{"id":3871,"date":"2015-12-15T12:59:24","date_gmt":"2015-12-15T03:59:24","guid":{"rendered":"http:\/\/inter-arteq.com\/?p=3871"},"modified":"2022-03-12T15:30:37","modified_gmt":"2022-03-12T06:30:37","slug":"implement-runge-kutta-method-in-c-plusplus","status":"publish","type":"post","link":"http:\/\/cms.inter-arteq.com\/implement-runge-kutta-method-in-c-plusplus\/","title":{"rendered":"[C++] Runge-Kutta\u6cd5\u3067\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u305f\u3089\u3061\u3087\u3063\u3068\u611f\u52d5\u3057\u305f\u8a71"},"content":{"rendered":"\n<p>\u5927\u5b66\u306e\u5f8c\u8f29\u304c\u7814\u7a76\u3068\u3057\u3066\u3001\u7269\u7406\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u3057\u3066\u3001VR\u3067\u904a\u3079\u308b\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3\u3092\u4f5c\u3063\u3066\u3044\u307e\u3057\u305f\u3002\u7a7a\u6c17\u62b5\u6297\u3092\u8003\u3048\u308b\u5fc5\u8981\u304c\u3042\u308a\u3001\u8907\u96d1\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304b\u306a\u304f\u3066\u306f\u306a\u3089\u306a\u3044\u3068\u306e\u3053\u3068\u3067\u3001\u5927\u5b66\u6642\u4ee3\u306b\u30ab\u30aa\u30b9\u7406\u8ad6\u3092\u5c02\u9580\u306b\u3057\u3066\u3044\u305f\u81ea\u5206\u306b<strong>Runge-Kutta\u6cd5<\/strong>\u3092\u6559\u3048\u3066\u307b\u3057\u3044\u3068\u76f8\u8ac7\u3055\u308c\u305f\u306e\u3067\u3001\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f5c\u3063\u3066\u307f\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_69_1 counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\">\u76ee\u6b21<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"http:\/\/cms.inter-arteq.com\/implement-runge-kutta-method-in-c-plusplus\/#Runge-Kutta%E6%B3%95%E3%81%A8%E3%81%AF\" title=\"Runge-Kutta\u6cd5\u3068\u306f\">Runge-Kutta\u6cd5\u3068\u306f<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"http:\/\/cms.inter-arteq.com\/implement-runge-kutta-method-in-c-plusplus\/#%E3%83%97%E3%83%AD%E3%82%B0%E3%83%A9%E3%83%A0\" title=\"\u30d7\u30ed\u30b0\u30e9\u30e0\">\u30d7\u30ed\u30b0\u30e9\u30e0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"http:\/\/cms.inter-arteq.com\/implement-runge-kutta-method-in-c-plusplus\/#%E4%BD%BF%E3%81%84%E6%96%B9\" title=\"\u4f7f\u3044\u65b9\">\u4f7f\u3044\u65b9<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"http:\/\/cms.inter-arteq.com\/implement-runge-kutta-method-in-c-plusplus\/#%E8%A7%A3%E8%AA%AC\" title=\"\u89e3\u8aac\">\u89e3\u8aac<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"http:\/\/cms.inter-arteq.com\/implement-runge-kutta-method-in-c-plusplus\/#%E7%B5%90%E6%9E%9C\" title=\"\u7d50\u679c\">\u7d50\u679c<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"http:\/\/cms.inter-arteq.com\/implement-runge-kutta-method-in-c-plusplus\/#%E5%8F%82%E8%80%83%E3%82%B5%E3%82%A4%E3%83%88\" title=\"\u53c2\u8003\u30b5\u30a4\u30c8\">\u53c2\u8003\u30b5\u30a4\u30c8<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Runge-Kutta%E6%B3%95%E3%81%A8%E3%81%AF\"><\/span>Runge-Kutta\u6cd5\u3068\u306f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Wikipedia\u3092\u53c2\u7167\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304b\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\"><p>\u6570\u5024\u89e3\u6790\u306b\u304a\u3044\u3066\u30eb\u30f3\u30b2\uff1d\u30af\u30c3\u30bf\u6cd5\uff08\u82f1: Runge\u2013Kutta method\uff09\u3068\u306f\u3001\u521d\u671f\u5024\u554f\u984c\u306b\u5bfe\u3057\u3066\u8fd1\u4f3c\u89e3\u3092\u4e0e\u3048\u308b\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5\u306b\u5bfe\u3059\u308b\u7dcf\u79f0\u3067\u3042\u308b\u3002\u3053\u306e\u6280\u6cd5\u306f1900\u5e74\u9803\u306b\u6570\u5b66\u8005\u30ab\u30fc\u30eb\u30fb\u30eb\u30f3\u30b2\u3068\u30de\u30eb\u30c6\u30a3\u30f3\u30fb\u30af\u30c3\u30bf\u306b\u3088\u3063\u3066\u767a\u5c55\u3092\u898b\u305f\u3002<\/p><cite><a href=\"\u30eb\u30f3\u30b2=\u30af\u30c3\u30bf\u6cd5\" target=\"_blank\" rel=\"noreferrer noopener\" title=\"https:\/\/ja.wikipedia.org\/wiki\/%E3%83%AB%E3%83%B3%E3%82%B2%EF%BC%9D%E3%82%AF%E3%83%83%E3%82%BF%E6%B3%95\">Wikipedia: \u30eb\u30f3\u30b2=\u30af\u30c3\u30bf\u6cd5<\/a><\/cite><\/blockquote>\n\n\n\n<p>\u7aef\u7684\u306b\u8a00\u3046\u3068\u521d\u671f\u5024\u3068\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u4e0e\u3048\u308b\u3053\u3068\u3067\u3001\u89e3\u3068\u306a\u308b\u95a2\u6570\u306e<strong>\u6570\u5024<\/strong>\u3092\u6c42\u3081\u308b\u65b9\u6cd5\u3067\u3059\u3002(\u89e3\u3068\u306a\u308b\u95a2\u6570\u3092\u5f97\u3089\u308c\u308b\u308f\u3051\u3067\u306f\u306a\u3044)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E3%83%97%E3%83%AD%E3%82%B0%E3%83%A9%E3%83%A0\"><\/span>\u30d7\u30ed\u30b0\u30e9\u30e0<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u8a00\u8a9e\u306fC++\u3067\u3059\u3002<\/p>\n\n\n\n<div class=\"hcb_wrap\"><pre class=\"prism line-numbers lang-cpp\" data-file=\" RungeKutta.hpp\" data-lang=\"C++\"><code>#ifndef RungeKutta_hpp\n#define RungeKutta_hpp\n#include &lt;stdio.h&gt;\nclass RungeKutta {\n    float * result;\n    int num;\n    void calculateNextStep(float (*f[])(float t, float *x), float t0, float *x, float tn, int num);\npublic:\n    RungeKutta(){};\n    RungeKutta(float (*f[])(float t, float *x), float t0, float tn, float *x, int div, int num);\n    float getValue(int step, int index);\n};\n#endif \/* RungeKutta_hpp *\/<\/code><\/pre><\/div>\n\n\n\n<div class=\"hcb_wrap\"><pre class=\"prism line-numbers lang-cpp\" data-file=\"RungeKutta.cpp\" data-lang=\"C++\"><code>#include &quot;RungeKutta.hpp&quot;\nfloat RungeKutta::getValue(int step, int index){\n    return result[step*num + index];\n}\nRungeKutta::RungeKutta(float (*f[])(float t, float *x), float t0, float tn, float *x, int div, int _num){\n    num = _num;\n    result = new float[div*num];\n    float h = (tn - t0)\/div;\n    for(int i=0; i&lt;div; ++i){\n        calculateNextStep(f, t0, x, t0+h, num);\n        for(int j=0; j&lt;num; ++j){\n            result[i*num+j] = x[j];\n            t0 += h;\n        }\n    }\n}\nvoid RungeKutta::calculateNextStep(float (*f[])(float t, float *x), float t0, float *x, float tn, int num){\n    float k1[num], k2[num], k3[num], k4[num], tmp[num];\n    