{"id":16217,"date":"2010-11-30T05:24:54","date_gmt":"2010-11-29T21:24:54","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=16217"},"modified":"2021-10-06T16:30:39","modified_gmt":"2021-10-06T08:30:39","slug":"%e6%8f%92%e5%80%bc%e5%a4%9a%e9%a0%85%e5%bc%8f%ef%bc%88interpolating-polynomial%ef%bc%89","status":"publish","type":"post","link":"http:\/\/localhost\/%e6%8f%92%e5%80%bc%e5%a4%9a%e9%a0%85%e5%bc%8f%ef%bc%88interpolating-polynomial%ef%bc%89\/","title":{"rendered":"\u63d2\u503c\u591a\u9805\u5f0f"},"content":{"rendered":"

\u63d2\u503c\u591a\u9805\u5f0f (Interpolating polynomial)<\/span><\/strong>
\n\u570b\u7acb\u81fa\u5357\u7b2c\u4e00\u9ad8\u7d1a\u4e2d\u5b78\u6578\u5b78\u79d1\u6797\u5009\u5104\u8001\u5e2b\/\u570b\u7acb\u81fa\u7063\u5e2b\u7bc4\u5927\u5b78\u6578\u5b78\u7cfb\u9000\u4f11\u6559\u6388\u6d2a\u842c\u751f\u8cac\u4efb\u7de8\u8f2f<\/strong><\/span><\/p>\n

\u4f8b\u984c1\uff1a\u5728\u5750\u6a19\u5e73\u9762\u4e0a\u7d66\u5b9a\u4e09\u500b\u9ede \\(A(1,-2)\\)\uff0c\\(B(2,3)\\) \u8207 \\(C(3,12)\\)\uff0c\u5982\u4f55\u627e\u5230\u4e00\u500b\u4e8c\u6b21\u591a\u9805\u5f0f\u4f7f\u5f97\u5176\u5716\u5f62\u901a\u904e\u9019\u4e09\u500b\u9ede\uff1f<\/span>\u00a0<\/span><\/p>\n

\u89e3\u6b64\u984c\u6700\u7c21\u55ae\u7684\u60f3\u6cd5\uff0c\u4e0d\u5916\u662f\u5047\u8a2d\u6240\u6c42\u591a\u9805\u5f0f\\(f(x)=ax^2+bx+c\\)\uff0c<\/span><\/p>\n

\u7136\u5f8c\u5c07 \\(A\\)\uff0c\\(B\\)\uff0c\\(C\\) \u4e09\u9ede\u4ee3\u5165\uff0c\u5f97\u4e09\u5143\u4e00\u6b21\u806f\u7acb\u65b9\u7a0b\u5f0f \\(\\begin{cases}-2=a+b+c\\\\3=4a+2b+c\\\\12=9a+3b+c\\end{cases}\\)\u00a0\uff0c<\/span><\/p>\n

\u5229\u7528\u52a0\u6e1b\u6d88\u53bb\u6cd5\u5373\u53ef\u6c42\u5f97\u00a0\\(\\begin{cases}a=2\\\\b=-1\\\\c=-3\\end{cases}\\)\uff0c\u5373\u6240\u6c42 \\(f(x)=2x^2-x-3\\)\u3002<\/span><\/p>\n

\u9019\u65b9\u6cd5\u4e0d\u932f\uff0c\u4f46\u6709\u500b\u7f3a\u9ede\uff0c\u5c31\u662f\u8981\u5047\u8a2d\u4e09\u500b\u672a\u77e5\u6578\uff0c\u7136\u5f8c\u8f9b\u82e6\u5730\u89e3\u806f\u7acb\u65b9\u7a0b\u5f0f\u3002\u4eca\u82e5\u5c07\u984c\u76ee\u6539\u6210\uff1a<\/p>\n

\u4f8b\u984c2\uff1a\u5728\u5750\u6a19\u5e73\u9762\u4e0a\u7d66\u5b9a\u56db\u500b\u9ede \\(A(1,10)\\)\uff0c\\(B(2,26)\\)\uff0c\\(C(3,58)\\) \u8207 \\(D(4,112)\\)\uff0c\u5982\u4f55\u627e\u5230\u4e00\u500b\u4e09\u6b21\u591a\u9805\u5f0f\u4f7f\u5f97\u5176\u5716\u5f62\u901a\u904e\u9019\u56db\u500b\u9ede\uff1f<\/p>\n

