{"id":4708,"date":"2025-03-31T17:13:12","date_gmt":"2025-03-31T08:13:12","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=4708"},"modified":"2025-08-20T13:28:18","modified_gmt":"2025-08-20T04:28:18","slug":"16-sparse","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/tutorial\/16-sparse\/","title":{"rendered":"16. \u758e\u884c\u5217\u306e\u7dda\u5f62\u8a08\u7b97"},"content":{"rendered":"\n

16.1 \u6982\u8981<\/h3>\n

\u884c\u5217\u306e\u8981\u7d20\u306e\u3046\u3061\u591a\u304f\u304c 0 \u3067\u3042\u308b\u3088\u3046\u306a\u884c\u5217\u3092\u758e\u884c\u5217\u3068\u3044\u3044\u307e\u3059. \u305d\u306e\u3088\u3046\u306a\u884c\u5217\u306f\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f\u305f\u3081\u306e\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306e\u4fc2\u6570\u306a\u3069\u3068\u3057\u3066\u3088\u304f\u73fe\u308f\u308c\u307e\u3059.<\/p>\n

\u758e\u884c\u5217\u3092\u901a\u5e38\u306e\u4e8c\u6b21\u5143\u914d\u5217\u3067\u8868\u3059\u3068, 0 \u3092\u683c\u7d0d\u3059\u308b\u305f\u3081\u306b\u591a\u304f\u306e\u30e1\u30e2\u30ea\u304c\u6d88\u8cbb\u3055\u308c, \u8a08\u7b97\u6642\u306b\u306f 0 \u3068\u306e\u6f14\u7b97\u306b\u591a\u304f\u306e CPU \u6642\u9593\u304c\u4f7f\u308f\u308c\u3066\u3057\u307e\u3044\u307e\u3059. \u305d\u308c\u3092\u907f\u3051\u308b\u305f\u3081\u306b\u758e\u884c\u5217\u5c02\u7528\u306e\u683c\u7d0d\u5f62\u5f0f\u304c\u4f7f\u308f\u308c\u307e\u3059.<\/p>\n

\u758e\u884c\u5217\u306e\u7dda\u5f62\u8a08\u7b97\u3067\u306f, \u683c\u7d0d\u5f62\u5f0f\u304c\u7570\u306a\u308b\u4ed6\u306b, \u9069\u7528\u3055\u308c\u308b\u554f\u984c\u306e\u6027\u8cea\u4e0a\u5927\u898f\u6a21 (\u4f8b\u3048\u3070, \u6570\u4e07\u5143\u4ee5\u4e0a) \u3067\u3042\u308b\u3053\u3068\u304c\u591a\u3044\u305f\u3081, \u9069\u3059\u308b\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3082\u7570\u306a\u3063\u3066\u304d\u307e\u3059. \u305d\u306e\u305f\u3081, \u5f93\u6765\u306e\u7dda\u5f62\u8a08\u7b97\u30d7\u30ed\u30b0\u30e9\u30e0\u306f\u305d\u306e\u307e\u307e\u4f7f\u3046\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093. \u5177\u4f53\u7684\u306b\u306f, \u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u3067\u306f\u53cd\u5fa9\u6cd5 (\u30af\u30ea\u30ed\u30d5\u90e8\u5206\u7a7a\u9593\u6cd5), \u56fa\u6709\u5024\u30fb\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3067\u306f\u30e9\u30f3\u30c1\u30e7\u30b9\u6cd5\u3084\u30a2\u30fc\u30ce\u30eb\u30c7\u30a3\u6cd5\u304c\u4f7f\u308f\u308c\u308b\u3053\u3068\u304c\u591a\u304f\u306a\u308a\u307e\u3059.<\/p>\n

\u4ee5\u4e0b, \u758e\u884c\u5217\u8a08\u7b97\u306e\u4f8b\u984c, XLPack \u3067\u4f7f\u308f\u308c\u3066\u3044\u308b\u758e\u884c\u5217\u306e\u683c\u7d0d\u5f62\u5f0f, \u57fa\u672c\u6f14\u7b97\u30d7\u30ed\u30b0\u30e9\u30e0, \u758e\u884c\u5217\u306e\u30c1\u30a7\u30c3\u30af, \u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5, \u30d5\u30a1\u30a4\u30eb\u5165\u51fa\u529b\u306a\u3069\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u307e\u3059.<\/p>\n

16.2 \u758e\u884c\u5217\u8a08\u7b97\u306e\u4f8b\u984c<\/h3>\n

5 \u70b9\u5dee\u5206\u8fd1\u4f3c\u3092\u4f7f\u3063\u3066\u30e9\u30d7\u30e9\u30b9\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u3092\u6c42\u3081\u308b\u4f8b\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u307e\u3059.<\/p>\n

\\(x = 0 \\sim 1, y = 0 \\sim 1\\) \u306e\u6b63\u65b9\u5f62\u9818\u57df\u306b\u304a\u3044\u3066, \u6b21\u306e\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3082\u306e\u3068\u3057\u307e\u3059.
\n\\[
\n-d^2u\/dx^2 – d^2u\/dy^2 = 0
\n\\]\n\u305f\u3060\u3057, \u5883\u754c\u6761\u4ef6\u306f
\n\\[
\nu(x, 0) = 0, u(0, y) = 0, u(x, 1) = x, u(1, y) = y \\space (\u30c7\u30a3\u30ea\u30af\u30ec\u6761\u4ef6)
\n\\]\n\u3068\u3057\u307e\u3059.<\/p>\n

\u3053\u306e\u3088\u3046\u306a\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f, \u4f8b\u3048\u3070\u71b1\u5e73\u8861\u72b6\u614b\u306b\u3042\u308b\u677f\u306e\u6e29\u5ea6\u5206\u5e03\u306e\u8a18\u8ff0\u306a\u3069\u306b\u9069\u7528\u3055\u308c\u307e\u3059.<\/p>\n

\u5dee\u5206\u8fd1\u4f3c\u3092\u884c\u3046\u305f\u3081\u306b \\(x, y\\) \u4e21\u65b9\u5411\u306b\u9818\u57df\u3092 \\(N\\) \u7b49\u5206\u3057\u307e\u3059. \u5206\u5272\u5e45\u3092 \\(h = 1\/N\\) \u3068\u3057\u3066, \u683c\u5b50\u70b9 \\(x = ih, y = jh (i = 0 \\sim N, j = 0 \\sim N)\\) \u306b\u304a\u3051\u308b \\(u\\) \u3092 \\(u[i,j]\\) \u3067\u8868\u3059\u3053\u3068\u306b\u3057\u307e\u3059. \u305d\u3057\u3066, \u5fae\u5206\u3092\u6b21\u306e\u3088\u3046\u306b\u5dee\u5206\u306b\u3088\u308a\u8fd1\u4f3c\u3057\u307e\u3059.
\n\\[
\nd^2u\/dx^2 = (u[i-1,j] – 2u[i,j] + u[i+1,j])\/h^2 \\\\
\nd^2u\/dy^2 = (u[i,j-1] – 2u[i,j] + u[i,j+1])\/h^2 \\\\
\n\\]\n\u3053\u308c\u3088\u308a, \u30e9\u30d7\u30e9\u30b9\u65b9\u7a0b\u5f0f\u306f\u6b21\u5f0f\u306b\u3088\u308a\u8fd1\u4f3c\u3055\u308c\u307e\u3059.
\n\\[
\n-u[i,j-1] – u[i-1,j] + 4u[i,j] – u[i+1,j] – u[i,j+1] = 0
\n\\]\n\u5883\u754c\u6761\u4ef6 (\u30c7\u30a3\u30ea\u30af\u30ec\u6761\u4ef6) \u306f\u6b21\u306e\u3088\u3046\u306b\u5883\u754c\u4e0a\u306e\u683c\u5b50\u70b9\u306e\u5024\u3092\u76f4\u63a5\u4e0e\u3048\u307e\u3059.
\n\\[
\nu[i,0] = 0 (i = 0 \\sim N) \\\\
\nu[0,j] = 0 (j = 0 \\sim N) \\\\
\nu[i,N] = ih (i = 0 \\sim N) \\\\
\nu[N,j] = jh (j = 0 \\sim N) \\\\
\n\\]\n\u3059\u306a\u308f\u3061, \u6b63\u65b9\u5f62\u9818\u57df\u306e\u3046\u3061\u8fba\u306e\u90e8\u5206\u306f\u65e2\u77e5\u3068\u306a\u308a, \u5185\u5074\u306e\u90e8\u5206\u3060\u3051\u6c42\u3081\u308c\u3070\u3088\u3044\u3053\u3068\u306b\u306a\u308a\u307e\u3059. \u305d\u3053\u3067, \\(m = N – 1\\) \u3068\u304a\u3044\u3066, \\(u_1 = u[1,1], u_2 = u[2,1], \\dots, u_m = u[m,1], u_{m+1} = u[1,2], \\dots, u_{m*m} = u[m,m]\\) \u3068\u8a18\u53f7\u3092\u3064\u3051\u76f4\u3059\u3053\u3068\u306b\u3057\u307e\u3059.<\/p>\n