float h = (tn - t0);\n    float t = t0;\n    for(int j=0; j&lt;num; j++){\n        k1[j] = (*f[j])(t, x);\n        tmp[j] = x[j] + h*k1[j]\/2;\n        k2[j] = (*f[j])(t+h\/2, tmp);\n        tmp[j] = x[j] + h*k2[j]\/2;\n        k3[j] = (*f[j])(t+h\/2, tmp);\n        tmp[j] = x[j] + h*k3[j];\n        k4[j] = (*f[j])(t+h, tmp);\n        x[j] += (k1[j] + 2*k2[j] + 2*k3[j] + k4[j])*h\/6;\n    }\n}<\/code><\/pre><\/div>\n\n\n\n<p> <\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E4%BD%BF%E3%81%84%E6%96%B9\"><\/span>\u4f7f\u3044\u65b9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u6e1b\u8870\u632f\u52d5\u306e\u4ee5\u4e0b\u306e\u5f0f\u3092\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3<\/p>\n\n\n\n<p><img decoding=\"async\" src=\"http:\/\/s0.wp.com\/latex.php?latex=%5Cfrac%7Bd%5E2x%7D%7Bdt%5E2%7D%3D-kx-b%5Cfrac%7Bdx%7D%7Bdt%7D&#038;bg=ffffff&#038;fg=000&#038;s=4&#038;c=20201002\" alt=\"&#92;frac{d^2x}{dt^2}=-kx-b&#92;frac{dx}{dt}\" class=\"latex\" \/><\/p>\n\n\n\n<p>\u4ee5\u4e0b\u306e\u3088\u3046\u306by\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p><img decoding=\"async\" src=\"http:\/\/s0.wp.com\/latex.php?latex=%5Cfrac%7Bdx%7D%7Bdt%7D%3Dy&#038;bg=ffffff&#038;fg=000&#038;s=4&#038;c=20201002\" alt=\"&#92;frac{dx}{dt}=y\" class=\"latex\" \/><\/p>\n\n\n\n<p>\u3059\u308b\u3068\u3001\u6e1b\u8870\u632f\u52d5\u306e\u65b9\u7a0b\u5f0f\u306f\u3053\u308c\u3092\u4ee3\u5165\u3059\u308b\u3053\u3068\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p><img decoding=\"async\" src=\"http:\/\/s0.wp.com\/latex.php?latex=%5Cfrac%7Bdy%7D%7Bdt%7D%3D-kx-by&#038;bg=ffffff&#038;fg=000&#038;s=4&#038;c=20201002\" alt=\"&#92;frac{dy}{dt}=-kx-by\" class=\"latex\" \/><\/p>\n\n\n\n<p>\u3053\u306e\u3088\u3046\u306b2\u3064\u306e\u4e00\u56de\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5206\u89e3\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p>\u3053\u306e\u4e8c\u3064\u306e\u95a2\u6570\u3092<code>_f0<\/code>, <code>_f1<\/code>\u3068\u3057\u3066\u5b9a\u7fa9\u3057\u3066\u3001Runge-Kutta\u306b\u304b\u3051\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"hcb_wrap\"><pre class=\"prism line-numbers lang-cpp\" data-lang=\"C++\"><code>#define LOOP 10000\nfloat _f0(float t, float * x){\n    return x[1];\n}\nfloat _f1(float t, float * x){\n    return - x[0] - 0.2 * x[1];\n}\nfloat (*f[2])(float, float*);\nf[0] = _f0;\nf[1] = _f1;\nfloat initialValues[] = {30.0, 0.0};\nRungeKutta rk = RungeKutta(f, 0, 30, initialValues, LOOP, 2);\nfor(int i=0; i&lt;LOOP; ++i){\n  cout &lt;&lt; rk.getValue(i, 0) &lt;&lt; endl;\n}<\/code><\/pre><\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E8%A7%A3%E8%AA%AC\"><\/span>\u89e3\u8aac<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u5909\u6570\u540d\u304c\u5909\u308f\u3063\u3066\u3044\u308b\u306e\u3067\u5206\u304b\u308a\u306b\u304f\u3044\u3067\u3059\u304c\u3001<code>_f0<\/code>, <code>_f1<\/code>\u306e\u4e2d\u306e<code>x[0]<\/code>\u304cx\u3001<code>x[1]<\/code>\u304cy\u306b\u76f8\u7b49\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>RungeKutta\u30