{"id":113436,"date":"2026-10-06T11:16:53","date_gmt":"2026-10-06T03:16:53","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/113436.html"},"modified":"2026-10-06T11:16:53","modified_gmt":"2026-10-06T03:16:53","slug":"%e7%ac%ac-18-%e7%ab%a0-%c2%b7-%e7%ba%bf%e6%80%a7%e6%96%b9%e7%a8%8b%e7%bb%84","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/113436.html","title":{"rendered":"\u7b2c 18 \u7ae0 \u00b7 \u7ebf\u6027\u65b9\u7a0b\u7ec4"},"content":{"rendered":"<p>\u77e9\u9635\u6700\u91cd\u8981\u7684\u5e94\u7528\u4e4b\u4e00&#xff0c;\u5c31\u662f\u8868\u793a\u548c\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002\u672c\u7ae0\u8bb2\u6e05\u695a&#034;Ax &#061; b&#034;\u5230\u5e95\u662f\u4ec0\u4e48\u610f\u601d&#xff0c;\u4ee5\u53ca\u5982\u4f55\u6c42\u89e3\u3002<\/p>\n<h3>18.1 \u4ec0\u4e48\u662f\u7ebf\u6027\u65b9\u7a0b\u7ec4<\/h3>\n<p>\u4f60\u80af\u5b9a\u89c1\u8fc7\u8fd9\u79cd\u65b9\u7a0b&#xff1a;<\/p>\n<p>2x &#043; y &#061; 5<br \/>\nx &#043; 3y &#061; 10<\/p>\n<p>\u4e24\u4e2a\u672a\u77e5\u6570 x\u3001y&#xff0c;\u4e24\u4e2a\u65b9\u7a0b\u3002\u6c42 x\u3001y \u7684\u503c&#xff0c;\u5c31\u662f\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002<\/p>\n<p>&#034;\u7ebf\u6027&#034;\u6307&#xff1a;\u6bcf\u4e2a\u65b9\u7a0b\u91cc\u672a\u77e5\u6570\u90fd\u662f\u4e00\u6b21\u7684&#xff08;\u6ca1\u6709 x\u00b2\u3001\u6ca1\u6709 xy\u3001\u6ca1\u6709 sin(x)&#xff09;\u3002<\/p>\n<h3>18.2 \u7528\u77e9\u9635\u8868\u793a\u65b9\u7a0b\u7ec4<\/h3>\n<p>\u4e0a\u9762\u7684\u65b9\u7a0b\u7ec4\u53ef\u4ee5\u5199\u6210\u77e9\u9635\u5f62\u5f0f&#xff1a;<\/p>\n<p>[ 2 1 ] [ x ]   [ 5  ]<br \/>\n[ 1 3 ] [ y ] &#061; [ 10 ]<\/p>\n<p>\u4e5f\u5c31\u662f&#xff1a;<\/p>\n<p>A     \u00b7 x   &#061;  b<\/p>\n<p>A &#061; [ 2 1 ]   x &#061; [ x ]   b &#061; [ 5  ]<br \/>\n    [ 1 3 ]       [ y ]       [ 10 ]<\/p>\n<ul>\n<li>A&#xff1a;\u7cfb\u6570\u77e9\u9635&#xff08;\u672a\u77e5\u6570\u524d\u9762\u7684\u7cfb\u6570&#xff09;<\/li>\n<li>x&#xff1a;\u672a\u77e5\u6570\u5411\u91cf<\/li>\n<li>b&#xff1a;\u5e38\u6570\u5411\u91cf&#xff08;\u7b49\u53f7\u53f3\u8fb9\u7684\u6570&#xff09;<\/li>\n<\/ul>\n<p>\u4e8e\u662f&#034;\u89e3\u65b9\u7a0b\u7ec4&#034;\u5c31\u53d8\u6210\u4e86&#034;\u5df2\u77e5 A \u548c b&#xff0c;\u6c42 x&#034;\u3002<\/p>\n<p>\u9a8c\u8bc1\u4e00\u4e0b&#xff1a;A\u00b7x \u5c55\u5f00\u5c31\u662f<\/p>\n<p>[ 2 1 ] [ x ]   [ 2x &#043; y  ]<br \/>\n[ 1 3 ] [ y ] &#061; [ x &#043; 3y  ]<\/p>\n<p>\u7b49\u4e8e b&#xff0c;\u6b63\u597d\u5c31\u662f\u539f\u6765\u7684\u4e24\u4e2a\u65b9\u7a0b&#xff01;\u6240\u4ee5 Ax &#061; b \u5c31\u662f\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u77e9\u9635\u5199\u6cd5\u3002<\/p>\n<h3>18.3 \u89e3\u65b9\u7a0b\u7ec4\u7684\u65b9\u6cd5<\/h3>\n<h4>\u65b9\u6cd5\u4e00&#xff1a;\u4ee3\u5165\u6d88\u5143&#xff08;\u4e2d\u5b66\u65b9\u6cd5&#xff09;<\/h4>\n<p>2x &#043; y &#061; 5      &#8230; \u2460<br \/>\nx &#043; 3y &#061; 10     &#8230; \u2461<\/p>\n<p>\u4ece \u2460 \u5f97 y &#061; 5 &#8211; 2x&#xff0c;\u4ee3\u5165 \u2461&#xff1a;<\/p>\n<p>x &#043; 3(5 &#8211; 2x) &#061; 10<br \/>\nx &#043; 15 &#8211; 6x &#061; 10<br \/>\n-5x &#061; -5<br \/>\nx &#061; 1<\/p>\n<p>\u4ee3\u56de y &#061; 5 &#8211; 2\u00d71 &#061; 3\u3002\u6240\u4ee5 x&#061;1&#xff0c;y&#061;3\u3002<\/p>\n<h4>\u65b9\u6cd5\u4e8c&#xff1a;\u9ad8\u65af\u6d88\u5143&#xff08;\u8ba1\u7b97\u673a\u7528&#xff09;<\/h4>\n<p>\u8ba1\u7b97\u673a\u7528\u7684\u662f\u9ad8\u65af\u6d88\u5143&#xff1a;\u901a\u8fc7&#034;\u884c\u53d8\u6362&#034;\u628a\u77e9\u9635\u53d8\u6210&#034;\u9636\u68af\u5f62&#034;&#xff0c;\u518d\u56de\u4ee3\u6c42\u89e3\u3002\u6838\u5fc3\u64cd\u4f5c&#xff1a;<\/p>\n<li>\u67d0\u4e00\u884c\u4e58\u4e00\u4e2a\u6570<\/li>\n<li>\u4e24\u884c\u4ea4\u6362<\/li>\n<li>\u67d0\u4e00\u884c\u52a0\u5230\u53e6\u4e00\u884c<\/li>\n<p>\u8fd9\u4e9b\u64cd\u4f5c\u4e0d\u6539\u53d8\u65b9\u7a0b\u7ec4\u7684\u89e3&#xff0c;\u4f46\u80fd\u628a\u77e9\u9635\u5316\u7b80\u6210\u5bb9\u6613\u6c42\u89e3\u7684\u5f62\u5f0f\u3002<\/p>\n<p>\u4f60\u4e0d\u9700\u8981\u624b\u7b97\u9ad8\u65af\u6d88\u5143&#xff0c;Eigen \u4f1a\u5e2e\u4f60\u505a\u3002\u4f46\u7406\u89e3&#034;\u89e3\u65b9\u7a0b\u7ec4 &#061; \u6d88\u5143&#034;\u8fd9\u4e2a\u601d\u60f3\u5f88\u91cd\u8981&#xff0c;\u56e0\u4e3a Eigen \u7684 solve() \u5185\u90e8\u5c31\u662f\u5728\u505a\u8fd9\u7c7b\u5206\u89e3\u3002<\/p>\n<h3>18.4 \u89e3\u7684\u5b58\u5728\u6027<\/h3>\n<p>\u65b9\u7a0b\u7ec4\u53ef\u80fd\u6709\u4e09\u79cd\u60c5\u51b5&#xff1a;<\/p>\n<table>\n<tr>\u60c5\u51b5\u4f8b\u5b50\u8bf4\u660e<\/tr>\n<tbody>\n<tr>\n<td>\u552f\u4e00\u89e3<\/td>\n<td>\u4e24\u6761\u4e0d\u5e73\u884c\u7684\u76f4\u7ebf<\/td>\n<td>\u4ea4\u4e8e\u4e00\u70b9<\/td>\n<\/tr>\n<tr>\n<td>\u65e0\u89e3<\/td>\n<td>\u4e24\u6761\u5e73\u884c\u76f4\u7ebf<\/td>\n<td>\u6c38\u8fdc\u4e0d\u76f8\u4ea4<\/td>\n<\/tr>\n<tr>\n<td>\u65e0\u7a77\u591a\u89e3<\/td>\n<td>\u4e24\u6761\u91cd\u5408\u76f4\u7ebf<\/td>\n<td>\u5904\u5904\u76f8\u4ea4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u51e0\u4f55\u4e0a\u770b&#xff0c;\u4e24\u4e2a\u672a\u77e5\u6570\u4e24\u4e2a\u65b9\u7a0b &#061; \u4e24\u6761\u76f4\u7ebf&#xff0c;\u89e3\u5c31\u662f\u5b83\u4eec\u7684\u4ea4\u70b9\u3002<\/p>\n<ul>\n<li>\u76f8\u4ea4 \u2192 \u552f\u4e00\u89e3<\/li>\n<li>\u5e73\u884c \u2192 \u65e0\u89e3<\/li>\n<li>\u91cd\u5408 \u2192 \u65e0\u7a77\u591a\u89e3<\/li>\n<\/ul>\n<p>\u5bf9\u5e94\u5230\u77e9\u9635&#xff1a;A \u662f\u5426\u53ef\u9006&#xff08;\u7b2c 19 \u7ae0&#xff09;\u51b3\u5b9a\u4e86\u662f\u5426\u6709\u552f\u4e00\u89e3\u3002<\/p>\n<h3>18.5 \u7528 Eigen \u89e3\u65b9\u7a0b\u7ec4<\/h3>\n<p><span class=\"token macro property\"><span class=\"token directive-hash\">#<\/span><span class=\"token directive keyword\">include<\/span> <span class=\"token string\">&lt;Eigen\/Dense&gt;<\/span><\/span><br \/>\n<span class=\"token macro property\"><span class=\"token directive-hash\">#<\/span><span class=\"token directive keyword\">include<\/span> <span class=\"token string\">&lt;iostream&gt;<\/span><\/span><\/p>\n<p><span class=\"token keyword\">int<\/span> <span class=\"token function\">main<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    Eigen<span class=\"token double-colon punctuation\">::<\/span>Matrix2d A<span class=\"token punctuation\">;<\/span><br \/>\n    A <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span><br \/>\n         <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">3<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    Eigen<span class=\"token double-colon punctuation\">::<\/span>Vector2d <span class=\"token function\">b<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">5<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">10<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u6c42\u89e3 Ax &#061; b<\/span><br \/>\n    Eigen<span class=\"token double-colon punctuation\">::<\/span>Vector2d x <span class=\"token operator\">&#061;<\/span> A<span class=\"token punctuation\">.<\/span><span class=\"token function\">colPivHouseholderQr<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">solve<\/span><span class=\"token punctuation\">(<\/span>b<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    std<span class=\"token double-colon punctuation\">::<\/span>cout <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;x &#061; &#034;<\/span> <span class=\"token operator\">&lt;&lt;<\/span> x<span class=\"token punctuation\">.