\\(N = 7 (m = 6)\\) \u306e\u5834\u5408\u3092\u56f3\u793a\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.<\/p>\n

\"\"<\/p>\n

\u3053\u306e\u5834\u5408, 64 \u683c\u5b50\u70b9\u306e\u3046\u3061 28 \u70b9\u306f\u5883\u754c\u6761\u4ef6\u3088\u308a\u65e2\u77e5\u3067, \u6b8b\u308a 36 \u70b9\u306b\u304a\u3051\u308b \\(\\boldsymbol{u}\\) \u306e\u5024\u304c\u672a\u77e5\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308a, \u6b21\u306e\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306b\u5e30\u7740\u3055\u308c\u307e\u3059.
\n\\[
\n\\boldsymbol{A}\\boldsymbol{u} = \\boldsymbol{b}
\n\\]\n\u305f\u3060\u3057, \\(\\boldsymbol{A}\\) \u306f \\(36 \\times 36\\) \u6b63\u65b9\u884c\u5217, \\(\\boldsymbol{u}\\) \u306f \\(u_1, u_2, \\dots, u_{36}\\) \u3092\u8981\u7d20\u3068\u3059\u308b\u30d9\u30af\u30c8\u30eb, \\(\\boldsymbol{b}\\) \u306f\u53f3\u8fba\u30d9\u30af\u30c8\u30eb\u3067\u3059.<\/p>\n

\u4e0a\u306e\u30e9\u30d7\u30e9\u30b9\u65b9\u7a0b\u5f0f\u306e\u8fd1\u4f3c\u5f0f\u306b\u304a\u3044\u3066, \\(i = 1 \\sim 6, j = 1 \\sim 6\\) \u3068\u3057\u305f \\(36\\) \u672c\u306e\u5f0f\u3088\u308a \\(\\boldsymbol{A}\\) \u3068 \\(\\boldsymbol{b}\\) \u3092\u5b9a\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059.
\n\\[
\n\\boldsymbol{A} =
\n\\begin{pmatrix}
\n 4&-1& & & & &-1 \\\\
\n-1& 4&-1& & & & &-1 \\\\
\n &-1& 4&-1& & & & &-1 \\\\
\n & &-1& 4&-1& & & & &-1 \\\\
\n & & &-1& 4&-1& & & & &-1 \\\\
\n & & & &-1& 4& & & & & &-1 \\\\
\n-1& & & & & & 4&-1& & & & &-1 \\\\
\n &-1& & & & &-1& 4&-1& & & & &-1 \\\\
\n & &-1& & & & &-1& 4&-1& & & & &-1 \\\\
\n & & &-1& & & & &-1& 4&-1& & & & &-1 \\\\
\n & & & &-1& & & & &-1& 4&-1& & & & &-1 \\\\
\n & & & & &-1& & & & &-1& 4& & & & & &-1 \\\\
\n & & & & & & & & & & & & & & & & & & \\\\
\n & & & & & & &\\ddots& & & & &&\\ddots& & & & & \\\\
\n & & & & & & & & & & & & & & & & & & \\\\
\n & & & & & & & & & &-1& & & & & & 4&-1 \\\\
\n & & & & & & & & & & &-1& & & & &-1& 4&-1 \\\\
\n & & & & & & & & & & & &-1& & & & &-1& 4&-1 \\\\
\n & & & & & & & & & & & & &-1& & & & &-1& 4&-1 \\\\
\n & & & & & & & & & & & & & &-1& & & & &-1& 4&-1 \\\\
\n & & & & & & & & & & & & & & &-1& & & & &-1& 4 \\\\
\n\\end{pmatrix}
\n\\]\n\\[
\n\\boldsymbol{b} =
\n\\begin{pmatrix}
\nu[1,0] + u[0,1] \\\\
\nu[2,0] \\\\
\nu[3,0] \\\\
\nu[4,0] \\\\
\nu[5,0] \\\\
\nu[6,0] + u[6,1] \\\\
\nu[0,2] \\\\
\n\\vdots \\\\
\nu[5,7] \\\\
\nu[6,7] + u[7,6] \\\\
\n\\end{pmatrix}
\n=
\n\\begin{pmatrix}
\n0 \\\\
\n0 \\\\
\n0 \\\\
\n0 \\\\
\n0 \\\\
\nh \\\\
\n0 \\\\
\n\\vdots \\\\
\n5h\\\\
\n12h \\\\
\n\\end{pmatrix}
\n\\]\n\u3053\u306e\u3088\u3046\u306b \\(\\boldsymbol{A}\\) \u306f 1 \u884c\u306e\u4e2d\u306b\u9ad8\u3005 5 \u500b\u306e\u975e\u30bc\u30ed\u8981\u7d20\u3057\u304b\u306a\u3044\u5bfe\u79f0\u758e\u884c\u5217\u306b\u306a\u308a\u307e\u3059.<\/p>\n

\\(N = 7\\) \u306e\u4f8b\u3092\u8aac\u660e\u3057\u307e\u3057\u305f\u304c, \u3088\u308a\u7cbe\u5bc6\u306b\u89e3\u3092\u6c42\u3081\u305f\u3051\u308c\u3070 \\(N\\) \u3092\u5927\u304d\u304f\u3057\u306a\u3051\u308c\u3070\u306a\u308a\u307e\u305b\u3093. \u4f8b\u3048\u3070, \\(N = 100\\) \u3068\u3059\u308c\u3070, \u65b9\u7a0b\u5f0f\u306f\u7d04 1 \u4e07\u5143\u306b\u306a\u308a\u307e\u3059. \u3055\u3089\u306b, 3 \u6b21\u5143\u306e\u554f\u984c\u3067\u3042\u308c\u3070\u7d04 100 \u4e07\u5143\u306b\u306a\u308a\u307e\u3059. \u305d\u3046\u306a\u308b\u3068, \u901a\u5e38\u306e\u7dda\u5f62\u8a08\u7b97\u30d7\u30ed\u30b0\u30e9\u30e0\u306f\u4f7f\u3048\u306a\u304f\u306a\u308a, \u5927\u898f\u6a21\u758e\u884c\u5217\u7528\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u304c\u5fc5\u8981\u306b\u306a\u308a\u307e\u3059.<\/p>\n

16.3 \u758e\u884c\u5217\u306e\u683c\u7d0d\u5f62\u5f0f<\/h3>\n

XLPack \u3067\u306f\u6b21\u306e3\u3064\u306e\u683c\u7d0d\u5f62\u5f0f\u3092\u4f7f\u7528\u3057\u3066\u3044\u307e\u3059.<\/p>\n

(1) \u5727\u7e2e\u884c\u683c\u7d0d (Compressed Sparse Row (CSR)) \u5f62\u5f0f
\n(2) \u5727\u7e2e\u5217\u683c\u7d0d (Compressed Sparse Column (CSC)) \u5f62\u5f0f
\n(3) \u5ea7\u6a19 (Coordinate (COO)) \u5f62\u5f0f<\/p>\n

(1) \u3068 (2) \u306f\u8a08\u7b97\u306b\u4f7f\u308f\u308c, (3) \u306f\u5165\u51fa\u529b\u306e\u969b\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059. XLPack \u57fa\u672c\u6a5f\u80fd\u3067\u306f (1) \u3068 (3) \u306e\u307f\u3092\u4f7f\u7528\u3057\u307e\u3059.<\/p>\n

\u4f8b\u3068\u3057\u3066\u6b21\u306e 3 x 4 \u884c\u5217\u3092\u8003\u3048\u307e\u3059.
\n\\[
\n\\begin{pmatrix}
\n1 & 0 & -2 & 4 \\\\
\n-1 & 2 & 0 & -9 \\\\
\n0 & 5 & 0 & -8 \\\\
\n\\end{pmatrix}
\n\\]\n

(1) \u5727\u7e2e\u884c\u683c\u7d0d (Compressed Sparse Row (CSR)) \u5f62\u5f0f<\/h4>\n