af\u30e9\u30b9\u306e\u30b3\u30f3\u30b9\u30c8\u30e9\u30af\u30bf\u306e\u5f15\u6570\u306e\u610f\u5473\u3092\u4ee5\u4e0b\u306b\u307e\u3068\u3081\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\u7b2c1\u5f15\u6570\u306f\u5fae\u5206\u65b9\u7a0b\u5f0f\u30921\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5206\u5272\u3057\u305f\u95a2\u6570\u7fa4<\/li><li>\u7b2c2,3\u5f15\u6570\u306f\u5b9a\u7fa9\u57df\u3092\u8868\u3057\u307e\u3059\u3002<\/li><li>\u7b2c4\u5f15\u6570\u306f\u5b9a\u7fa9\u57df\u3092\u3069\u306e\u304f\u3089\u3044\u5206\u5272\u3057\u3066\u8a08\u7b97\u3059\u308b\u304b\u3092\u8868\u3057\u307e\u3059\u3002<\/li><li>\u7b2c5\u5f15\u6570\u306f\u5fae\u5206\u65b9\u7a0b\u5f0f\u30921\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5206\u5272\u3057\u305f\u3068\u304d\u306e\u65b9\u7a0b\u5f0f\u306e\u500b\u6570\u3067\u3059\u3002<\/li><\/ul>\n\n\n\n<p>\u4eca\u56de\u306e\u30b5\u30f3\u30d7\u30eb\u30b3\u30fc\u30c9\u3067\u306f2\u3064\u306e\u521d\u671f\u4f4d\u7f6e30cm\u3001\u521d\u901f0cm\/s\u3068\u3057\u3066\u3044\u307e\u3059\u3002\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u30920\u301c30\u79d2\u306e\u6642\u9593\u309210000\u5206\u5272(0.003\u79d2\u306e\u7c92\u5ea6)\u3067\u3001\u8a08\u7b97\u3059\u308b\u3068\u3044\u3046\u610f\u5473\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p>rk.getValue\u306e\u5f15\u6570\u306e\u610f\u5473\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3059\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\u7b2c1\u5f15\u6570\u306f\u4f55\u756a\u76ee\u306e\u5024\u304b\uff1f<\/li><li>\u7b2c2\u5f15\u6570\u306f\u4f55\u756a\u76ee\u306e\u5909\u6570\u304b\uff1f<\/li><\/ul>\n\n\n\n<p>\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u4eca\u56de\u306fx(<code>x[0]<\/code>)\u306e\u5024\u304c\u6b32\u3057\u3044\u306e\u3067\u3001\u7b2c2\u5f15\u6570\u30920\u3068\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u3082\u3057\u3001\u7b2c2\u5f15\u6570\u30921\u306b\u3059\u308b\u3068\u3001y(<code>x[1]<\/code>)\u306e\u5024\u304c\u53d6\u308c\u308b\u306e\u3067\u3001\u901f\u5ea6\u304c\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E7%B5%90%E6%9E%9C\"><\/span>\u7d50\u679c<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>openFrameworks\u3067\u30b0\u30e9\u30d5\u3092\u63cf\u753b\u3057\u3066\u3084\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"301\" src=\"http:\/\/cms.inter-arteq.com\/wp-content\/uploads\/2022\/03\/Runge-Kutta-sample-1024x301.png\" alt=\"\" class=\"wp-image-208423\" srcset=\"http:\/\/cms.inter-arteq.com\/wp-content\/uploads\/2022\/03\/Runge-Kutta-sample-1024x301.png 1024w, http:\/\/cms.inter-arteq.com\/wp-content\/uploads\/2022\/03\/Runge-Kutta-sample-300x88.png 300w, http:\/\/cms.inter-arteq.com\/wp-content\/uploads\/2022\/03\/Runge-Kutta-sample-768x226.png 768w, http:\/\/cms.inter-arteq.com\/wp-content\/uploads\/2022\/03\/Runge-Kutta-sample.png 1136w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>\u3061\u3083\u3093\u3068\u6e1b\u8870\u632f\u52d5\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" 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