u5f85\u305f\u305a\u306b \\(x_i\\) \u3092\u3059\u3050\u306b\u66f4\u65b0\u3057\u3066\u73fe\u5728\u306e\u8fd1\u4f3c\u89e3\u3068\u3059\u308b. \u3059\u306a\u308f\u3061, \\(i = 1, \\dots, n\\) \u306e\u9806\u306b\u66f4\u65b0\u3059\u308b.
\n\\[
\nx_i^{(k+1)} = (b_i – \\sum_{j \\lt i} a_{ij} x_j^{(k+1)} – \\sum_{i \\lt j} a_{ij} x_j^{(k)}) \/ a_{ii}
\n\\]\n\u884c\u5217\u8868\u793a\u3067\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.
\n\\[
\n\\begin{align}
\nx^{(k+1)} & = D^{-1}(b – Lx^{(k+1)} – Ux^{(k)}) \\\\
\n& = -(D + L)^{-1} Ux^{(k)} + (D + L)^{-1}b \\\\
\n\\end{align}
\n\\]\n\u53cd\u5fa9\u884c\u5217 M \u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b.
\n\\[
\nM = -(D + L)^{-1}U
\n\\]\n

A \u304c\u72ed\u7fa9\u306e\u5bfe\u89d2\u512a\u4f4d\u3067\u3042\u308b\u3068\u304d, \u30ac\u30a6\u30b9\u30fb\u30b6\u30a4\u30c7\u30eb\u6cd5\u306e\u53cd\u5fa9\u884c\u5217 M \u306e\u30b9\u30da\u30af\u30c8\u30eb\u534a\u5f84 \\(\\rho\\)(M) \u306f 1 \u3088\u308a\u5c0f\u3055\u304f, \u30ac\u30a6\u30b9\u30fb\u30b6\u30a4\u30c7\u30eb\u6cd5\u306f\u53ce\u675f\u3059\u308b.<\/p>\n

\u3053\u306e\u65b9\u6cd5\u306f, \u5f0f\u3067\u66f8\u304f\u3068\u30e4\u30b3\u30d3\u6cd5\u3088\u308a\u9762\u5012\u306b\u601d\u3048\u308b\u304c, x \u3092\u3059\u3050\u306b\u66f8\u304d\u63db\u3048\u3066\u3082\u3088\u3044\u306e\u3067\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u66f8\u304f\u306e\u306f\u3053\u3061\u3089\u306e\u65b9\u304c\u697d\u3067\u3042\u308b.<\/p>\n

5.2.3 SOR (\u9010\u6b21\u904e\u5270\u7de9\u548c) \u6cd5<\/h3>\n

\u30ac\u30a6\u30b9\u30fb\u30b6\u30a4\u30c7\u30eb\u6cd5\u306e\u53cd\u5fa9\u306b\u304a\u3051\u308b\u672c\u6765\u306e\u66f4\u65b0\u91cf\u3092 \\(\\omega\\) \u500d\u3057\u3066\u52a0\u901f\u3059\u308b\u65b9\u6cd5\u3067\u3042\u308b. \\(\\omega\\) \u306f\u52a0\u901f\u30d1\u30e9\u30e1\u30fc\u30bf\u3068\u547c\u3070\u308c\u308b.<\/p>\n

\u884c\u5217\u8868\u793a\u3067\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b.
\n\\[
\nx^{(k+1)} = (I + \\omega D^{-1}L)^{-1} ((1 – \\omega)I – \\omega D^{-1}U) x^{(k)} + \\omega(D + \\omega L)^{-1}b
\n\\]\n\u53cd\u5fa9\u884c\u5217 M \u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b.
\n\\[
\nM = (I + \\omega D^{-1}L)^{-1} ((1 – \\omega)I – \\omega D^{-1}U)
\n\\]\n\\(\\omega\\) \u306e\u6700\u9069\u5024\u3092\u524d\u3082\u3063\u3066\u77e5\u308b\u3053\u3068\u306f\u96e3\u3057\u304f, \u7d4c\u9a13\u7684\u306b\u6c7a\u3081\u3089\u308c\u308b\u3053\u3068\u304c\u591a\u3044. \u6700\u9069\u5024\u3092\u4f7f\u3048\u3070 SOR \u6cd5\u306f\u30ac\u30a6\u30b9\u30fb\u30b6\u30a4\u30c7\u30eb\u6cd5\u3088\u308a\u304b\u306a\u308a\u901f\u304f\u53ce\u675f\u3059\u308b.<\/p>\n