1 \u672c\u306e\u6d6e\u52d5\u5c0f\u6570\u914d\u5217 val \u3068 2 \u672c\u306e\u6574\u6570\u914d\u5217 rowptr \u304a\u3088\u3073 colind \u306b\u975e\u30bc\u30ed\u8981\u7d20\u3060\u3051\u3092\u683c\u7d0d\u3057\u307e\u3059. val \u306b\u306f\u975e\u30bc\u30ed\u8981\u7d20\u306e\u5024\u3092\u884c\u65b9\u5411\u9806\u306b\u683c\u7d0d\u3057\u307e\u3059. rowptr \u306f\u5404\u884c\u306e\u5148\u982d\u8981\u7d20\u306e val \u914d\u5217\u4e2d\u306e\u4f4d\u7f6e\u3092\u8868\u3057\u307e\u3059. colind \u306f\u5bfe\u5fdc\u3059\u308b val \u306e\u8981\u7d20\u306e\u5217\u756a\u53f7\u3067\u3059. rowptr \u306e\u6700\u5f8c\u306b\u306f val \u914d\u5217\u306e\u6700\u5f8c + 1 \u3092\u5165\u308c\u3066\u304a\u304d\u307e\u3059.<\/p>\n

\u884c\u756a\u53f7\u3068\u5217\u756a\u53f7\u306f, Fortran \u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u5834\u5408\u306b\u306f 1 \u304b\u3089, C \u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u5834\u5408\u306b\u306f 0 \u304b\u3089\u59cb\u307e\u308b\u3053\u3068\u304c\u591a\u3044\u306e\u3067\u4e92\u63db\u6027\u306b\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3059.<\/p>\n

C\u8a00\u8a9e\u5f62\u5f0f<\/p>\n

\"\"<\/p>\n

Fortran\u5f62\u5f0f<\/p>\n

\"\"<\/p>\n

\u5bfe\u79f0\u884c\u5217\u306e\u5834\u5408\u306b\u306f, \u5bfe\u89d2\u8981\u7d20\u3068\u305d\u306e\u5de6\u5074\u306e\u307f, \u3042\u308b\u3044\u306f, \u5bfe\u89d2\u8981\u7d20\u3068\u305d\u306e\u53f3\u5074\u306e\u307f\u3092\u683c\u7d0d\u3057\u3066\u4f7f\u7528\u30e1\u30e2\u30ea\u3092\u6e1b\u3089\u3059\u3053\u3068\u304c\u3042\u308a\u307e\u3059. \u3053\u3053\u3067\u306f, \u305d\u306e\u3088\u3046\u306a\u884c\u5217\u3092 SSR \u5f62\u5f0f\u3068\u8868\u8a18\u3057\u307e\u3059 (\u4e00\u822c\u7684\u306a\u547c\u3073\u65b9\u3067\u306f\u3042\u308a\u307e\u305b\u3093).<\/p>\n

\u4e0a\u306e\u4f8b\u3067\u306f\u5404\u884c\u306e\u4e2d\u3067\u5217\u756a\u53f7\u306e\u6607\u9806\u306b\u4e26\u3093\u3067\u3044\u307e\u3059. \u30d7\u30ed\u30b0\u30e9\u30e0\u306b\u3088\u3063\u3066\u306f\u884c\u5185\u306e\u4e26\u3073\u9806\u306f\u4efb\u610f\u3067\u3088\u3044\u5834\u5408\u3082\u3042\u308a\u307e\u3059\u304c, XLPack \u3067\u306f\u8a08\u7b97\u52b9\u7387\u306e\u305f\u3081\u6607\u9806\u306b\u4e26\u3093\u3067\u3044\u306a\u3051\u308c\u3070\u306a\u308a\u307e\u305b\u3093.<\/p>\n

(2) \u5727\u7e2e\u5217\u683c\u7d0d (Compressed Sparse Column (CSC)) \u5f62\u5f0f<\/h4>\n

XLPack (\u57fa\u672c\u6a5f\u80fd) \u3067\u306f\u4f7f\u7528\u3057\u306a\u3044\u305f\u3081\u8aac\u660e\u306f\u7701\u7565\u3057\u307e\u3059\u304c, \u975e\u30bc\u30ed\u8981\u7d20\u306e\u5024\u3092\u5217\u65b9\u5411\u9806\u306b\u683c\u7d0d\u3057\u305f\u3082\u306e\u3067\u3059.<\/p>\n

(3) \u5ea7\u6a19 (Coordinate (COO)) \u5f62\u5f0f<\/h4>\n

1 \u672c\u306e\u6d6e\u52d5\u5c0f\u6570\u914d\u5217 val \u3068 2 \u672c\u306e\u6574\u6570\u914d\u5217 rowind \u304a\u3088\u3073 colind \u306b\u975e\u30bc\u30ed\u8981\u7d20\u3060\u3051\u3092\u683c\u7d0d\u3057\u307e\u3059. val \u306b\u306f\u975e\u30bc\u30ed\u8981\u7d20\u306e\u5024\u3092\u683c\u7d0d\u3057\u307e\u3059. rowind, colind \u306b\u306f\u5bfe\u5fdc\u3059\u308b\u884c\u756a\u53f7, \u5217\u756a\u53f7\u3092\u5165\u308c\u307e\u3059.<\/p>\n

\u884c\u756a\u53f7\u3068\u5217\u756a\u53f7\u306f, Fortran \u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u5834\u5408\u306b\u306f 1 \u304b\u3089, C \u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u5834\u5408\u306b\u306f 0 \u304b\u3089\u59cb\u307e\u308b\u3053\u3068\u304c\u591a\u3044\u306e\u3067\u4e92\u63db\u6027\u306b\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3059.<\/p>\n

C\u8a00\u8a9e\u5f62\u5f0f<\/p>\n

\"\"<\/p>\n

Fortran\u5f62\u5f0f<\/p>\n

\"\"<\/p>\n

\u4e0a\u306e\u4f8b\u3067\u306f\u884c\u756a\u53f7, \u5217\u756a\u53f7\u306e\u6607\u9806\u306b\u4e26\u3093\u3067\u3044\u307e\u3059\u304c, \u30d7\u30ed\u30b0\u30e9\u30e0\u306b\u3088\u3063\u3066\u306f\u683c\u7d0d\u306e\u9806\u756a\u306f\u4efb\u610f\u3067\u3088\u3044\u5834\u5408\u304c\u3042\u308a\u307e\u3059.<\/p>\n

16.4 \u57fa\u672c\u6f14\u7b97\u30d7\u30ed\u30b0\u30e9\u30e0<\/h3>\n

XLPack (\u57fa\u672c\u6a5f\u80fd)\u3067\u306f, CSR \u5f62\u5f0f\u306e\u884c\u5217\u306b\u5bfe\u3057\u3066\u6b21\u306e\u3088\u3046\u306a\u57fa\u672c\u6f14\u7b97\u30d7\u30ed\u30b0\u30e9\u30e0\u304c\u7528\u610f\u3055\u308c\u3066\u3044\u307e\u3059.<\/p>\n

 \r\n  CsrDusmv<\/strong>  y <- \u03b1Ax + \u03b2y \u307e\u305f\u306f y <- \u03b1A^Tx + \u03b2y\r\n  CsrDussv<\/strong>  Ax = b \u307e\u305f\u306f A^Tx = b \u306e\u89e3 (\u4e09\u89d2\u884c\u5217)\r\n  CsrDusmm<\/strong>  C <- \u03b1AB + \u03b2C \u307e\u305f\u306f C <- \u03b1A^TB + \u03b2C\r\n  CsrDussm<\/strong>  AX = B \u307e\u305f\u306f A^TX = B \u306e\u89e3 (\u4e09\u89d2\u884c\u5217)\r\n  SsrDusmv<\/strong>  y <- \u03b1Ax + \u03b2y (\u5bfe\u79f0\u884c\u5217)\r\n  CsrTrans<\/strong>  \u758e\u884c\u5217\u306e\u8ee2\u7f6e\r\n\r\n<\/pre>\n
 
