{"id":2025,"date":"2022-06-08T09:30:00","date_gmt":"2022-06-08T00:30:00","guid":{"rendered":"https:\/\/www.insilico.jp\/blog\/?p=2025"},"modified":"2022-06-08T09:49:35","modified_gmt":"2022-06-08T00:49:35","slug":"lalibrary_eigen_part2","status":"publish","type":"post","link":"https:\/\/www.insilico.jp\/blog\/2022\/06\/08\/lalibrary_eigen_part2\/","title":{"rendered":"C++\u306e\u7dda\u578b\u4ee3\u6570\u30e9\u30a4\u30d6\u30e9\u30eaEigen\u306e\u4f7f\u3044\u65b9\u5099\u5fd8\u9332 \u301c\u5fdc\u7528\u7de8\u301c"},"content":{"rendered":"\n<p>\u524d\u56de\u306e\u30d6\u30ed\u30b0\u3067\u306fEigen\u3092\u5229\u7528\u3057\u305f\u30d9\u30af\u30c8\u30eb\u3068\u884c\u5217\u306e\u57fa\u672c\u7684\u306a\u4f7f\u3044\u65b9\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u307e\u3057\u305f\u3002\u4eca\u56de\u306f\u7dda\u578b\u65b9\u7a0b\u5f0f\u3084\u56fa\u6709\u5024\u554f\u984c\u306e\u89e3\u3092\u6c42\u3081\u308bEigen\u306e\u5229\u7528\u65b9\u6cd5\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u307e\u3059\u3002\u3053\u306e\u5fdc\u7528\u7de8\u306b\u3064\u3044\u3066\u3082\u81ea\u5206\u7528\u306e\u5099\u5fd8\u9332\u3068\u3057\u3066\u307e\u3068\u3081\u305f\u3082\u306e\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">LU\u5206\u89e3\u306a\u3069<\/h2>\n\n\n\n<p>LU\u5206\u89e3\u3068\u306f\u3042\u308b\u6b63\u65b9\u884c\u5217A\u3092\u4e0b\u4e09\u89d2\u884c\u5217L\u3068\u4e0a\u4e09\u89d2\u884c\u5217U\u306e\u7a4d\u306b\u5206\u89e3\uff08\u3064\u307e\u308aA=LU\uff09\u3059\u308b\u3053\u3068\u3067\u3059\u3002\u3053\u3053\u3067\u306f\u8a73\u3057\u304f\u89e3\u8aac\u3057\u307e\u305b\u3093\u304c\u3001WEB\u691c\u7d22\u3059\u308b\u3068\u5c71\u307b\u3069\u8a73\u3057\u3044\u8a18\u4e8b\u304c\u30d2\u30c3\u30c8\u3059\u308b\u306f\u305a\u3067\u3059\u3002LU\u5206\u89e3\u3059\u308b\u3068\u3001\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u304c\u89e3\u304f\u3053\u3068\u304c\u51fa\u6765\u305f\u308a\u3001\u884c\u5217\u5f0f\u304c\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002\u3042\u3068Eigen\u3067\u306f\u9006\u884c\u5217\u306e\u8a08\u7b97\u3082\u51fa\u6765\u307e\u3059\u3002\u3082\u3063\u3068\u3082\u3001\u9006\u884c\u5217\u3092\u967d\u306b\u8a08\u7b97\u3059\u308b\u3088\u3046\u306a\u5834\u9762\u306b\u51fa\u304f\u308f\u3059\u3053\u3068\u306a\u3069\u7121\u3044\u3067\u3057\u3087\u3046\u3002\u6570\u5024\u8a08\u7b97\u306e\u30d7\u30ed\u30b0\u30e9\u30df\u30f3\u30b0\u3067\u306f\u9006\u884c\u5217\u3092\u8a08\u7b97\u3057\u305f\u3089\u8ca0\u3051\u3067\u3059\u3002<\/p>\n\n\n\n<p>\u3042\u308b5\u6b21\u5143\u6b63\u65b9\u884c\u5217A\u306eLU\u5206\u89e3\