s=1+{2.7771904733328640131}/{\sqrt{x}}-{3.1087580197427311379}/{x+2 \ln (x)}
相关系数能达到几个9？拟合公式貌似与mathe的预测有出入。这个可能只管较小的n吧？
我的预测也只是大概，估计出来的$log(x)$项不是很合理，看来主项还是应该$c/{sqrt(n)}$,
52# hujunhua
相关系数是0.99997。
这是两个待定参数的公式，如果用三个参数来拟合，相关系数可以达到六个9
wayne 发表于 2010-7-13 14:30
52# hujunhua
相关系数是0.99997。
这么好的相关系数,倒使我很想知道这个精确的公式是啥了
考虑`\D f(x)=\ln(x+\frac{1}{x})`，或许下面的定理有用。
根据http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=89035中8#，Vasc说他在某本代数不等式的书上看到有下列定理：
LCRCF-定理（左凹右凸函数定理） `a,b,c`是满足`a < c < b`的实数，`f`是区间`I=`上是凹的，在`[c,b)`上是凸的。
若`x_1,x_2,x_3,\ldots,x_n \in I`满足`x_1+x_2+\cdots+x_n=S`(常数)，其中`S<(n-1)c+b`，那么
`f(x_1)+f(x_2)+\cdots+f(x_n)`在`x_1=x_2=\cdots=x_{n-1} \leqslant x_n`时取最大值。
SIP-定理（单拐点定理） `f`是`R`上只有一个拐点的二次可微函数，`S`是某个固定的实数，令`\D g(x)=f(x)+(n-1)f(\frac{S-x}{n-1})`.
若`x_1,x_2,...,x_n`是满足`x_1+x_2+\cdots+x_n=S`的实数，那么
$$\inf_{x \in R}g(x) \leqslant f(x_1)+f(x_2)+\cdots+f(x_n)\leqslant \sup_{x \in R}g(x)$$.
lcrcf显然是错误的，比如我们选择函数f在x<c时为x(c-x)在x>c时为(x-c)(x-2c).于是我们选择任意大的b对应的函数都符合题目要求，但是这样的函数显然不符合条件，我们可以选S使得S/n落在函数最小值点
@hujunhua
"漂亮的弧度，力的感觉"
一点也不漂亮!
(*非线性方程的拟合*)
Clear["Global`*"];(*Clear all variables*)
data={{3,2.00286740806350017941198515943545931361},
{4,1.9283589164749804283600849366519560639},
{5,1.86400765000052485077735500471126683613},
{6,1.81051438122816316692944707545183728148},
{7,1.76578235001873982989444491931146614181},
{8,1.72786431204913392367633970225491638819},
{9,1.69526942444040933881970369205445408976},
{10,1.66689198313550214392865903283543517526},
{11,1.64190904616094851366101664075402220374},
{12,1.6196998599244209974512574541584287912},
{13,1.59978869842201869056004669676183358389},
{14,1.58180517574496658723209695065387815411},
{15,1.56545660481407284533740801524947779306},
{16,1.55050851817644197976392356354789658309},
{17,1.5367707372478112927676021766159055014},
{18,1.52408725542126641935994748503824288176},
{19,1.51232877977090028689582251305994783841},
{20,1.50138715374182425550739674900868882105},
{21,1.49117113037198696816232030789158595774},
{22,1.481603128981051478805513257481434365},
{23,1.47261671766567601608562219697005149641},
{24,1.46415463821023838316030988522039252819},
{25,1.45616724114422507854121331602445094798},
{26,1.4486112343465122080296824494356282472},
{27,1.44144867381000076000046233040898435983},
{28,1.43464614322487864607990982178408997221},
{29,1.42817408210585004403922210801221054461},
{30,1.4220062317567129945412344765141001355},
{31,1.41611917544533369254102429082160513084},
{32,1.41049195445244880077344760779527345639},
{33,1.4051057456479795593700060551553949446},
{34,1.39994358928469090337863094895696596845},
{35,1.39499015802841523529377394252379049117},
{36,1.39023156004523883530353774388849366078},
{37,1.38565517036913390218387987164360902442},
{38,1.38124948587417599494888974270272683602},
{39,1.37700400004464983512682767443775686515},
{40,1.37290909442703835047823373266554024708},
{41,1.36895594420004818247311699643331377899},
{42,1.36513643574275156839766072988517148305},
{43,1.36144309443977055373412981089021928638},
{44,1.35786902125399984336720056851594591129},
{45,1.35440783683543930473427573602498607559},
{46,1.3510536321300155887091520853891945445},
{47,1.34780092461321684380654451302931688756},
{48,1.34464461940656131097555379617152740538},
{49,1.34157997464560704966451256923364150444},
{50,1.33860257056055295892954001842319703868}};
out=NonlinearModelFit
Normal
Show,{x,0,52},PlotRange->{{0,54},{0,2.5}},AspectRatio->Automatic],ListPlot,ImageSize->1200]
(*直线方程的拟合*)
Clear["Global`*"];(*Clear all variables*)
data={{3,2.00286740806350017941198515943545931361},
{4,1.9283589164749804283600849366519560639},
{5,1.86400765000052485077735500471126683613},
{6,1.81051438122816316692944707545183728148},
{7,1.76578235001873982989444491931146614181},
{8,1.72786431204913392367633970225491638819},
{9,1.69526942444040933881970369205445408976},
{10,1.66689198313550214392865903283543517526},
{11,1.64190904616094851366101664075402220374},
{12,1.6196998599244209974512574541584287912},
{13,1.59978869842201869056004669676183358389},
{14,1.58180517574496658723209695065387815411},
{15,1.56545660481407284533740801524947779306},
{16,1.55050851817644197976392356354789658309},
{17,1.5367707372478112927676021766159055014},
{18,1.52408725542126641935994748503824288176},