\n\u307e\u305f, \u6b21\u306e\u5909\u63db\u30d7\u30ed\u30b0\u30e9\u30e0\u304c\u7528\u610f\u3055\u308c\u3066\u3044\u307e\u3059.<\/p>\n
 \r\n  CooCsr<\/strong>    COO -> CSR\r\n  CsrCoo<\/strong>    CSR -> COO\r\n  SsrCsr<\/strong>    SSR (CSR \u5bfe\u79f0\u884c\u5217\u5f62\u5f0f) -> CSR (\u5bfe\u79f0\u306a\u30d5\u30eb\u884c\u5217)\r\n  CsrSsr<\/strong>    CSR (\u5bfe\u79f0\u306a\u30d5\u30eb\u884c\u5217) -> SSR (CSR \u5bfe\u79f0\u884c\u5217\u5f62\u5f0f)\r\n  CsrDense<\/strong>  CSR -> \u5bc6\u884c\u5217\r\n  DenseCsr<\/strong>  \u5bc6\u884c\u5217 -> CSR\r\n\r\n<\/pre>\n
 <\/p>\n
16.5 \u758e\u884c\u5217\u306e\u30c1\u30a7\u30c3\u30af<\/h3>\n
CSR \u5f62\u5f0f\u306e\u884c\u5217\u3092\u4f7f\u7528\u3059\u308b\u969b\u306b\u6ce8\u610f\u304c\u5fc5\u8981\u306a\u306e\u306f\u30dd\u30a4\u30f3\u30bf\u3068\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u306e\u8aa4\u308a\u3067\u3059. \u3053\u308c\u3089\u306f\u5909\u6570\u306e\u30a2\u30c9\u30ec\u30b9\u306b\u7b49\u3057\u3044\u306e\u3067, \u9593\u9055\u3044\u304c\u3042\u308b\u3068\u4e88\u671f\u305b\u306c\u30a2\u30c9\u30ec\u30b9\u306e\u8aad\u307f\u66f8\u304d\u3092\u3057\u3066\u3057\u307e\u3044\u30a2\u30af\u30bb\u30b9\u30a8\u30e9\u30fc\u3092\u5f15\u304d\u8d77\u3053\u3057\u307e\u3059. \u901a\u5e38 VBA \u3067\u306f\u30dd\u30a4\u30f3\u30bf\u306f\u6271\u3048\u305a, \u914d\u5217\u306e\u7bc4\u56f2\u3082\u5e38\u306b\u30c1\u30a7\u30c3\u30af\u3055\u308c\u308b\u306e\u3067\u30af\u30e9\u30c3\u30b7\u30e5\u3059\u308b\u3053\u3068\u306f\u3042\u308a\u307e\u305b\u3093\u304c, CSR \u5f62\u5f0f\u884c\u5217\u306e\u30dd\u30a4\u30f3\u30bf\u3084\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u306e\u8aa4\u308a\u3067\u306f Excel \u304c\u30af\u30e9\u30c3\u30b7\u30e5\u3059\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059. \u30d7\u30ed\u30b0\u30e9\u30e0\u4f5c\u6210\u6642\u306b\u306f\u7d30\u5fc3\u306e\u6ce8\u610f\u304c\u5fc5\u8981\u306b\u306a\u308a\u307e\u3059.<\/p>\n
\u3053\u306e\u3088\u3046\u306a\u30a8\u30e9\u30fc\u3092\u5c11\u3057\u3067\u3082\u6e1b\u3089\u3059\u305f\u3081\u306b\u758e\u884c\u5217\u306e\u30c1\u30a7\u30c3\u30af\u30d7\u30ed\u30b0\u30e9\u30e0 CsrCheck<\/strong> \u304c\u63d0\u4f9b\u3055\u308c\u3066\u3044\u307e\u3059. \u30a8\u30e9\u30fc\u304c\u5b8c\u5168\u306b\u9632\u3052\u308b\u308f\u3051\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u304c, \u6d3b\u7528\u3059\u308b\u3068\u3088\u3044\u3067\u3057\u3087\u3046. \u4e0b\u306b\u4f7f\u7528\u4f8b\u3092\u793a\u3057\u307e\u3059.<\/p>\n
\n
Sub Start_2()\r\n    Const M = 3, N = 4\r\n    Dim Val(M * N) As Double, Ptr(M) As Long, Ind(M * N) As Long\r\n    Dim Result(9) As Long, Info As Long\r\n    '-- Check (1)\r\n    Debug.Print \"** Check 1 **\"\r\n    Val(0) = 1: Val(1) = -2: Val(2) = 4: Val(3) = -1: Val(4) = 2\r\n    Val(5) = -9: Val(6) = 5: Val(7) = -8\r\n    Ptr(0) = 0: Ptr(1) = 3: Ptr(2) = 6: Ptr(3) = 8\r\n    Ind(0) = 0: Ind(1) = 2: Ind(2) = 3: Ind(3) = 0: Ind(4) = 1\r\n    Ind(5) = 3: Ind(6) = 1: Ind(7) = 3\r\n    Call CsrCheck(M, N, Val(), Ptr(), Ind(), Result(), Info)\r\n    Call PrintResult(Info, Result())\r\n    '-- Check (2)\r\n    Debug.Print \"** Check 2 **\"\r\n    Val(0) = 0: Val(1) = 4: Val(2) = -2: Val(3) = -1: Val(4) = 2\r\n    Val(5) = -9: Val(6) = 5: Val(7) = -8\r\n    Ptr(0) = 0: Ptr(1) = 3: Ptr(2) = 6: Ptr(3) = 8\r\n    Ind(0) = 0: Ind(1) = 3: Ind(2) = 2: Ind(3) = 0: Ind(4) = 1\r\n    Ind(5) = 3: Ind(6) = 1: Ind(7) = 5\r\n    Call CsrCheck(M, N, Val(), Ptr(), Ind(), Result(), Info)\r\n    Call PrintResult(Info, Result())\r\nEnd Sub\r\n\r\nSub PrintResult(Info As Long, Result() As Long)\r\n    Debug.Print \"Info =\" + Str(Info)\r\n    Debug.Print \"  Sym =\" + Str(Result(0))\r\n    Debug.Print \"  Nnz =\" + Str(Result(1));\r\n    Debug.Print \"  Nnz(L) =\" + Str(Result(2));\r\n    Debug.Print \"  Nnz(U) =\" + Str(Result(3));\r\n    Debug.Print \"  Nnz(D) =\" + Str(Result(4))\r\n    Debug.Print \"  Zero =\" + Str(Result(5));\r\n    Debug.Print \"  Zero(D) =\" + Str(Result(6))\r\n    Debug.Print \"  Empty rows =\" + Str(Result(7));\r\n    Debug.Print \"  Unsorted rows =\" + Str(Result(8))\r\n    Debug.Print \"  Invalid inds =\" + Str(Result(9))\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
\u51fa\u529b\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3057\u305f.<\/p>\n
 \r\n** Check 1 **\r\nInfo = 0\r\n  Sym = 0\r\n  Nnz = 8  Nnz(L) = 2  Nnz(U) = 4  Nnz(D) = 2\r\n  Zero = 0  Zero(D) = 0\r\n  Empty rows = 0  Unsorted rows = 0\r\n  Invalid inds = 0\r\n** Check 2 **\r\nInfo = 14\r\n  Sym = 0\r\n  Nnz = 8  Nnz(L) = 2  Nnz(U) = 4  Nnz(D) = 2\r\n  Zero = 1  Zero(D) = 1\r\n  Empty rows = 0  Unsorted rows = 1\r\n  Invalid inds = 1\r\n\r\n<\/pre>\n
 