\u9019\u6642\u518d\u8981\u7528\u4e0a\u8ff0\u7684\u65b9\u6cd5\uff0c\u5149\u7528\u60f3\u7684\u5c31\u6709\u9ede\u7d2f\u4e86\uff0c\u66f4\u4f55\u6cc1\u5982\u679c\u63a8\u5ee3\u5230\u4e00\u822c\u60c5\u6cc1\uff1a\u7d66\u5b9a \\(n+1\\) \u500b\u9ede\uff0c\u627e\u4e00\u500b \\(n\\) \u6b21\u591a\u9805\u5f0f\u4f7f\u5f97\u5176\u5716\u5f62\u901a\u904e\u9019 \\(n+1\\) \u500b\u9ede\u3002\u6bd4\u5982\u8aaa \\(n=100\\) \u6642\uff0c\u82e5\u771f\u8981\u7528\u4e0a\u8ff0\u7684\u65b9\u6cd5\uff0c\u4e0d\u4f46\u9700\u8981\u5f88\u5927\u7684\u52c7\u6c23\u3001\u6bc5\u529b\uff0c\u66f4\u9700\u8981\u4e00\u5f35\u5f88\u5927\u5f88\u5927\u7684\u8a08\u7b97\u7d19\uff0c\u624d\u80fd\u5bb9\u7d0d\u6709 \\(101\\) \u500b\u65b9\u7a0b\u5f0f\u7684\u806f\u7acb\u65b9\u7a0b\u7d44\uff0c\u800c\u9019\u624d\u53ea\u662f\u7b2c\u4e00\u6b65\uff0c\u66f4\u7d2f\u7684\u9084\u5728\u5f8c\u982d\u2026\u2026\u3002<\/p>\n

\u70ba\u4e86\u61c9\u4ed8\u4e0a\u8ff0\u9019\u7a2e\u554f\u984c\uff0c\u5169\u500b\u5049\u5927\u7684\u6578\u5b78\u5bb6\u2500\u725b\u9813\uff08Issac Newton, 1643~1727\uff09\u8207\u62c9\u683c\u6717\u65e5\uff08Joseph Louis Lagrange\uff09\u5206\u5225\u60f3\u51fa\u4e86\u4e0d\u540c\u7684\u65b9\u6cd5\u3002\u4ee5\u4f8b\u984c1\u70ba\u4f8b\uff0c\u725b\u9813\u7684\u65b9\u6cd5\u662f\u5047\u8a2d \\(f(x)=a+b(x-1)+c(x-1)(x-2)\\)\uff0c\u518d\u5c07 \\(A\\)\uff0c\\(B\\)\uff0c\\(C\\) \u4e09\u9ede\u4ee3\u5165\uff0c\u5f97<\/p>\n

\\(\\begin{cases}-2=a\\\\3=a+b\\\\12=a+2b+2c\\end{cases}\\Rightarrow\\begin{cases}a=-2\\\\b=5\\\\c=2\\end{cases}\\)<\/p>\n

\\(\\Rightarrow f(x)=-2+5(x-1)+2(x-1)(x-2)=2x^2-x-3\\)<\/p>\n

\u725b\u9813\u65b9\u6cd5\u7684\u597d\u8655\u5728\u65bc\u5de7\u5999\u7684\u5047\u8a2d\u591a\u9805\u5f0f\uff0c\u5927\u5e45\u6e1b\u5316\u4e86\u8a08\u7b97\u3002<\/p>\n

\u725b\u9813\u7684\u5de7\u5999\u5df2\u5920\u4ee4\u4eba\u9a5a\u8c54\u4e86\uff0c\u4f46\u62c9\u683c\u6717\u65e5\u66f4\u9752\u51fa\u65bc\u85cd\uff0c\u9023\u5047\u8a2d\u90fd\u4e0d\u7528\uff0c\u76f4\u63a5\u5beb\u51fa\u6240\u6c42\u7684\u591a\u9805\u5f0f<\/p>\n

\\(f(x)=\\displaystyle -2\\cdot\\frac{(x-2)(x-3)}{(1-2)(1-3)}+3\\cdot\\frac{(x-1)(x-3)}{(2-1)(2-3)}+12\\cdot\\frac{(x-1)(x-2)}{(3-1)(3-2)}\\)\u3002<\/p>\n