A \u304c\u72ed\u7fa9\u306e\u5bfe\u89d2\u512a\u4f4d\u884c\u5217\u3067\u3042\u308b\u3068\u304d, \\(\\omega\\) \u306e\u5024\u304c \\(0 < \\omega < 2\\) \u3067\u3042\u308c\u3070 SOR \u6cd5\u306f\u53ce\u675f\u3059\u308b.\n\n\n\n

5.2.3.1 SOR \u6cd5\u306e ω \u306e\u6700\u9069\u5024<\/h4>\n

\u30b9\u30da\u30af\u30c8\u30eb\u534a\u5f84 \\(\\rho\\)(M) \u306f \\(\\omega\\) \u306b\u3088\u308a\u5909\u5316\u3059\u308b. 5.2.4.3 \u6570\u5024\u5b9f\u9a13 (1) \u306e\u4f8b\u984c\u306b\u304a\u3044\u3066, \u53cd\u5fa9\u884c\u5217 M \u306e\u56fa\u6709\u5024\u3092\u6c42\u3081\u3066\u30ce\u30eb\u30e0\u304c\u6700\u5927\u306e\u3082\u306e\u3059\u306a\u308f\u3061\u30b9\u30da\u30af\u30c8\u30eb\u534a\u5f84 ρ(M) \u3092\u30d7\u30ed\u30c3\u30c8\u3059\u308b. \u6a2a\u8ef8\u306f ω, \u7e26\u8ef8\u306f\u30b9\u30da\u30af\u30c8\u30eb\u534a\u5f84 ρ(M) \u3067\u3042\u308b.<\/p>\n

\"\"<\/p>\n

\u5c16\u3063\u305f\u5e95\u306b\u306a\u3063\u3066\u3044\u308b\u70b9\u306f ω \u306e\u6700\u9069\u5024\u3067 1.53 \u3067\u3042\u3063\u305f. ω \u304c 2 \u3092\u8d85\u3048\u308b\u3068\u30b9\u30da\u30af\u30c8\u30eb\u534a\u5f84\u304c 1 \u3088\u308a\u5927\u304d\u304f\u306a\u308b (\u53ce\u675f\u3057\u306a\u304f\u306a\u308b).<\/p>\n

ω \u306e\u6700\u9069\u5024\u306f\u6b21\u5f0f\u304b\u3089\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b (\u6587\u732e[6]\u53c2\u7167).
\n\\[
\n\\omega_{opt} = \\frac{2}{1 + (1 – \\mu_0^2)^{1\/2}}
\n\\]\n\u3053\u3053\u3067 \\(\\mu_0\\) \u306f\u30e4\u30b3\u30d3\u6cd5\u306e\u53cd\u5fa9\u884c\u5217\u306e\u30b9\u30da\u30af\u30c8\u30eb\u534a\u5f84\u3067\u3042\u308b. \u3053\u306e\u5834\u5408, \\(\\mu_0\\) = 0.951 \u3067\u3042\u308b\u304b\u3089, \u4e0a\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068 \\(\\omega_{opt}\\) = 1.53 \u3068\u306a\u308a\u5b9f\u9a13\u7d50\u679c\u3068\u4e00\u81f4\u3059\u308b.<\/p>\n

\u306a\u304a, ω = 1, \u3059\u306a\u308f\u3061\u30ac\u30a6\u30b9\u30fb\u30b6\u30a4\u30c7\u30eb\u6cd5\u306e\u5834\u5408\u306f ρ(M) = 0.9045 \u3067\u3042\u3063\u305f\u304c, \u3053\u308c\u306f\u30e4\u30b3\u30d3\u6cd5\u306e\u53cd\u5fa9\u884c\u5217\u306e ρ(M) (= \\(\\mu_0\\)) = 0.951 \u306e 2 \u4e57\u306b\u4e00\u81f4\u3059\u308b. \u307e\u305f, \\(\\omega_{opt}\\) \u3092\u8d85\u3048\u308b\u3068 ρ(M) \u3068 ω \u306e\u95a2\u4fc2\u306f\u76f4\u7dda\u306b\u306a\u308a, ρ(M) = ω – 1 \u306b\u306a\u308b.<\/p>\n