\nCheck 2 \u3067\u306f, \u5024\u304c 0 \u306e\u8981\u7d20 1 (\u5b9f\u5bb3\u306f\u306a\u3044), \u6607\u9806\u306b\u306a\u3063\u3066\u3044\u306a\u3044\u884c 1 (\u57fa\u672c\u6f14\u7b97\u30d7\u30ed\u30b0\u30e9\u30e0\u306a\u3069\u591a\u304f\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u304c\u8aa4\u52d5\u4f5c\u3059\u308b), \u4e0d\u6b63\u306a\u30a4\u30f3\u30c7\u30c3\u30af\u30b9 1 (\u30a2\u30af\u30bb\u30b9\u30a8\u30e9\u30fc\u3092\u8d77\u3053\u3059) \u304c\u691c\u51fa\u3055\u308c\u307e\u3057\u305f.<\/p>\n
16.6 \u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f<\/h3>\n
16.6.1 \u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5<\/h4>\n
\u758e\u884c\u5217\u306e\u7dda\u5f62\u8a08\u7b97\u3067\u306f\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u3068\u3057\u3066\u306f\u4e3b\u306b\u53cd\u5fa9\u6cd5 (\u30af\u30ea\u30ed\u30d5\u90e8\u5206\u7a7a\u9593\u6cd5) \u304c\u4f7f\u7528\u3055\u308c\u307e\u3059.<\/p>\n
\u4e00\u822c\u306e\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u3068\u3057\u3066\u4f7f\u308f\u308c\u308b LU \u5206\u89e3\u3084\u30b3\u30ec\u30b9\u30ad\u30fc\u5206\u89e3\u3067\u306f, \u4fc2\u6570\u884c\u5217\u306b\u30bc\u30ed\u8981\u7d20\u304c\u3042\u3063\u3066\u3082\u8a08\u7b97\u3092\u9032\u3081\u308b\u904e\u7a0b\u3067 0 \u3067\u306a\u304f\u306a\u3063\u3066\u3044\u304d\u307e\u3059. \u3053\u308c\u306b\u5bfe\u3057\u3066\u53cd\u5fa9\u6cd5\u3067\u306f, \u4fc2\u6570\u884c\u5217\u306e\u30bc\u30ed\u8981\u7d20\u306f 0 \u306e\u307e\u307e\u3067\u8a08\u7b97\u3092\u9032\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u3044\u3046\u7279\u9577\u304c\u3042\u308a, \u758e\u884c\u5217\u306e\u884c\u5217\u683c\u7d0d\u5f62\u5f0f\u306b\u5411\u3044\u3066\u3044\u307e\u3059.<\/p>\n
16.6.1.1 \u5171\u5f79\u52fe\u914d (CG) \u6cd5<\/h5>\n
\u4fc2\u6570\u884c\u5217 \\(\\boldsymbol{A}\\) \u304c\u6b63\u5b9a\u5024\u5bfe\u79f0\u306e\u5834\u5408\u306b\u306f\u5171\u5f79\u52fe\u914d (CG) \u6cd5\u304c\u7528\u3044\u3089\u308c\u307e\u3059.<\/p>\n
\\(\\boldsymbol{A}\\) \u304c\u6b63\u5b9a\u5024\u5bfe\u79f0\u884c\u5217\u306e\u3068\u304d, \\(\\boldsymbol{x}\\) \u304c\u65b9\u7a0b\u5f0f
\n\\[
\n\\boldsymbol{A}\\boldsymbol{x} = \\boldsymbol{b}
\n\\]\n\u306e\u89e3\u3067\u3042\u308b\u3053\u3068\u3068, \\(\\boldsymbol{x}\\) \u304c\u95a2\u6570
\n\\[
\nf(\\boldsymbol{x}) = (1\/2)(\\boldsymbol{x}, \\boldsymbol{Ax}) – (\\boldsymbol{x}, \\boldsymbol{b})
\n\\]\n\u3092\u6700\u5c0f\u306b\u3059\u308b\u3053\u3068\u306f\u540c\u5024\u306b\u306a\u308a\u307e\u3059.<\/p>\n
\u305d\u3053\u3067 CG \u6cd5\u3067\u306f, \\(f(\\boldsymbol{x})\\) \u3092\u76ee\u7684\u95a2\u6570\u3068\u3059\u308b\u975e\u7dda\u5f62\u6700\u9069\u5316\u554f\u984c\u3092\u89e3\u304d\u6700\u5c0f\u70b9\u3092\u6c42\u3081\u308b\u3053\u3068\u306b\u3088\u308a\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u3092\u6c42\u3081\u307e\u3059. \u305d\u306e\u305f\u3081\u306b\u306f\u964d\u4e0b\u6cd5 (10 \u7ae0\u3092\u53c2\u7167) \u3092\u9069\u7528\u3057\u307e\u3059.<\/p>\n
\u3053\u3053\u3067, \\(\\boldsymbol{p_k}\\) \u3092\u65b9\u5411\u30d9\u30af\u30c8\u30eb, \\(\\alpha_k\\) \u3092\u30b9\u30c6\u30c3\u30d7\u9577\u3068\u3057\u3066\u964d\u4e0b\u6cd5\u306e\u53cd\u5fa9\u3092\u884c\u3046\u3053\u3068\u306b\u3057\u307e\u3059.<\/p>\n
\u30b9\u30c6\u30c3\u30d7\u9577\u306f \\(f(\\boldsymbol{x})\\) \u304c\u6700\u5c0f\u306b\u306a\u308b\u305f\u3081\u306e\u6761\u4ef6
\n\\[
\n{\\partial}f(\\boldsymbol{x_k})\/{\\partial}\\alpha_k = 0
\n\\]\n\u304b\u3089\u6b21\u306e\u3088\u3046\u306b\u6c42\u3081\u3089\u308c\u307e\u3059.
\n\\[
\n\\alpha_k = (\\boldsymbol{p_k}, \\boldsymbol{r_k})\/(\\boldsymbol{p_k}, \\boldsymbol{Ap_k})
\n\\]\n\u3053\u308c\u3092\u4f7f\u3063\u3066 \\(\\boldsymbol{x_{k+1}}\\) \u3092\u66f4\u65b0\u3059\u308b\u3053\u3068\u3067\u304d\u307e\u3059. \u307e\u305f, \u6b8b\u5dee \\(\\boldsymbol{r_k} = \\boldsymbol{b} – \\boldsymbol{Ax}\\) \u3082\u66f4\u65b0\u3067\u304d\u307e\u3059.
\n\\[
\n\\begin{align}
\n& \\boldsymbol{x_{k+1}} = \\boldsymbol{x_k} + \\alpha_k \\boldsymbol{p_k} \\\\
\n& \\boldsymbol{r_{k+1}} = \\boldsymbol{r_k} – \\alpha_k \\boldsymbol{Ap_k} \\\\
\n\\end{align}
\n\\]\n
\u65b9\u5411\u30d9\u30af\u30c8\u30eb\u3068\u3057\u3066\u6b21\u306e\u3088\u3046\u306b\u6700\u5927\u52fe\u914d\u65b9\u5411\u3092\u3068\u308b\u306e\u304c\u3088\u3044\u3088\u3046\u306b\u601d\u308f\u308c\u307e\u3059 (\u6700\u6025\u964d\u4e0b\u6cd5).
\n\\[
\n-\\nabla f(\\boldsymbol{x_k}) = \\boldsymbol{b} – \\boldsymbol{Ax} (= \\boldsymbol{r_k})
\n\\]\n\u3057\u304b\u3057, \u6700\u6025\u964d\u4e0b\u6cd5\u3067\u306f\u5b9f\u969b\u306e\u8a08\u7b97\u306b\u304a\u3044\u3066\u306f\u53ce\u675f\u6027\u304c\u60aa\u304f\u306a\u308b\u3053\u3068\u304c\u3042\u308b\u305f\u3081, \u3053\u308c\u306b\u4fee\u6b63\u3092\u52a0\u3048\u305f\u65b9\u6cd5\u304c\u4f7f\u308f\u308c\u307e\u3059. \u306a\u304a, \u6700\u5927\u52fe\u914d\u65b9\u5411\u306f\u6b8b\u5dee \\(\\boldsymbol{r_k}\\) \u306b\u4e00\u81f4\u3057\u3066\u3044\u307e\u3059.<\/p>\n
\u521d\u671f\u5024\u306f\u6700\u5927\u52fe\u914d\u65b9\u5411 \\(\\boldsymbol{p_0} = \\boldsymbol{r_0}\\) \u304b\u3089\u59cb\u3081\u3066, \\(\\boldsymbol{p_k}\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u66f4\u65b0\u3057\u3066\u3044\u304d\u307e\u3059.
\n\\[
\n\\begin{align}
\n& \\beta_k = -(\\boldsymbol{r_{k+1}}, \\boldsymbol{Ap_k})\/(\\boldsymbol{p_k}, \\boldsymbol{Ap_k}) \\\\
\n& \\boldsymbol{p_{k+1}} = \\boldsymbol{r_{k+1}} + \\beta_k \\boldsymbol{p_k} \\\\
\n\\end{align}
\n\\]\n\u3053\u306e\u3088\u3046\u306b\u3059\u308b\u3068 \\(\\boldsymbol{p_{k+1}}\\) \u306f\u3059\u3079\u3066\u306e \\(\\boldsymbol{p_i} (i = 1 \\sim k)\\) \u3068 A-\u5171\u5f79\u306e\u95a2\u4fc2\u306b\u306a\u308a\u307e\u3059. \u3059\u306a\u308f\u3061, \u6b21\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059.
\n\\[
\n(\\boldsymbol{p_i}, \\boldsymbol{Ap_j}) = 0 \\space (i \\neq j)
\n\\]\n\u307e\u305f, \u6b8b\u5dee \\(\\boldsymbol{r_{k+1}}\\) \u306f\u3059\u3079\u3066\u306e \\(\\boldsymbol{r_i} (i = 1 \\sim k)\\) \u3068\u76f4\u4ea4\u3057\u307e\u3059.
\n\\[
\n(\\boldsymbol{r_i}, \\boldsymbol{r_j}) = 0 \\space (i \\neq j)