u3059\u308b\u30b5\u30f3\u30d7\u30eb\u30d7\u30ed\u30b0\u30e9\u30e0\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\u30d7\u30ed\u30b0\u30e9\u30e0\u306e10\u884c\u76ee\u3067LU\u5206\u89e3\u3092\u5b9f\u884c\u3057\u3066\u3044\u307e\u3059\u3002\u30c6\u30f3\u30d7\u30ec\u30fc\u30c8\u30d1\u30e9\u30e1\u30fc\u30bf\uff08&lt;&gt;\u306e\u4e2d\u306e\u5024\uff09\u3068\u3057\u3066\u306fMatrixXd\u4ee5\u5916\u3082\u6307\u5b9a\u3067\u304d\u308b\u3002QR\u5206\u89e3\u3084\u30b3\u30ec\u30b9\u30ad\u30fc\u5206\u89e3\u306a\u3069\u4ed6\u306e\u65b9\u6cd5\u3092\u7528\u3044\u308b\u5834\u5408\u3082\u3053\u306e\u884c\u3068\u4f3c\u305f\u3088\u3046\u306a\u66f8\u5f0f\u3067\u3059\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code> 1 #include &lt;Eigen\/Dense&gt;\n 2 #include &lt;iostream&gt;\n 3 \n 4 using namespace std;\n 5 using namespace Eigen;\n 6 \n 7 int main() {\n 8   MatrixXd A(5, 5);\n 9   A &lt;&lt; 5, 4, 2, 8, 4, 4, 18, 9, 10, 9, 2, 9, 6, 5, 6, 8, 10, 5, 18, 9, 4, 9, 6, 9, 10;\n10   FullPivLU&lt;MatrixXd&gt; LU(A);\n11   MatrixXd X = LU.matrixLU(); L\u3068U\u3092\u4e00\u3064\u306b\u307e\u3068\u3081\u305f\u884c\u5217\u3092\u53d6\u5f97\u3059\u308b\n12   MatrixXd P = LU.permutationP(); A\u3092\u30d4\u30dc\u30c3\u30c6\u30a3\u30f3\u30b0\u3059\u308b\u884c\u5217P\u3092\u53d6\u5f97\u3059\u308b\n13   MatrixXd Q = LU.permutationQ(); A\u3092\u30d4\u30dc\u30c3\u30c6\u30a3\u30f3\u30b0\u3059\u308b\u884c\u5217Q\u3092\u53d6\u5f97\u3059\u308b\n14   MatrixXd L = LU.matrixLU().triangularView&lt;UnitLower&gt;(); L\u3092\u53d6\u5f97\u3059\u308b\n15   MatrixXd U = LU.matrixLU().triangularView&lt;Upper&gt;(); U\u3092\u53d6\u5f97\u3059\u308b\n16 \n17   cout &lt;&lt; \"A=\" &lt;&lt; endl;\n18   cout &lt;&lt; A &lt;&lt; endl &lt;&lt; endl;\n19   cout &lt;&lt; \"L*U=\" &lt;&lt; endl;\n20   cout &lt;&lt; L * U &lt;&lt; endl &lt;&lt; endl; A\u304c\u30d4\u30dc\u30c3\u30c6\u30a3\u30f3\u30b0\u3055\u308c\u3066\u3044\u308b\u306e\u3067A=LU\u3068\u306f\u306a\u3089\u306a\u3044\n21   cout &lt;&lt; \"P*A*Q=\" &lt;&lt; endl;\n22   cout &lt;&lt; P * A * Q &lt;&lt; endl &lt;&lt; endl; A\u3092\u30d4\u30dc\u30c3\u30c6\u30a3\u30f3\u30b0\u3057\u305f\u5f8c\u306a\u3089LU\u3068\u4e00\u81f4\u3059\u308b\n23 }<\/code><\/pre>\n\n\n\n<p>QR\u5206\u89e3\u306e\u5834\u5408\u306f\u3001<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>FullPivHouseholderQR&lt;MatrixXd&gt; QR(A);<\/code><\/pre>\n\n\n\n<p>\u6b63\u5b9a\u5024\u5bfe\u79f0\u884c\u5217\u306e\u5834\u5408\u306f\u30b3\u30ec\u30b9\u30ad\u30fc\u5206\u89e3\u3067\u3059\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>LLT&lt;MatrixXd&gt; CD(A);<\/code><\/pre>\n\n\n\n<p>\u6b63\u5b9a\u5024\u3058\u3083\u306a\u3044\u3068\u304d\u306f\u4fee\u6b63\u30b3\u30ec\u30b9\u30ad\u30fc\u5206\u89e3\u3067\u3059\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>LDLT&lt;MatrixXd&gt; MCD(A);<\/code><\/pre>\n\n\n\n<h2 