5.2.3.2 \u9069\u5fdc\u7684 SOR \u6cd5<\/h4>\n

SOR \u6cd5\u306b\u304a\u3044\u3066 ω \u306e\u6700\u9069\u306a\u5024\u3092\u524d\u3082\u3063\u3066\u77e5\u308b\u3053\u3068\u306f\u96e3\u3057\u3044\u304c, \u53cd\u5fa9\u4e2d\u306b\u53ce\u675f\u7387\u3092\u898b\u7a4d\u308a ω \u306e\u5024\u3092\u81ea\u52d5\u7684\u306b\u8abf\u7bc0\u3059\u308b\u9069\u5fdc\u7684\u30d7\u30ed\u30b0\u30e9\u30e0\u3082\u7814\u7a76\u3055\u308c\u3066\u3044\u308b. \u3053\u306e\u3088\u3046\u306b, ω \u306e\u5024\u3092\u81ea\u52d5\u7684\u306b\u6700\u9069\u5024\u306b\u8abf\u7bc0\u3059\u308b\u6a5f\u80fd\u3092\u6301\u3063\u305f SOR \u6cd5\u3092\u9069\u5fdc\u7684 SOR \u6cd5 (Adaptive SOR method) \u3068\u3044\u3046.<\/p>\n

\u4e00\u4f8b\u3068\u3057\u3066, SOR \u6cd5\u306e\u53cd\u5fa9\u4e2d\u306b\u5f97\u3089\u308c\u308b\u60c5\u5831\u3092\u4f7f\u3063\u3066\u30b9\u30da\u30af\u30c8\u30eb\u534a\u5f84\u3092\u898b\u7a4d\u3082\u308b\u3053\u3068\u306b\u3088\u308a ω \u306e\u5024\u3092\u81ea\u52d5\u7684\u306b\u8abf\u7bc0\u3059\u308b\u65b9\u6cd5\u304c\u63d0\u6848\u3055\u308c\u3066\u3044\u308b (\u8a73\u7d30\u306f\u6587\u732e[10]\u53c2\u7167). \u3053\u306e\u65b9\u6cd5\u306f\u4fc2\u6570\u884c\u5217\u304c\u5bfe\u79f0\u6b63\u5b9a\u5024\u884c\u5217\u3067\u3042\u308b\u3053\u3068\u306a\u3069\u9069\u7528\u3067\u304d\u308b\u554f\u984c\u306b\u5236\u9650\u304c\u3042\u308b\u304c\u3046\u307e\u304f\u52d5\u4f5c\u3059\u308b\u3088\u3046\u3067\u3042\u308b.<\/p>\n

5.2.4 \u5b9a\u5e38\u53cd\u5fa9\u6cd5\u306e\u6bd4\u8f03<\/h3>\n

\u4f8b\u3068\u3057\u3066\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u306e\u5883\u754c\u5024\u554f\u984c\u3092\u89e3\u304f.<\/p>\n

5.2.4.1 \u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u306e\u5883\u754c\u5024\u554f\u984c\u306e\u4f8b<\/h4>\n

2 \u6b21\u5143\u306e\u6b63\u65b9\u5f62\u9818\u57df (0 ≤ x ≤ 1, 0 ≤ y ≤ 1) \u306b\u304a\u3051\u308b\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u3092\u8003\u3048\u308b.
\n\\[
\n-\\Delta u = f
\n\\]\n\u3053\u3053\u3067, \\(\\Delta u = \\nabla \\cdot(\\nabla u) = \\partial^2u\/\\partial x^2 + \\partial^2u\/\\partial y^2\\) \u3067\u3042\u308b.<\/p>\n

\u5883\u754c\u6761\u4ef6\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.
\n\\[
\n\\begin{align}
\n& u(x, 0) = 0, \\space u(x, 1) = 0 \\space (0 \\le x \\le 1) \\\\
\n& u(0, y) = 0, \\space u(1, y) = 0 \\space (0 \\le y \\le 1) \\\\
\n\\end{align}
\n\\]\n\u3059\u306a\u308f\u3061, \u6b63\u65b9\u5f62\u9818\u57df\u306e\u5883\u754c\u4e0a\u3067\u306f 0 \u3068\u3059\u308b. <\/p>\n

5.2.4.2 5 \u70b9\u5dee\u5206\u8fd1\u4f3c\u306b\u3088\u308b\u96e2\u6563\u5316<\/h4>\n