\n\\]\n\u5f93\u3063\u3066, \u8a08\u7b97\u8aa4\u5dee\u304c\u306a\u3051\u308c\u3070 \\(\\boldsymbol{r_{n+1}}\\) \u4ee5\u964d\u306f \\(0\\) \u306b\u306a\u308b\u306f\u305a\u306a\u306e\u3067, \u3053\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f \\(n\\) \u56de\u306e\u53cd\u5fa9\u3067\u53ce\u675f\u3059\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059. \u5b9f\u969b\u306b\u306f\u305d\u3046\u306f\u306a\u3089\u305a, \u3080\u3057\u308d \\(n\\) \u56de\u4ee5\u4e0b\u3067\u53ce\u675f\u3059\u308b\u3053\u3068\u304c\u591a\u3044\u3088\u3046\u3067\u3059.<\/p>\n
\u4e0a\u306e\u8a08\u7b97\u624b\u9806\u3092\u3088\u304f\u898b\u308b\u3068\u4fc2\u6570\u884c\u5217 \\(\\boldsymbol{A}\\) \u306f\u5c02\u3089 \\(\\boldsymbol{Ax}\\) \u306e\u3088\u3046\u306b\u884c\u5217\u00d7\u30d9\u30af\u30c8\u30eb\u306e\u6f14\u7b97\u306b\u4f7f\u308f\u308c\u3066\u3044\u308b\u3060\u3051\u3067\u3059. \u3059\u306a\u308f\u3061, \\(\\boldsymbol{A}\\) \u81ea\u4f53\u306f\u5909\u66f4\u3059\u308b\u3053\u3068\u304c\u306a\u3044\u306e\u3067\u758e\u884c\u5217\u683c\u7d0d\u5f62\u5f0f\u306e\u884c\u5217\u3068\u76f8\u6027\u304c\u3088\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059.<\/p>\n
CG \u6cd5\u306b\u306f\u5225\u306e\u89e3\u91c8\u3082\u3042\u308a\u307e\u3059. \u8a73\u7d30\u306f\u8ff0\u3079\u307e\u305b\u3093\u304c, \u30af\u30ea\u30ed\u30d5\u90e8\u5206\u7a7a\u9593\u3068\u547c\u3070\u308c\u308b\u7a7a\u9593\u3092\u751f\u6210\u3057\u3066\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306e\u8fd1\u4f3c\u89e3\u3092\u6c42\u3081\u308b\u89e3\u6cd5\u3092\u30af\u30ea\u30ed\u30d5\u90e8\u5206\u7a7a\u9593\u6cd5\u3068\u3044\u3044\u307e\u3059. CG \u6cd5\u306f\u3053\u306e\u65b9\u6cd5\u306e 1 \u3064\u306e\u30d0\u30ea\u30a8\u30fc\u30b7\u30e7\u30f3\u3068\u307f\u306a\u3059\u3053\u3068\u3082\u3067\u304d\u307e\u3059.<\/p>\n
16.6.1.2 \u53cc\u5171\u5f79\u52fe\u914d (BICG) \u6cd5<\/h5>\n
\u4fc2\u6570\u884c\u5217 \\(\\boldsymbol{A}\\) \u304c\u4e00\u822c\u884c\u5217 (\u975e\u5bfe\u79f0\u884c\u5217) \u306e\u5834\u5408\u306b\u306f CG \u6cd5\u304c\u4f7f\u3048\u307e\u305b\u3093. \u305d\u306e\u305f\u3081, \u7a2e\u3005\u306e\u89e3\u6cd5\u304c\u63d0\u6848\u3055\u308c\u3066\u3044\u307e\u3059\u304c, \u6709\u529b\u306a\u89e3\u6cd5\u306e\u3072\u3068\u3064\u304c\u53cc\u5171\u5f79\u52fe\u914d (BICG) \u6cd5\u3067\u3059.<\/p>\n
\u65b9\u7a0b\u5f0f \\(\\boldsymbol{Ax} = \\boldsymbol{b}\\) \u306e\u4ed6\u306b\u88dc\u52a9\u7684\u306a\u65b9\u7a0b\u5f0f \\(\\boldsymbol{A}^T\\boldsymbol{x}^* = \\boldsymbol{b}^*\\) \u3092\u8003\u3048\u307e\u3059.<\/p>\n
\u3053\u308c\u3092\u4f7f\u3063\u3066\u6b21\u306e\u3088\u3046\u306a\u884c\u5217\u3068\u30d9\u30af\u30c8\u30eb\u3092\u5b9a\u7fa9\u3057\u307e\u3059.
\n\\[
\n\\boldsymbol{\\hat{A}} =
\n\\begin{pmatrix}
\n\\boldsymbol{A} & \\boldsymbol{0} \\\\
\n\\boldsymbol{0} & \\boldsymbol{A}^T \\\\
\n\\end{pmatrix}
\n\\]\n\\[
\n\\boldsymbol{\\hat{x}} =
\n\\begin{pmatrix}
\n\\boldsymbol{x} \\\\
\n\\boldsymbol{x}^* \\\\
\n\\end{pmatrix}
\n\\]\n\\[
\n\\boldsymbol{\\hat{b}} =
\n\\begin{pmatrix}
\n\\boldsymbol{b} \\\\
\n\\boldsymbol{b}^* \\\\
\n\\end{pmatrix}
\n\\]\n\\[
\n\\boldsymbol{\\hat{H}} =
\n\\begin{pmatrix}
\n\\boldsymbol{0} & \\boldsymbol{I} \\\\
\n\\boldsymbol{I} & \\boldsymbol{0} \\\\
\n\\end{pmatrix}
\n\\]\n\u6b21\u306e\u3088\u3046\u306b \\(\\boldsymbol{\\hat{H}}\\) \u5185\u7a4d\u3092\u5b9a\u7fa9\u3057\u307e\u3059.
\n\\[
\n<\\boldsymbol{\\hat{x}}, \\boldsymbol{\\hat{y}}> \\equiv (\\boldsymbol{\\hat{x}}, \\boldsymbol{\\hat{H}}\\boldsymbol{\\hat{y}}) = (\\boldsymbol{\\hat{H}}\\boldsymbol{\\hat{x}}, \\boldsymbol{\\hat{y}})
\n\\]\n\u305d\u3046\u3059\u308b\u3068, \\(\\boldsymbol{\\hat{x}}\\) \u304c\u65b9\u7a0b\u5f0f
\n\\[
\n\\boldsymbol{\\hat{A}}\\boldsymbol{\\hat{x}} = \\boldsymbol{\\hat{b}}
\n\\]\n\u306e\u89e3\u3067\u3042\u308b\u3053\u3068\u3068, \u95a2\u6570
\n\\[
\nf(\\boldsymbol{\\hat{x}}) = (1\/2)<\\boldsymbol{\\hat{x}}, \\boldsymbol{\\hat{A}}\\boldsymbol{\\hat{x}}> – <\\boldsymbol{\\hat{x}}, \\boldsymbol{\\hat{b}}>
\n\\]\n\u3092\u6700\u5c0f\u306b\u3059\u308b\u3053\u3068\u306f\u540c\u5024\u306b\u306a\u308a\u307e\u3059.<\/p>\n
\u3053\u308c\u306b CG \u6cd5\u3068\u540c\u69d8\u306e\u8a08\u7b97\u624b\u9806\u3092\u9069\u7528\u3057\u3066\u5f97\u3089\u308c\u308b\u306e\u304c BICG \u6cd5\u306b\u306a\u308a\u307e\u3059.<\/p>\n
BICG \u6cd5\u3082\u30af\u30ea\u30ed\u30d5\u90e8\u5206\u7a7a\u9593\u6cd5\u306e 1 \u3064\u306e\u30d0\u30ea\u30a8\u30fc\u30b7\u30e7\u30f3\u3068\u307f\u306a\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059.<\/p>\n
16.6.2 XLPack \u3092\u4f7f\u3063\u305f\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u304d\u65b9<\/h4>\n
XLPack (\u57fa\u672c\u6a5f\u80fd) \u3067\u306f, \u5bfe\u79f0\u884c\u5217\u7528\u306e\u5171\u5f79\u52fe\u914d (CG) \u6cd5 (\u30b5\u30d6\u30eb\u30fc\u30c1\u30f3 Cg1<\/strong>) \u3068\u4e00\u822c (\u975e\u5bfe\u79f0) \u884c\u5217\u7528\u306e\u53cc\u5171\u5f79\u52fe\u914d (BICG) \u6cd5 (\u30b5\u30d6\u30eb\u30fc\u30c1\u30f3 Bicg1<\/strong>)\u3092\u4f7f\u7528\u3057\u3066\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059.<\/p>\n
\u4e0a\u3067\u8aac\u660e\u3057\u305f\u30e9\u30d7\u30e9\u30b9\u65b9\u7a0b\u5f0f\u3092 5 \u70b9\u5dee\u5206\u8fd1\u4f3c\u306b\u3088\u308a\u89e3\u304f\u4f8b\u3092\u8aac\u660e\u3057\u307e\u3059. \u4fc2\u6570\u884c\u5217\u304c\u5bfe\u79f0\u306a\u306e\u3067 CG \u6cd5\u3092\u4f7f\u7528\u3057\u307e\u3059.<\/p>\n
16.6.2.1 \u4fc2\u6570\u884c\u5217\u304a\u3088\u3073\u53f3\u8fba\u884c\u5217\u306e\u751f\u6210<\/h5>\n
\u4f8b\u984c\u306e\u8aac\u660e\u306b\u5f93\u3063\u3066\u4fc2\u6570\u884c\u5217 \\(\\boldsymbol{A}\\) \u304a\u3088\u3073\u53f3\u8fba\u30d9\u30af\u30c8\u30eb \\(\\boldsymbol{b}\\) \u3092\u7528\u610f\u3057\u307e\u3059.<\/p>\n
\n