class=\"wp-block-heading\">\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f<\/h2>\n\n\n\n<p>\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u3092\u6c42\u3081\u308b\u5834\u5408\u306fLU\u5206\u89e3\uff08\u3042\u308b\u3044\u306f\u30b3\u30ec\u30b9\u30ad\u30fc\u5206\u89e3\uff09\u3057\u3066\u304b\u3089\u8a08\u7b97\u3057\u307e\u3059\u3002\u4ee5\u4e0b\u306b\u30b5\u30f3\u30d7\u30eb\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u793a\u3057\u307e\u3059\u300214\u884c\u76ee\u306b\u3042\u308b\u3088\u3046\u306bsolve\u30e1\u30bd\u30c3\u30c9\u3092\u5229\u7528\u3057\u307e\u3059\u3002LAPACK\u3067\u306fDGESV\u3068\u3044\u3046\u95a2\u6570\u3092\u30b3\u30fc\u30eb\u3059\u308c\u3070\u826f\u3044\u3060\u3051\u306a\u306e\u3067\u3001Eigen\u306e\u65b9\u304c1\u30b9\u30c6\u30c3\u30d7\u624b\u9593\u304c\u5897\u3048\u3066\u3044\u308b\u611f\u3058\u306b\u306a\u308a\u307e\u3059\u3002\u3057\u304b\u3057\u306a\u304c\u3089\u3001\u884c\u5217\u5206\u89e3\u3057\u3066\u304b\u3089\u89e3\u3092\u6c42\u3081\u308b\u3068\u3044\u3046\u30b9\u30c6\u30c3\u30d7\u304c\u6570\u5b66\u7684\u306b\u3082\u308f\u304b\u308a\u3084\u3059\u3044\u306e\u3067\u3001API\u306e\u8a2d\u8a08\u3068\u3057\u3066\u306f\u3053\u3061\u3089\u306e\u65b9\u304c\u79c1\u306f\u597d\u304d\u3067\u3059\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code> 1 #include &lt;Eigen\/Dense&gt;\n 2 #include &lt;iostream&gt;\n 3 \n 4 using namespace std;\n 5 using namespace Eigen;\n 6 \n 7 int main() {\n 8   MatrixXd A(5, 5);\n 9   A &lt;&lt; 5, 4, 2, 8, 4, 4, 18, 9, 10, 9, 2, 9, 6, 5, 6, 8, 10, 5, 18, 9, 4, 9, 6,\n10       9, 10;\n11   VectorXd b(5);\n12   b &lt;&lt; 1, 2, 3, 4, 5;\n13   FullPivLU&lt;MatrixXd&gt; LU(A);\n14   VectorXd x = LU.solve(b);\n15   cout &lt;&lt; \"x=\" &lt;&lt; endl;\n16   cout &lt;&lt; x.transpose() &lt;&lt; endl &lt;&lt; endl;\n17 }<\/code><\/pre>\n\n\n\n<p>\u4e0a\u8a18\u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u884c\u5217\u306f\u5bfe\u79f0\u884c\u5217\u3067\u3059\u304c\u3001\u5b9f\u969b\u306e\u8a08\u7b97\u306b\u51fa\u3066\u304f\u308b\u884c\u5217\u306f\u5bfe\u79f0\u884c\u5217\u304c\u591a\u3044\u3067\u3059\u3002\u305d\u306e\u3088\u3046\u306a\u5834\u5408\u306f\u3001\u30b3\u30ec\u30b9\u30ad\u30fc\u5206\u89e3\u3092\u5229\u7528\u3059\u308b\u65b9\u304c\u9ad8\u901f\u3067\u3059\u3002\u305d\u