\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5fae\u5206\u3092\u5dee\u5206\u306b\u7f6e\u304d\u63db\u3048\u3066\u5dee\u5206\u65b9\u7a0b\u5f0f\u3092\u4f5c\u308a, \u305d\u308c\u3092\u89e3\u304f\u3053\u3068\u306b\u3088\u308a\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u8fd1\u4f3c\u89e3\u3092\u6c42\u3081\u308b\u65b9\u6cd5\u304c\u5dee\u5206\u6cd5\u3067\u3042\u308b.<\/p>\n

2 \u968e\u5fae\u5206\u306e\u5dee\u5206\u8fd1\u4f3c\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b.
\n\\[
\n\\begin{align}
\n& \\partial^2 u\/\\partial x^2 = (u(x-h, y) – 2u(x, y) + u(x+h, y)) \/ h^2 \\\\
\n& \\partial^2 u\/\\partial y^2 = (u(x, y-h) – 2u(x, y) + u(x, y+h)) \/ h^2 \\\\
\n\\end{align}
\n\\]\n\u3053\u3053\u3067, h \u306f\u5c0f\u3055\u306a\u5024\u306e\u523b\u307f\u3067\u3042\u308b\u304c, \u6b63\u65b9\u5f62\u306e\u5b9a\u7fa9\u57df\u306e x, y \u65b9\u5411\u305d\u308c\u305e\u308c\u3092 N \u7b49\u5206\u3057\u3066\u683c\u5b50\u3092\u4f5c\u308a\u305d\u306e\u5404\u8fba\u3092\u523b\u307f\u3068\u3057\u3066\u4f7f\u3046\u3053\u3068\u306b\u3059\u308b\u3068, \u3053\u306e\u5834\u5408 h = 1 \/ N \u306b\u306a\u308b. \u3055\u3089\u306b, \u683c\u5b50\u70b9\u3067\u306e\u5024\u3092 \\(u_{ij} = u(i\/N, j\/N) \\space (i = 0, \\dots, N, \\space j = 0, \\dots, N)\\) \u3068\u8868\u3059\u3068, \u683c\u5b50\u70b9 (i, j) \u3067\u306e\u5dee\u5206\u8fd1\u4f3c\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.
\n\\[
\n\\begin{align}
\n& \\partial^2 u\/\\partial x^2 = (u_{i-1,j} – 2u_{i,j} + u_{i+1,j}) \/ h^2 \\\\
\n& \\partial^2 u\/\\partial y^2 = (u_{i,j-1} – 2u_{i,j} + u_{i,j+1}) \/ h^2 \\\\
\n\\end{align}
\n\\]\n\u3053\u308c\u3092\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\u6b21\u306e\u8fd1\u4f3c\u5f0f\u304c\u5f97\u3089\u308c\u308b.
\n\\[
\nu_{i,j} = (u_{i-1,j} + u_{i+1,j} + u{i,j-1} + u{i,j+1})\/4
\n\\]\n\u56f3\u3067\u8868\u3059\u3068, \u4e0b\u306e\u3088\u3046\u306b\u524d\u5f8c\u5de6\u53f3\u306e\u5024\u306e\u5e73\u5747\u5024\u3067\u771f\u3093\u4e2d\u306e\u5024\u3092\u8fd1\u4f3c\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3053\u308c\u306f 5 \u70b9\u5dee\u5206\u8fd1\u4f3c\u3068\u3088\u3070\u308c\u308b.<\/p>\n

\"\"<\/p>\n

\u3053\u3053\u3067 N = 10 \u3068\u3059\u308b\u3068, \u683c\u5b50\u70b9\u306f 11\u00d711 = 121 \u70b9\u306b\u306a\u308b. \u305d\u306e\u3046\u3061\u5883\u754c\u4e0a\u306e 40 \u70b9 (\u4e00\u756a\u5916\u5074\u306e 4 \u8fba\u4e0a\u306e\u70b9) \u306f\u5883\u754c\u6761\u4ef6\u3067\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u304b\u3089, \u672a\u77e5\u306e \\(u_{i,j}\\) \u306f\u5185\u5074\u306e 81 \u70b9 (i = 1, …, 9, j = 1, …, 9) \u3067\u3042\u308b. \u307e\u305f, \u4e0a\u306e\u8fd1\u4f3c\u5f0f\u3082\u3053\u308c\u3089\u306e 81 \u70b9\u306b\u3064\u3044\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u304b\u3089 81 \u672c\u3042\u308b. \u5f93\u3063\u3066, \u3053\u306e\u8fd1\u4f3c\u5f0f\u304b\u3089\u306a\u308b 81 \u5143\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u306b\u3088\u308a\u5168\u3066\u306e \\(u_{i,j}\\) \u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b. \u3059\u306a\u308f\u3061, \u5dee\u5206\u6cd5\u306b\u3088\u308a\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5883\u754c\u5024\u554f\u984c\u306f\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f (\u5dee\u5206\u65b9\u7a0b\u5f0f) \u306b\u5e30\u7740\u3055\u308c\u305f.<\/p>\n