Sub Mat5DF(M As Long, Val() As Double, Ptr() As Long, Ind() As Long)\r\n    Dim MM As Long, II As Long, I As Long, J As Long, K As Long\r\n    MM = M * M\r\n    K = 0\r\n    For II = 0 To MM - 1\r\n        I = Int(II \/ M)\r\n        J = II - I * M\r\n        Ptr(II) = K\r\n        If I > 0 Then\r\n            Ind(K) = II - M\r\n            Val(K) = -1\r\n            K = K + 1\r\n        End If\r\n        If J > 0 Then\r\n            Ind(K) = II - 1\r\n            Val(K) = -1\r\n            K = K + 1\r\n        End If\r\n        Ind(K) = II\r\n        Val(K) = 4\r\n        K = K + 1\r\n        If J < M - 1 Then\r\n            Ind(K) = II + 1\r\n            Val(K) = -1\r\n            K = K + 1\r\n        End If\r\n        If I < M - 1 Then\r\n            Ind(K) = II + M\r\n            Val(K) = -1\r\n            K = K + 1\r\n        End If\r\n    Next\r\n    Ptr(MM) = K\r\nEnd Sub\r\n\r\nSub Rhs5DF(M As Long, B() As Double)\r\n    Dim H As Double, MM As Long, I As Long\r\n    H = 1# \/ (M + 1)\r\n    MM = M * M\r\n    For I = 0 To MM - 1\r\n        B(I) = 0\r\n    Next\r\n    For I = M - 1 To MM - 1 Step M\r\n        B(I) = H * (Int(I \/ M) + 1)\r\n    Next\r\n    For I = M * (M - 1) To MM - 1\r\n        B(I) = B(I) + H * (I - M * (M - 1) + 1)\r\n    Next\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
16.6.2.2 CG \u6cd5\u306b\u3088\u308b\u89e3\u306e\u8a08\u7b97<\/h5>\n
\\(\\boldsymbol{A}\\) \u3068 \\(\\boldsymbol{b}\\) \u304c\u7528\u610f\u3067\u304d\u305f\u3089 Cg1<\/strong> \u3092\u547c\u3073\u51fa\u3057\u3066\u89e3\u3092\u6c42\u3081\u307e\u3059. \u521d\u671f\u5024\u306f\u3059\u3079\u3066 0 \u3068\u3057\u3066\u307f\u307e\u3057\u305f. \u89e3\u304c\u6c42\u3081\u3089\u308c\u305f\u3089\u9069\u5f53\u306a\u30d5\u30a9\u30fc\u30de\u30c3\u30c8\u3067\u51fa\u529b\u3057\u307e\u3059.<\/p>\n
\n
Sub Start()\r\n    Const N = 7, M = N - 1, MM = M * M\r\n    Dim Val(5 * MM) As Double, Ptr(MM) As Long, Ind(5 * MM) As Long\r\n    Dim B(MM - 1) As Double, X(MM - 1) As Double\r\n    Dim Resid As Double, Iter As Long, Info As Long\r\n    Dim I As Long, J As Long\r\n    '-- Set coefficients\r\n    Call Mat5DF(M, Val(), Ptr(), Ind())\r\n    Call Rhs5DF(M, B())\r\n    '-- Set initial approximation\r\n    For I = 0 To MM - 1\r\n        X(I) = 0\r\n    Next\r\n    '-- Solve equations\r\n    Call Cg1(MM, Val(), Ptr(), Ind(), B(), X(), Info, Iter, Resid)\r\n    '-- Output result\r\n    If Info = 0 Then\r\n        Call Output(N, X())\r\n    Else\r\n        MsgBox (\"Error in Cg1: Info =\" + Str(Info))\r\n    End If\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
\u3053\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u5b9f\u884c\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306a\u7d50\u679c\u304c\u5f97\u3089\u308c\u307e\u3057\u305f. \u3053\u3053\u3067\u306f, \u5883\u754c\u70b9\u3092\u542b\u3081\u5168\u683c\u5b50\u70b9\u306e\u5024\u3092\u8868\u793a\u3057\u3066\u3044\u307e\u3059.<\/p>\n
\"\"<\/p>\n
16.7 \u30d5\u30a1\u30a4\u30eb\u5165\u51fa\u529b<\/h3>\n
\u758e\u884c\u5217\u3092\u30c6\u30ad\u30b9\u30c8\u30d5\u30a1\u30a4\u30eb\u3068\u3057\u3066\u683c\u7d0d\u3059\u308b\u969b\u306b\u3088\u304f\u4f7f\u7528\u3055\u308c\u308b\u5f62\u5f0f\u306e\u4e00\u3064\u306b MM \u5f62\u5f0f (Matrix Market Exchange Format) \u304c\u3042\u308a\u307e\u3059. \u30c6\u30b9\u30c8\u30de\u30c8\u30ea\u30c3\u30af\u30b9\u3092\u6d41\u901a\u3055\u305b\u308b\u969b\u306a\u3069\u306b\u4f7f\u7528\u3055\u308c\u3066\u3044\u307e\u3059.<\/p>\n
XLPack (\u57fa\u672c\u6a5f\u80fd) \u3067\u306f, MM \u5f62\u5f0f\u30d5\u30a1\u30a4\u30eb\u306b\u5bfe\u3057\u3066\u6b21\u306e\u3088\u3046\u306a\u5165\u51fa\u529b\u30d7\u30ed\u30b0\u30e9\u30e0\u304c\u7528\u610f\u3055\u308c\u3066\u3044\u307e\u3059.<\/p>\n
 \r\n  MMRead<\/strong>      Matrix Market\u5f62\u5f0f\u30d5\u30a1\u30a4\u30eb\u306e\u8aad\u307f\u8fbc\u307f\r\n  MMReadInfo<\/strong>  Matrix Market\u5f62\u5f0f\u30d5\u30a1\u30a4\u30eb\u306e\u884c\u5217\u60c5\u5831\u306e\u8aad\u307f\u8fbc\u307f\r\n  MMWrite<\/strong>     Matrix Market\u5f62\u5f0f\u30d5\u30a1\u30a4\u30eb\u3078\u306e\u66f8\u304d\u8fbc\u307f\r\n\r\n<\/pre>\n
 <\/p>\n\n
\u4ed8\u9332. \u6709\u9650\u8981\u7d20\u6cd5\u306b\u3088\u308b\u8a08\u7b97\u4f8b<\/h3>\n
XLPack \u3067\u306f\u758e\u884c\u5217\u8a08\u7b97\u6a5f\u80fd\u306e\u4e00\u90e8\u3068\u3057\u3066\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\u95a2\u9023\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u304c\u542b\u307e\u308c\u3066\u3044\u307e\u3059 (\u73fe\u6642\u70b9\u3067\u306f\u5b9f\u9a13\u30d0\u30fc\u30b8\u30e7\u30f3\u306e\u305f\u3081\u5c06\u6765\u5909\u66f4\u3055\u308c\u308b\u53ef\u80fd\u6027\u304c\u3042\u308a\u307e\u3059). \u305d\u306e\u4e2d\u306e\u6709\u9650\u8981\u7d20\u6cd5\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f7f\u7528\u3057\u3066\u4e0a\u306e\u30e9\u30d7\u30e9\u30b9\u65b9\u7a0b\u5f0f\u306e\u4f8b\u984c\u3092\u89e3\u3044\u3066\u307f\u307e\u3059.<\/p>\n
\u4e09\u89d2\u8981\u7d20\u306b\u3088\u308b\u5358\u7d14\u30e1\u30c3\u30b7\u30e5\u3092\u4f7f\u3063\u305f\u4f8b\u3067\u3059. \u683c\u5b50\u70b9, \u6709\u9650\u8981\u7d20\u304a\u3088\u3073\u5883\u754c\u8981\u7d20\u306e\u60c5\u5831\u3092 Fem2p<\/strong> \u306b\u5165\u529b\u3059\u308b\u3068, CSR \u5f62\u5f0f\u306e\u6709\u9650\u8981\u7d20\u884c\u5217\u3068\u53f3\u8fba\u30d9\u30af\u30c8\u30eb\u3092\u51fa\u529b\u3057\u307e\u3059. \u3053\u308c\u306f\u5bfe\u79f0\u884c\u5217\u306a\u306e\u3067 CG \u6cd5\u3067\u89e3\u3051\u3070\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u304c\u6c42\u3081\u3089\u308c\u307e\u3059.<\/p>\n
\n