<\/span><span class=\"token function\">transpose<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&lt;&lt;<\/span> std<span class=\"token double-colon punctuation\">::<\/span>endl<span class=\"token punctuation\">;<\/span>   <span class=\"token comment\">\/\/ 1 3<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u9a8c\u8bc1<\/span><br \/>\n    std<span class=\"token double-colon punctuation\">::<\/span>cout <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;A*x &#061; &#034;<\/span> <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token punctuation\">(<\/span>A <span class=\"token operator\">*<\/span> x<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">transpose<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&lt;&lt;<\/span> std<span class=\"token double-colon punctuation\">::<\/span>endl<span class=\"token punctuation\">;<\/span>  <span class=\"token comment\">\/\/ 5 10<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>A.\u5206\u89e3\u65b9\u6cd5().solve(b) \u5c31\u662f Eigen \u89e3\u65b9\u7a0b\u7ec4\u7684\u6807\u51c6\u5199\u6cd5&#xff08;\u7b2c 29 \u7ae0\u8be6\u89e3\u5404\u5206\u89e3\u65b9\u6cd5\u7684\u9009\u62e9&#xff09;\u3002<\/p>\n<h3>18.6 \u5b9e\u9645\u5e94\u7528<\/h3>\n<p>\u7ebf\u6027\u65b9\u7a0b\u7ec4\u662f\u79d1\u5b66\u8ba1\u7b97\u7684\u6838\u5fc3&#xff1a;<\/p>\n<ul>\n<li>\u7535\u8def\u5206\u6790&#xff1a;\u57fa\u5c14\u970d\u592b\u5b9a\u5f8b\u5217\u51fa\u7684\u65b9\u7a0b<\/li>\n<li>\u7ed3\u6784\u529b\u5b66&#xff1a;\u53d7\u529b\u5e73\u8861\u65b9\u7a0b<\/li>\n<li>\u7ecf\u6d4e\u6a21\u578b&#xff1a;\u6295\u5165\u4ea7\u51fa\u65b9\u7a0b<\/li>\n<li>\u6700\u5c0f\u4e8c\u4e58\u62df\u5408&#xff1a;\u62df\u5408\u6570\u636e&#xff08;\u7b2c 36 \u7ae0\u9879\u76ee\u4f1a\u7528\u5230&#xff09;<\/li>\n<\/ul>\n<p>\u5f88\u591a\u5b9e\u9645\u95ee\u9898\u6700\u7ec8\u90fd\u5f52\u7ed3\u4e3a&#034;\u89e3\u4e00\u4e2a Ax &#061; b&#034;\u3002<\/p>\n<h3>18.7 \u5c0f\u7ed3<\/h3>\n<ul>\n<li>\u7ebf\u6027\u65b9\u7a0b\u7ec4 &#061; Ax &#061; b&#xff08;A \u7cfb\u6570\u3001x \u672a\u77e5\u3001b \u5e38\u6570&#xff09;\u3002<\/li>\n<li>\u51e0\u4f55\u610f\u4e49&#xff1a;\u4e24\u6761&#xff08;\u6216\u591a\u6761&#xff09;\u76f4\u7ebf\u7684\u4ea4\u70b9\u3002<\/li>\n<li>\u89e3\u6709\u552f\u4e00\u89e3\u3001\u65e0\u89e3\u3001\u65e0\u7a77\u591a\u89e3\u4e09\u79cd\u60c5\u51b5\u3002<\/li>\n<li>Eigen \u7528 A.\u5206\u89e3().solve(b) \u6c42\u89e3\u3002<\/li>\n<\/ul>\n<p>\u4e0b\u4e00\u7ae0&#xff0c;\u5b66\u4e60\u884c\u5217\u5f0f\u4e0e\u9006\u77e9\u9635&#xff0c;\u7406\u89e3&#034;\u53ef\u9006&#034;\u548c&#034;\u552f\u4e00\u89e3&#034;\u7684\u5173\u7cfb\u3002<\/p>\n<hr \/>\n<h4>\u7ec3\u4e60\u9898<\/h4>\n<li>\u628a\u65b9\u7a0b\u7ec4 3x &#043; y &#061; 9; x &#043; 2y &#061; 8 \u5199\u6210 Ax &#061; b \u7684\u77e9\u9635\u5f62\u5f0f\u3002<\/li>\n<li>\u7528\u4ee3\u5165\u6d88\u5143\u624b\u7b97\u4e0a\u9762\u65b9\u7a0b\u7ec4\u7684\u89e3\u3002<\/li>\n<li>\u7528 Eigen \u6c42\u89e3\u4e0a\u9762\u65b9\u7a0b\u7ec4&#xff0c;\u5e76\u9a8c\u8bc1\u7ed3\u679c\u3002<\/li>\n<li>\u51e0\u4f55\u89e3\u91ca&#xff1a;\u4e3a\u4ec0\u4e48\u4e24\u4e2a\u672a\u77e5\u6570\u4e24\u4e2a\u65b9\u7a0b\u5bf9\u5e94\u4e24\u6761\u76f4\u7ebf&#xff1f;<\/li>\n<li>\u8bf4\u8bf4\u7ebf\u6027\u65b9\u7a0b\u7ec4\u65e0\u89e3\u3001\u552f\u4e00\u89e3\u3001\u65e0\u7a77\u591a\u89e3\u5206\u522b\u5bf9\u5e94\u76f4\u7ebf\u7684\u4ec0\u4e48\u4f4d\u7f6e\u5173\u7cfb\u3002<\/li>\n","protected":false},"excerpt":{"rendered":"<p>\u77e9\u9635\u6700\u91cd\u8981\u7684\u5e94\u7528\u4e4b\u4e00&#xff0c;\u5c31\u662f\u8868\u793a\u548c\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002\u672c\u7ae0\u8bb2\u6e05\u695a\\&#8221;Ax  b\\&#8221;\u5230\u5e95\u662f\u4ec0\u4e48\u610f\u601d&#xff0c;\u4ee5\u53ca\u5982\u4f55\u6c42\u89e3\u300218.1 \u4ec0\u4e48\u662f\u7ebf\u6027\u65b9\u7a0b\u7ec4<br \/>\n\u4f60\u80af\u5b9a\u89c1\u8fc7\u8fd9\u79cd\u65b9\u7a0b&#xff1a;<br \/>\n2x  y  5<br \/>\nx  3y  10\u4e24\u4e2a\u672a\u77e5\u6570 x\u3001y&#xff0c;\u4e24\u4e2a\u65b9\u7a0b\u3002\u6c42 x\u3001y \u7684\u503c&#xff0c;\u5c31\u662f\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002<br \/>\n\\&#8221;\u7ebf\u6027\\&#8221;\u6307&#xff1a;\u6bcf\u4e2a\u65b9\u7a0b\u91cc\u672a\u77e5\u6570\u90fd\u662f\u4e00\u6b21\u7684&#xff08;\u6ca1\u6709 x\u00b2\u3001\u6ca1\u6709 xy\u3001\u6ca1\u6709 sin(x)&#xff09;\u3002<br \/>\n18.2 \u7528\u77e9\u9635\u8868\u793a\u65b9\u7a0b<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[55],"topic":[],"class_list":["post-113436","post","type-post","status-publish","format-standard","hentry","category-server","tag-c"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u7b2c 18 \u7ae0 \u00b7 \u7ebf\u6027\u65b9\u7a0b\u7ec4 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.wsisp.com\/helps\/113436.html\" \/>\n<meta property=\"og:locale\" content=\"zh_CN\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u7b2c 18 \u7ae0 \u00b7 \u7ebf\u6027\u65b9\u7a0b\u7ec4 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\" \/>\n<meta property=\"og:description\" content=\"\u77e9\u9635\u6700\u91cd\u8981\u7684\u5e94\u7528\u4e4b\u4e00&#xff0c;\u5c31\u662f\u8868\u793a\u548c\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002\u672c\u7ae0\u8bb2\u6e05\u695a&quot;Ax b&quot;\u5230\u5e95\u662f\u4ec0\u4e48\u610f\u601d&#xff0c;\u4ee5\u53ca\u5982\u4f55\u6c42\u89e3\u300218.1 \u4ec0\u4e48\u662f\u7ebf\u6027\u65b9\u7a0b\u7ec4 \u4f60\u80af\u5b9a\u89c1\u8fc7\u8fd9\u79cd\u65b9\u7a0b&#xff1a; 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