\u3053\u3046\u3057\u3066\u5f97\u3089\u308c\u305f\u5dee\u5206\u65b9\u7a0b\u5f0f\u306e\u4fc2\u6570\u884c\u5217\u306f, \u5909\u6570\u3092 \\(u_{i,j}\\) ((i,j) = (1,1), (2,1), …, (9,1), (1,2), (2,2), …, (9,9)) \u306e\u9806\u306b\u4e26\u3079\u308b\u3068, \u4e0b\u306b\u793a\u3059\u30d1\u30bf\u30fc\u30f3\u306e\u884c\u5217\u306b\u306a\u308b. \u9752\u3067\u8868\u3055\u308c\u3066\u3044\u308b\u306e\u304c\u975e\u30bc\u30ed\u8981\u7d20 (\u3053\u306e\u5834\u5408\u306f 369 \u500b\u3067, \u5168\u4f53\u306e 5.6% \u306b\u3042\u305f\u308b) \u3067, \u305d\u306e\u4ed6\u306e\u8981\u7d20\u306f\u3059\u3079\u3066 0 \u306e\u758e\u884c\u5217\u3067\u3042\u308b. \u307e\u305f, \u3053\u306e\u5834\u5408, \u6b63\u5b9a\u5024\u5bfe\u79f0\u884c\u5217\u3067\u3042\u308b.<\/p>\n

\"\"<\/p>\n

\u5dee\u5206\u6cd5\u306e\u5834\u5408, \u7cbe\u5ea6\u3092\u4e0a\u3052\u308b\u305f\u3081\u306b\u683c\u5b50\u3092\u7d30\u304b\u304f\u3057\u3066\u3044\u304f\u3068\u5dee\u5206\u65b9\u7a0b\u5f0f\u306f N \u306e 2 \u4e57\u306b\u6bd4\u4f8b\u3057\u3066\u5927\u898f\u6a21\u306b\u306a\u3063\u3066\u3044\u304f. \u3053\u306e\u4f8b\u3067\u306f, N = 20 \u3067\u306f 361 \u5143, N = 100 \u3067\u306f 9801 \u5143\u3068\u3044\u3046\u3088\u3046\u306b\u3059\u3050\u306b\u5927\u304d\u304f\u306a\u308b.<\/p>\n

5.2.4.3 \u6570\u5024\u5b9f\u9a13 (1) \u5dee\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f<\/h4>\n

\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u306e\u4f8b\u984c\u306e\u5dee\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f. \u3053\u3053\u3067\u306f N = 10 (81 \u5143\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f) \u3068\u3057, \u89e3\u304c (1, 1, …, 1) \u3068\u306a\u308b\u3088\u3046\u306b\u53f3\u8fba\u3092\u8a2d\u5b9a\u3057\u305f.<\/p>\n

\u4fc2\u6570\u884c\u5217 A \u306e\u5bfe\u89d2\u8981\u7d20\u306f\u5168\u3066 4 \u3067\u3042\u308b\u304b\u3089 \\(|a_ii|\\) = 4 \u3067\u3042\u308b. \u305d\u3057\u3066, \u5404\u884c\u306e\u5bfe\u89d2\u8981\u7d20\u4ee5\u5916\u306e\u975e\u30bc\u30ed\u8981\u7d20\u306e\u5024\u306f -1 \u3067\u305d\u306e\u500b\u6570\u306f\u6700\u5927\u3067 4 \u500b\u3067\u3042\u308b\u304b\u3089, \\(\\sum |a_ij| \\le 4\\) \u3067\u3042\u308b. \u5f93\u3063\u3066, \u3053\u306e\u4fc2\u6570\u884c\u5217 A \u306f\u72ed\u7fa9\u306e\u5bfe\u89d2\u512a\u4f4d\u3067\u3042\u308b. \u3059\u306a\u308f\u3061, \u30e4\u30b3\u30d3\u6cd5, \u30ac\u30a6\u30b9\u30fb\u30b6\u30a4\u30c7\u30eb\u6cd5, SOR \u6cd5 (0 < ω < 2) \u306f\u53ce\u675f\u3059\u308b.\n\n\u30e4\u30b3\u30d3\u6cd5\u3068 ω = 1, 1.43 \uff5e 1.73 \u306b\u5909\u5316\u3055\u305b\u305f SOR \u6cd5 (ω = 1 \u306e\u3068\u304d\u30ac\u30a6\u30b9\u30fb\u30b6\u30a4\u30c7\u30eb\u6cd5\u306b\u4e00\u81f4) \u3067\u8a08\u7b97\u3057\u3066\u307f\u308b. \u521d\u671f\u5024\u3092 (0, 0, ..., 0) \u3068\u3057\u305f\u3068\u304d\u306e\u53ce\u675f\u306e\u69d8\u5b50\u3092\u6b21\u306b\u793a\u3059. \u6a2a\u8ef8\u306f\u53cd\u5fa9\u56de\u6570, \u7e26\u8ef8\u306f\u6b8b\u5dee\u30ce\u30eb\u30e0 (\u53f3\u8fba\u30d9\u30af\u30c8\u30eb\u306e\u30ce\u30eb\u30e0\u3067\u6b63\u898f\u5316\u3057\u305f\u3082\u306e) \u3067\u3042\u308b.<\/br><\/p>\n