Sub Start()\r\n    Const Nx = 7, Ny = 7, N = (Nx + 1) * (Ny + 1), Ne = 2 * Nx * Ny\r\n    Const Nb = 2 * (Nx + Ny), Nb2 = 0, LdKnc = 4, LdKs = 3\r\n    Dim X(N - 1) As Double, Y(N - 1) As Double\r\n    Dim P(N - 1) As Double, Q(N - 1) As Double, F(N - 1) As Double\r\n    Dim Knc(LdKnc - 1, Ne - 1) As Long, Ks(LdKs - 1, Nb - 1) As Long\r\n    Dim Lb(Nb - 1) As Long, Ib(Nb - 1) As Long, Bv(Nb - 1) As Double\r\n    Dim Ks2() As Long, Alpha() As Double, Beta() As Double\r\n    Dim Val(20 * N) As Double, Ptr(N) As Long, Ind(20 * N) As Long\r\n    Dim B(N - 1) As Double, U(N - 1) As Double\r\n    Dim Ux As Double, Err As Double\r\n    Dim Info As Long\r\n    Dim I As Long, J As Long\r\n    '-- Set mesh data\r\n    Call SetData(Nx, Ny, N, Ne, Nb, X(), Y(), Knc(), Ks(), Lb(), P(), Q(), F(), Ib(), Bv())\r\n    '-- Assemble FEM matrix\r\n    Call Fem2p(N, Ne, X(), Y(), Knc(), P(), Q(), F(), Nb, Ib(), Bv(), Nb2, Ks2(), Alpha(), Beta(), Val(), Ptr(), Ind(), B(), Info)\r\n    '-- Solve equation\r\n    Dim Iter As Long, Res As Double\r\n    Call Cg1(N, Val(), Ptr(), Ind(), B(), U(), Info, Iter, Res)\r\n    Debug.Print \"Info =\" + Str(Info) + \", Iter =\" + Str(Iter) + \", Res =\" + Str(Res)\r\n    '-- Display solution\r\n    For I = 0 To Nx\r\n        For J = 0 To Ny\r\n            Cells(4 + Nx - J, 1 + I) = U((Ny + 1) * J + I)\r\n        Next\r\n    Next\r\nEnd Sub\r\n\r\nSub SetData(Nx As Long, Ny As Long, N As Long, Ne As Long, Nb As Long, X() As Double, Y() As Double, Knc() As Long, Ks() As Long, Lb() As Long, P() As Double, Q() As Double, F() As Double, Ib() As Long, Bv() As Double)\r\n    Dim Id() As Long\r\n    Dim I As Long, J As Long, K As Long\r\n    Call Mesh23(Nx, Ny, X(), Y(), Knc(), Ks(), Lb())\r\n    For I = 0 To N - 1\r\n        P(I) = 1\r\n        Q(I) = 0\r\n        F(I) = 0\r\n    Next\r\n    '-- Set boundary condition 1\r\n    ReDim Id(N - 1)\r\n    For I = 0 To Nb - 1\r\n        For J = 1 To 2\r\n            Id(Ks(J, I) - 1) = Lb(I)\r\n        Next\r\n    Next\r\n    K = 0\r\n    For I = 0 To N - 1\r\n        If Id(I) = 1 Or Id(I) = 4 Then\r\n            Bv(K) = 0\r\n        ElseIf Id(I) = 2 Then\r\n            Bv(K) = Y(I)\r\n        ElseIf Id(I) = 3 Then\r\n            Bv(K) = X(I)\r\n        Else\r\n            GoTo Continue\r\n        End If\r\n        Ib(K) = I + 1\r\n        K = K + 1\r\nContinue:\r\n    Next\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
\u3053\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u5b9f\u884c\u3059\u308b\u3068\u5dee\u5206\u6cd5\u306b\u3088\u308b\u89e3\u3068 (\u8aa4\u5dee\u306e\u7bc4\u56f2\u3067) \u540c\u3058\u7d50\u679c\u304c\u5f97\u3089\u308c\u307e\u3059.<\/p>\n
\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\u95a2\u9023\u6a5f\u80fd\u3068\u3057\u3066 VTK \u30d5\u30a1\u30a4\u30eb\u3078\u306e\u51fa\u529b\u30d7\u30ed\u30b0\u30e9\u30e0 WriteVtkug<\/strong> \u304c\u63d0\u4f9b\u3055\u308c\u3066\u3044\u307e\u3059. \u3053\u308c\u3092\u4f7f\u7528\u3057\u3066\u4e0a\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u6700\u5f8c\u306e\u90e8\u5206\u3092\u5909\u66f4\u3057\u307e\u3059.<\/p>\n
\n
Sub Start2()\r\n    Const Nx = 7, Ny = 7, N = (Nx + 1) * (Ny + 1), Ne = 2 * Nx * Ny\r\n    Const Nb = 2 * (Nx + Ny), Nb2 = 0, LdKnc = 4, LdKs = 3\r\n    Dim X(N - 1) As Double, Y(N - 1) As Double\r\n    Dim P(N - 1) As Double, Q(N - 1) As Double, F(N - 1) As Double\r\n    Dim Knc(LdKnc - 1, Ne - 1) As Long, Ks(LdKs - 1, Nb - 1) As Long\r\n    Dim Lb(Nb - 1) As Long, Ib(Nb - 1) As Long, Bv(Nb - 1) As Double\r\n    Dim Ks2() As Long, Alpha() As Double, Beta() As Double\r\n    Dim Val(20 * N) As Double, Ptr(N) As Long, Ind(20 * N) As Long\r\n    Dim B(N - 1) As Double, U(N - 1) As Double\r\n    Dim Ux As Double, Err As Double\r\n    Dim Info As Long\r\n    Dim I As Long, J As Long\r\n    '-- Set mesh data\r\n    Call SetData(Nx, Ny, N, Ne, Nb, X(), Y(), Knc(), Ks(), Lb(), P(), Q(), F(), Ib(), Bv())\r\n    '-- Assemble FEM matrix\r\n    Call Fem2p(N, Ne, X(), Y(), Knc(), P(), Q(), F(), Nb, Ib(), Bv(), Nb2, Ks2(), Alpha(), Beta(), Val(), Ptr(), Ind(), B(), Info)\r\n    '-- Solve equation\r\n    Dim Iter As Long, Res As Double\r\n    Call Cg1(N, Val(), Ptr(), Ind(), B(), U(), Info, Iter, Res)\r\n    Debug.Print \"Info =\" + Str(Info) + \", Iter =\" + Str(Iter) + \", Res =\" + Str(Res)\r\n    '-- Output solution\r\n    Dim Z(N - 1) As Double\r\n    Call WriteVtkug(\"Test.vtk\", N, X(), Y(), Z(), Ne, Knc(), U(), Info)\r\n    Debug.Print \"Info =\" + Str(Info)\r\nEnd Sub<\/code><\/pre>\n<\/div>\n
\u3053\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u5b9f\u884c\u3059\u308b\u3068 Test.vtk \u30d5\u30a1\u30a4\u30eb\u304c\u5f97\u3089\u308c\u307e\u3059. vtk \u30d5\u30a1\u30a4\u30eb\u3092\u5916\u90e8\u30d7\u30ed\u30b0\u30e9\u30e0\u306b\u5165\u529b\u3057\u3066\u7d50\u679c\u3092\u8868\u793a\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059. \u4f8b\u3048\u3070, ParaView \u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u8868\u793a\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.<\/p>\n
\"\"<\/p>\n
XLPack(\u57fa\u672c\u6a5f\u80fd)\u3067\u306f\u4ed6\u306b\u3082\u30e1\u30c3\u30b7\u30e5\u751f\u6210\u30d7\u30ed\u30b0\u30e9\u30e0 gmsh \u306e\u30d5\u30a1\u30a4\u30eb\u306e\u5165\u51fa\u529b\u30d7\u30ed\u30b0\u30e9\u30e0 (ReadGmsh22<\/strong>, WriteGmsh22<\/strong>) \u304c\u63d0\u4f9b\u3055\u308c\u307e\u3059. \u3053\u308c\u3092\u4f7f\u7528\u3059\u308b\u3068 gmsh \u3067\u4f5c\u6210\u3057\u305f\u3044\u308d\u3044\u308d\u306a\u5f62\u306e\u9818\u57df\u3067\u306e\u8a08\u7b97\u3092\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059. \u7c21\u5358\u306a\u8a08\u7b97\u7d50\u679c\u4f8b\u3092\u6b21\u306b\u793a\u3057\u307e\u3059.<\/p>\n
\"\"<\/p>\n","protected":false},"excerpt":{"rendered":"
\u758e\u884c\u5217\u306e\u7dda\u5f62\u8a08\u7b97\u306e\u89e3\u6cd5\u3068XLPack\u3092\u4f7f\u3063\u305f\u89e3\u304d\u65b9<\/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,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[10],"tags":[],"class_list":["post-4708","post","type-post","status-publish","format-standard","hentry","category-tutorial"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/4708","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/comments?post=4708"}],"version-history":[{"count":5,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/4708\/revisions"}],"predecessor-version":[{"id":5140,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/4708\/revisions\/5140"}],"wp:attachment":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/media?parent=4708"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/categories?post=4708"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/tags?post=4708"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}