查看“︁格里旺克函数”︁的源代码

'''格里旺克函数'''（Griewank function）是[[數學]]上常用于测试[[优化]]程序效率的[[函数]]，定义如下：<ref>Griewank, A. O. "Generalized Decent for Global Optimization." J. Opt. Th. Appl. 34, 11-39, 1981</ref><ref>{{Cite journal |last=Bosse |first=Torsten F. |last2=Bücker |first2=H. Martin |date=2024-10-29 |title=A piecewise smooth version of the Griewank function |url=https://www.tandfonline.com/doi/full/10.1080/10556788.2024.2414186 |journal=Optimization Methods and Software |language=en |pages=1–11 |doi=10.1080/10556788.2024.2414186 |issn=1055-6788|doi-access=free }}</ref>
<math>G(x_1,x_2,\cdots,x_n)=1+\frac{1}{4000}\sum_{1}^{n}x_{i}^2-\prod_{i=1}^{n}cos(\frac{x_i)}{\sqrt(i)}</math>

==一阶格里旺函数==
[[File:1st order Griewank function.png|thumb|400px| 一阶格里旺函数]]
<math>g := 1+(1/4000)*x[1]^2-cos(x[1])</math>

如图所示，一阶格里旺函数有许多极点。<ref>Locatelli, M. "A Note on the Griewank Test Function." J. Global Opt. 25, 169-174, 2003</ref>取上述函数的一阶导数，令其为0：

<math>\frac{1}{2000}*x[1]+sin(x[1]) = 0</math>

用数值解法，求其中在实数域[-100..100]之间的解，共得62个，列出如下：

[-97.438110610025603200, -94.200661844477748520, -91.151778636270389965, -87.920619819329359985, -84.865447114417916660, -81.640577359698817225, -78.579116013127494725, -75.360534496781834905, -72.292785301096032350, -69.080491261738200370, -66.006454947055212230, -62.800447685694571355, -59.720124919768677970, -56.520403799747265470, -53.433795188029228405, -50.240359634965042195, -47.147465720656019171, -43.960315222391878044, -40.861136486491770843, -37.680270593049735600, -34.574807454399982858, -31.400225777941327138, -28.288478593262152626, -25.120180808052873462, -22.002149871974999083, -18.840135714356858698, -15.715821259447690012, -12.560090527814781691, -9.4294927245990724370, -6.2800452793799046870, -3.1431642363549054240, 3.1431642363549054240, 6.2800452793799046870, 9.4294927245990724370, 12.560090527814781691, 15.715821259447690012, 18.840135714356858698, 22.002149871974999083, 25.120180808052873462, 28.288478593262152626, 31.400225777941327138, 34.574807454399982858, 37.680270593049735600, 40.861136486491770847, 43.960315222391878044, 47.147465720656019171, 50.240359634965042195, 53.433795188029228405, 56.520403799747265470, 59.720124919768677970, 62.800447685694571355, 66.006454947055212230, 69.080491261738200370, 72.292785301096032350, 75.360534496781834905, 78.579116013127494725, 81.640577359698817225, 84.865447114417916660, 87.920619819329359985, 91.151778636270389965, 94.200661844477748520, 97.438110610025603200, 0.]

在[-10000,10000]区间，极点个数=6365

==二阶格里旺函数==
[[File:Griewank function 3D plot.png|thumb|left|330px|2nd order Griewank function 3D plot]]
[[File:2nd order Griewank function contour plot.png|thumb|300px|2nd order Griewank function contour plot]]
<math>g2 := 1+(1/4000)*x[1]^2+(1/4000)*x[2]^2-cos(x[1])*cos((1/2)*x[2]*\sqrt(2))</math>

==三阶格里旺函数==
[[File:Third order Griewank function Maple animation.gif|thumb|300px|Third order Griewank function Maple animation]]
<math>{1+(1/4000)*x[1]^2+(1/4000)*x[2]^2+(1/4000)*x[3]^2-cos(x[1])*cos((1/2)*x[2]*\sqrt(2))*cos((1/3)*x[3]*sqrt(3))}  </math>

==相關條目==
* {{link-en|Himmelblau函數|Himmelblau's function}}
* {{link-en|Rastrigin函數|Rastrigin function}}
* [[Rosenbrock函數]]

==参考文献==
<references/>

[[Category:特殊函数]]
[[Category:數學最佳化]]