\"\"<\/p>\n

\u56f3\u306e\u3088\u3046\u306b\u30e4\u30b3\u30d3\u6cd5\u3088\u308a\u30ac\u30a6\u30b9\u30fb\u30b6\u30a4\u30c7\u30eb\u6cd5\u306e\u65b9\u304c\u901f\u3044\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n

SOR \u6cd5\u3067\u306f, \u3053\u306e\u5834\u5408\u306f ω = 1 \u304b\u3089 ω \u304c\u5897\u52a0\u3059\u308b\u306b\u4f34\u3063\u3066\u53ce\u675f\u304c\u901f\u304f\u306a\u3063\u3066\u3044\u304f\u304c, \u6700\u9069\u5024 (\u3053\u306e\u5834\u5408\u306f ω = 1.53)\u3092\u8d85\u3048\u308b\u3068\u9006\u306b\u9045\u304f\u306a\u3063\u3066\u3044\u304f\u306e\u304c\u308f\u304b\u308b. \u6700\u9069\u5024\u3092\u8d85\u3048\u308b\u3068\u53ce\u675f\u304c\u4e0d\u5b89\u5b9a\u306b\u306a\u308b\u50be\u5411\u304c\u307f\u3048\u308b.<\/p>\n

5.2.4.4 \u6570\u5024\u5b9f\u9a13 (2) \u9069\u5fdc\u7684 SOR \u6cd5\u306b\u3088\u308a\u5dee\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f<\/h4>\n

\u4e0a\u3068\u540c\u3058\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f\u306e\u4f8b\u984c\u306b\u9069\u5fdc\u7684 SOR \u6cd5\u3092\u9069\u7528\u3057\u305f\u6570\u5024\u5b9f\u9a13\u7d50\u679c\u3092\u793a\u3059.<\/p>\n

\"\"<\/p>\n

\u6bd4\u8f03\u306e\u305f\u3081, ω = 1.53 (\u6700\u9069\u5024) \u304a\u3088\u3073 1.63 \u306e\u5834\u5408\u3082\u793a\u3057\u305f. \u3053\u306e\u4f8b\u3067\u306f, \u6700\u9069\u306a ω \u3092\u4f7f\u7528\u3057\u305f\u5834\u5408\u3068\u905c\u8272\u306a\u3044\u7d50\u679c\u3092\u793a\u3057\u305f.<\/p>\n","protected":false},"excerpt":{"rendered":"

n \u5143\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f Ax = b \u306b\u5bfe\u3057\u3066\u6b21\u306e\u3088\u3046\u306a\u5f62\u5f0f\u306e\u53cd\u5fa9\u6cd5\u3092\u5b9a\u5e38\u53cd\u5fa9\u89e3\u6cd5\u3068\u3044\u3046. \u4e0a\u4ed8\u304d\u306e (k) \u306f k \u56de\u76ee\u306e\u53cd\u5fa9\u3092\u8868\u3059. \\[ x^{(k+1)} = M x^{(k)} + N b \\] \u3053\u3053\u3067, M \u3068 […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"footnotes":""},"categories":[13],"tags":[],"class_list":["post-3281","post","type-post","status-publish","format-standard","hentry","category-num"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t