\u4e0d\u76f8\u4fe1\u7684\u8b80\u8005\u53ef\u4ee5\u62ff\u51fa\u7d19\u7b46\u7b97\u4e00\u7b97\uff0c\u770b\u770b\u7528\u62c9\u683c\u6717\u65e5\u65b9\u6cd5\u6240\u5f97\u5230\u7684\u591a\u9805\u5f0f\u5316\uff0c\u5728\u5316\u7c21\u5f8c\u662f\u5426\u771f\u7684\u662f\\(2x^2-x-3\\)\u3002\u4f9d\u62c9\u683c\u6717\u65e5\u4e4b\u6cd5\uff0c\u4f8b\u984c2\u6240\u6c42\u7684\u591a\u9805\u5f0f<\/p>\n

\\(\\begin{multline*}f(x)=10\\cdot\\frac{(x-2)(x-3)(x-4)}{(1-2)(1-3)(1-4)}+26\\cdot\\frac{(x-1)(x-3)(x-4)}{(2-1)(2-3)(2-4)}\\\\+58\\cdot\\frac{(x-1)(x-2)(x-4)}{(3-1)(3-2)(3-4)}+112\\cdot\\frac{(x-1)(x-2)(x-3)}{(4-1)(4-2)(4-3)}\\end{multline*}\\)<\/p>\n

\u73fe\u7528\u6578\u5b78\u7b26\u865f\u5c07\u62c9\u683c\u6717\u65e5\u7684\u65b9\u6cd5\u5beb\u5728\u4e0b\u9762\uff1a<\/p>\n

(1)\u5728\u5750\u6a19\u5e73\u9762\u4e0a\u901a\u904e \\(A(x_1,y_1)\\)\uff0c\\(B(x_2,y_2)\\) \u8207 \\(C(x_3,y_3)\\) \u4e09\u9ede\u7684\u4e8c\u6b21\u591a\u9805\u5f0f\u70ba<\/p>\n

\\(f(x)=\\displaystyle y_1\\cdot\\frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)}+y_2\\cdot\\frac{(x_1-x)(x_3-x)}{(x_2-x_1)(x_2-x_3)}+y_3\\cdot\\frac{(x_1-x)(x_2-x)}{(x-x_3)(x_1-x_3)}\\)<\/p>\n

(2)\u5728\u5750\u6a19\u5e73\u9762\u4e0a\u901a\u904e \\(A(x_1,y_1)\\)\uff0c\\(B(x_2,y_2)\\)\uff0c\\(C(x_3,y_3)\\) \u8207 \\(D(x_4,y_4)\\) \u56db\u9ede\u7684\u4e09\u6b21\u591a\u9805\u5f0f\u70ba<\/p>\n

\\(\\begin{multline*}f(x)=y_1\\cdot\\frac{(x-x_2)(x-x_3)(x-x_4)}{(x_1-x_2)(x_1-x_3)(x_1-x_4)}+y_2\\cdot\\frac{(x-x_1)(x-x_3)(x-x_4)}{(x_2-x_1)(x_2-x_3)(x_2-x_4)}\\\\+y_3\\cdot\\frac{(x-x_1)(x-x_2)(x-x_4)}{(x_3-x_1)(x_3-x_2)(x_3-x_4)}+y_4\\cdot\\frac{(x-x_1)(x-x_2)(x-x_3)}{(x_4-x_1)(x_4-x_2)(x_4-x_3)}\\end{multline*}\\)<\/p>\n

(3)\u5716\u5f62\u901a\u904e\u9019 \\(n+1\\) \u500b\u9ede\u7684 \\(n\\) \u6b21\u591a\u9805\u5f0f\u53ef\u4eff\u4e0a\u8ff0\u4e4b\u5f62\u5f0f\u5beb\u51fa\u3002<\/p>\n

\u4e0a\u8ff0\u9019\u500b\u65b9\u6cd5\u5c31\u7a31\u70ba\u300c\u62c9\u683c\u6717\u65e5\u63d2\u503c\u591a\u9805\u5f0f\u300d\u6216\u300c\u62c9\u683c\u6717\u65e5\u63d2\u503c\u6cd5\u300d\uff0c\u6b64\u65b9\u6cd5\u4e0d\u4f46\u4fdd\u6709\u984c\u76ee\u6240\u7d66\u7684\u9ede\u5750\u6a19\uff0c\u800c\u4e14\u9084\u6709\u898f\u5f8b\u53ef\u5faa\uff0c\u5728\u96fb\u8166\u767c\u9054\u7684\u4eca\u65e5\u4f86\u8aaa\uff0c\u66f4\u662f\u5341\u5206\u9069\u65bc\u5beb\u6210\u96fb\u8166\u7a0b\u5f0f\uff0c\u9019\u9ebc\u4e00\u4f86\uff0c\u6709\u518d\u591a\u500b\u9ede\u4e5f\u4e0d\u7528\u6015\u4e86\uff01<\/p>\n