306e\u5834\u5408\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u306f13\u884c\u76ee\u306e\u884c\u5217\u5206\u89e3\u306e\u65b9\u6cd5\u3092LU\u5206\u89e3\u304b\u3089\u30b3\u30ec\u30b9\u30ad\u30fc\u5206\u89e3\u306b\u7f6e\u304d\u63db\u3048\u308c\u3070\u826f\u3044\u3060\u3051\u3067\u3059\u3002\u6b63\u5b9a\u5024\u3067\u306a\u3044\u5834\u5408\u3082\u3042\u308b\u306e\u3067\u3001\u79c1\u306e\u5834\u5408\u306f\u4fee\u6b63\u30b3\u30ec\u30b9\u30ad\u30fc\u5206\u89e3\u306e\u65b9\u3092\u5e38\u7528\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u56fa\u6709\u5024\u554f\u984c<\/h2>\n\n\n\n<p>\u524d\u8ff0\u3057\u305f\u3088\u3046\u306b\u3001\u5b9f\u969b\u306e\u8a08\u7b97\u3067\u306f\u307b\u3068\u3093\u3069\u5bfe\u79f0\u884c\u5217\uff08\u3082\u3057\u304f\u306f\u30a8\u30eb\u30df\u30fc\u30c8\u884c\u5217\uff09\u3070\u304b\u308a\u306a\u306e\u3067\u3001\u30b5\u30f3\u30d7\u30eb\u30d7\u30ed\u30b0\u30e9\u30e0\u3082\u5bfe\u79f0\u884c\u5217\u306e\u56fa\u6709\u5024\u554f\u984c\u3092\u89e3\u304f\u3082\u306e\u3068\u3057\u3066\u3044\u307e\u3059\u300211\u884c\u76ee\u306eSelfAdjointEigenSolver\u3067\u8a08\u7b97\u3057\u3066\u3044\u307e\u3059\u3002\u305d\u3057\u3066\u3001eigenvalues\u30e1\u30bd\u30c3\u30c9\u3067\u56fa\u6709\u5024\u3092\u53d6\u5f97\u3067\u304d\u3066\u3001eigenvectors\u30e1\u30bd\u30c3\u30c9\u3067\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u304c\u53d6\u5f97\u3067\u304d\u307e\u3059\u3002\u7d50\u679c\u306f\u56fa\u6709\u5024\u304c\u5c0f\u3055\u3044\u9806\u306b\u683c\u7d0d\u3055\u308c\u307e\u3059\u3002\u4f8b\u3048\u3070\u4e0b\u8a18\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u6700\u5c0f\u56fa\u6709\u5024\u306f<code>e(0)<\/code>\u3067\u3042\u308a\u3001\u305d\u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306f<code>U.col(0)<\/code>\u3068\u306a\u308a\u307e\u3059\u3002SelfAdjoint\u304c\u4ed8\u304b\u306a\u3044&#8221;EigenSolver&#8221;\u3092\u4f7f\u3048\u3070\u975e\u5bfe\u79f0\u884c\u5217\u3067\u3082\u5bfe\u89d2\u5316\u3067\u304d\u308b\u307f\u305f\u3044\u3067\u3059\u304c\u3001\u51fa\u756a\u306f\u5c11\u306a\u3044\u3067\u3057\u3087\u3046\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code> 1 #include &lt;Eigen\/Dense&gt;\n 2 #include &lt;iostream&gt;\n 3 \n 4 using namespace std;\n 5 using namespace Eigen;\n 6 \n 7 int main() {\n 8   MatrixXd A(5, 5);\n 9   A &lt;&lt; 5, 4, 2, 8, 4, 4, 18, 9, 10, 9, 2, 9, 6, 5, 6, 8, 10, 5, 18, 9, 4, 9, 6,\n10       9, 10;\n11   SelfAdjointEigenSolver&lt;MatrixXd&gt; ES(A);\n12   VectorXd e = ES.eigenvalues();\n13   MatrixXd U = ES.eigenvectors();\n14   cout &lt;&lt; \"e=\" &lt;&lt; endl;\n15   cout &lt;&lt; e.transpose() &lt;&lt; endl &lt;&lt; endl;\n16   cout &lt;&lt; \"U=\" &lt;&lt; endl;\n17   cout &lt;&lt; U &lt;&lt; endl &lt;&lt; endl;\n18 }<\/code><\/pre>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e00\u822c\u5316\u56fa\u6709\u5024\u554f\u984c<\/h2>\n\n\n\n<p>\u4e00\u822c\u5316\u56fa\u6709\u5024\u554f\u984c\u306b\u3064\u3044\u3066\u3082\u56fa\u6709\u5024\u554f\u984c\u3068\u307b\u3068\u3093\u3069\u540c\u3058\u3088\u3046\u306a\u4f7f\u3044\u52dd\u624b\u3067\u3059\u3002\u91cf\u5b50\u5316\u5b66\u306a\u3069\u3067\u306f\u826f\u304f\u51fa\u3066\u304f\u308b\u8a08\u7b97\u306a\u306e\u3067\u3059\u304c\u3001\u73fe\u5728\u306e\u79c1\u304c\u3053\u306e\u3088\u3046\u306a\u8a08\u7b97\u3092\u3059\u308b\u6a5f\u4f1a\u306f\u591a\u5206\u7121\u3044\u3067\u3057\u3087\u3046\u306d\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code> 1 #include &lt;Eigen\/Dense&gt;\n 2 #include &lt;iostream&gt;\n 3 \n 4 using namespace std;\n 5 using namespace Eigen;\n 6 \n 7 int main() {\n 8   MatrixXd A(5, 5);\n 9   A &lt;&lt; 5, 4, 2, 8, 4, 4, 18, 9, 10, 9, 2, 9, 6, 5, 6, 8, 10, 5, 18, 9, 4, 9, 6,\n10       9, 10;\n11   MatrixXd B(5, 5);\n12   B &lt;&lt; 2, 2, 1, 3, 3, 2, 6, 1, 4, 5, 1, 1, 3, 2, 2, 3, 4, 2, 9, 6, 3, 5, 2, 6,\n13       7;\n14   GeneralizedSelfAdjointEigenSolver&lt;MatrixXd&gt; ES(A, B);\n15   VectorXd e = ES.eigenvalues();\n16   MatrixXd U = ES.eigenvectors();\n17   cout &lt;&lt; \"e=\" &lt;&lt; endl;\n18   cout &lt;&lt; e.transpose() &lt;&lt; endl &lt;&lt; endl;\n19   cout &lt;&lt; \"U=\" &lt;&lt; endl;\n20   cout &lt;&lt; U &lt;&lt; endl &lt;&lt; endl;\n21 \n22 }<\/code><\/pre>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u524d\u56de\u306e\u30d6\u30ed\u30b0\u3067\u306fEigen\u3092\u5229\u7528\u3057\u305f\u30d9\u30af\u30c8\u30eb\u3068\u884c\u5217\u306e\u57fa\u672c\u7684\u306a\u4f7f\u3044\u65b9\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u307e\u3057\u305f\u3002\u4eca\u56de\u306f\u7dda\u578b\u65b9\u7a0b\u5f0f\u3084\u56fa\u6709\u5024\u554f\u984c\u306e\u89e3\u3092\u6c42\u3081\u308bEigen\u306e\u5229\u7528\u65b9\u6cd5\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u307e\u3059\u3002\u3053\u306e\u5fdc\u7528\u7de8\u306b\u3064\u3044\u3066\u3082\u81ea\u5206\u7528\u306e\u5099\u5fd8\u9332\u3068\u3057\u3066\u307e\u3068\u3081\u305f\u3082\u306e&#8230;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[75],"class_list":["post-2025","post","type-post","status-publish","format-standard","hentry","category-15","tag-75"],"yoast_head":"<!-- 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