\u62c9\u683c\u6717\u65e5\u7684\u65b9\u6cd5\u96d6\u7136\u795e\u5999\uff0c\u4f46\u76f8\u4fe1\u6709\u8b80\u8005\u4e00\u5b9a\u5fc3\u6709\u4e0d\u670d\uff0c\u8a8d\u70ba\u62c9\u683c\u6717\u65e5\u7684\u65b9\u6cd5\u4e26\u672a\u5c07\u7b54\u6848\u5beb\u6210\u4e00\u822c\u5f0f\uff0c\u5373 \\(f(x)=ax^2+bx+c\\) \u6216 \\(f(x)=ax^3+bx^2+cx+d\\) \u7b49\u7b49\uff0c\u8aaa\u767d\u8a71\u4e00\u9ede\uff0c\u6709\u7a2e\u5c1a\u672a\u300c\u7b97\u5b8c\u300d\u7684\u611f\u89ba\u3002\u62c9\u683c\u6717\u65e5\u63d2\u503c\u591a\u9805\u5f0f\u78ba\u6703\u8b93\u4eba\u6709\u9019\u7a2e\u300c\u534a\u6210\u54c1\u300d\u7684\u611f\u89ba\uff0c\u4f46\u8a66\u60f3\uff0c\u82e5\u4eca\u65e5\u6211\u5011\u91cd\u9ede\u4e26\u4e0d\u5728\u65bc\u6c42\u51fa\u591a\u9805\u5f0f\uff0c\u800c\u5728\u65bc\u9810\u6e2c\u4e0b\u4e00\u500b\u6578\u503c\uff0c\u90a3\u9ebc\uff0c\u62c9\u683c\u6717\u65e5\u63d2\u503c\u591a\u9805\u5f0f\u4e0d\u5c31\u65b9\u4fbf\u591a\u4e86\u55ce\uff1f\u4f8b\u5982\u5c07\u4f8b\u984c2\u6539\u6210\uff1a<\/p>\n

\u4f8b\u984c3<\/strong>\uff1a\u5df2\u77e5\u4e09\u6b21\u591a\u9805\u5f0f \\(f(x)\\) \u7684\u5716\u5f62\u901a\u904e \\(A(1,10)\\)\uff0c\\(B(2,26)\\)\uff0c\\(C(3,58)\\) \u8207 \\(D(4,112)\\) \u56db\u9ede\uff0c\u6c42 \\(f(5)\\) \u4e4b\u503c\u3002<\/p>\n

\u9019\u6642\uff0c\u50c5\u9808\u5c07 \\(x=5\\) \u4ee3\u5165\u4f8b\u984c2<\/strong>\u6240\u5f97\u7684\u62c9\u683c\u6717\u65e5\u63d2\u503c\u591a\u9805\u5f0f\uff0c\u5f88\u8f15\u6613\u5730\u5c31\u53ef\u4ee5\u6c42\u51fa \\(f(5)=194\\)\u3002\u63db\u500b\u89d2\u5ea6\u4f86\u770b\uff0c\u62c9\u683c\u6717\u65e5\u63d2\u503c\u591a\u9805\u5f0f\u7684\u7cbe\u795e\uff0c\u5373\u5728\u65bc\u5148\u5229\u7528\u7d66\u5b9a\u7684\u8cc7\u6599\u5efa\u7acb\u6a21\u5f0f\uff0c\u518d\u7531\u6a21\u5f0f\u4f86\u9810\u6e2c\u5176\u4ed6\u7684\u503c\u3002\u85c9\u52a9\u4eca\u65e5\u96fb\u8166\u79d1\u6280\u7684\u5f37\u5927\u904b\u7b97\u80fd\u529b\uff0c\u62c9\u683c\u6717\u65e5\u63d2\u503c\u591a\u9805\u5f0f\u5728\u8af8\u591a\u9818\u57df\u5c07\u6709\u66f4\u5927\u63ee\u7051\u7a7a\u9593\u3002\u00a0<\/span><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

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