{19,1.51232877977090028689582251305994783841},
{20,1.50138715374182425550739674900868882105},
{21,1.49117113037198696816232030789158595774},
{22,1.481603128981051478805513257481434365},
{23,1.47261671766567601608562219697005149641},
{24,1.46415463821023838316030988522039252819},
{25,1.45616724114422507854121331602445094798},
{26,1.4486112343465122080296824494356282472},
{27,1.44144867381000076000046233040898435983},
{28,1.43464614322487864607990982178408997221},
{29,1.42817408210585004403922210801221054461},
{30,1.4220062317567129945412344765141001355},
{31,1.41611917544533369254102429082160513084},
{32,1.41049195445244880077344760779527345639},
{33,1.4051057456479795593700060551553949446},
{34,1.39994358928469090337863094895696596845},
{35,1.39499015802841523529377394252379049117},
{36,1.39023156004523883530353774388849366078},
{37,1.38565517036913390218387987164360902442},
{38,1.38124948587417599494888974270272683602},
{39,1.37700400004464983512682767443775686515},
{40,1.37290909442703835047823373266554024708},
{41,1.36895594420004818247311699643331377899},
{42,1.36513643574275156839766072988517148305},
{43,1.36144309443977055373412981089021928638},
{44,1.35786902125399984336720056851594591129},
{45,1.35440783683543930473427573602498607559},
{46,1.3510536321300155887091520853891945445},
{47,1.34780092461321684380654451302931688756},
{48,1.34464461940656131097555379617152740538},
{49,1.34157997464560704966451256923364150444},
{50,1.33860257056055295892954001842319703868}};
out=NonlinearModelFit;
Normal
Show,{x,0,52},PlotRange->{{0,54},{0,2.5}},AspectRatio->Automatic],ListPlot,ImageSize->1200]
(*第一列乘以第二列,然后取代第二列的值*)
Clear["Global`*"];(*Clear all variables*)
data={{3,2.00286740806350017941198515943545931361},
{4,1.9283589164749804283600849366519560639},
{5,1.86400765000052485077735500471126683613},
{6,1.81051438122816316692944707545183728148},
{7,1.76578235001873982989444491931146614181},
{8,1.72786431204913392367633970225491638819},
{9,1.69526942444040933881970369205445408976},
{10,1.66689198313550214392865903283543517526},
{11,1.64190904616094851366101664075402220374},
{12,1.6196998599244209974512574541584287912},
{13,1.59978869842201869056004669676183358389},
{14,1.58180517574496658723209695065387815411},
{15,1.56545660481407284533740801524947779306},
{16,1.55050851817644197976392356354789658309},
{17,1.5367707372478112927676021766159055014},
{18,1.52408725542126641935994748503824288176},
{19,1.51232877977090028689582251305994783841},
{20,1.50138715374182425550739674900868882105},
{21,1.49117113037198696816232030789158595774},
{22,1.481603128981051478805513257481434365},
{23,1.47261671766567601608562219697005149641},
{24,1.46415463821023838316030988522039252819},
{25,1.45616724114422507854121331602445094798},
{26,1.4486112343465122080296824494356282472},
{27,1.44144867381000076000046233040898435983},
{28,1.43464614322487864607990982178408997221},
{29,1.42817408210585004403922210801221054461},
{30,1.4220062317567129945412344765141001355},
{31,1.41611917544533369254102429082160513084},
{32,1.41049195445244880077344760779527345639},
{33,1.4051057456479795593700060551553949446},
{34,1.39994358928469090337863094895696596845},
{35,1.39499015802841523529377394252379049117},
{36,1.39023156004523883530353774388849366078},
{37,1.38565517036913390218387987164360902442},
{38,1.38124948587417599494888974270272683602},
{39,1.37700400004464983512682767443775686515},
{40,1.37290909442703835047823373266554024708},
{41,1.36895594420004818247311699643331377899},
{42,1.36513643574275156839766072988517148305},
{43,1.36144309443977055373412981089021928638},
{44,1.35786902125399984336720056851594591129},
{45,1.35440783683543930473427573602498607559},
{46,1.3510536321300155887091520853891945445},
{47,1.34780092461321684380654451302931688756},
{48,1.34464461940656131097555379617152740538},
{49,1.34157997464560704966451256923364150444},
{50,1.33860257056055295892954001842319703868}};
(*第二列换成第一列与第二列的乘积*)
data[]*=data[];
out=NonlinearModelFit;
Normal
Show,{x,0,52},PlotRange->{{0,60},{0,70}},AspectRatio->Automatic],ListPlot,ImageSize->800]
Print["拟合优度值是:"];
out["RSquared"]
拟合方程是:
-1.208100914943779516587560800750535150 + 2.42523844131677074670895367755557525 Sqrt +1.019655051803581738323392673297013864 x
拟合优度值是:
0.99999995799099676515334782556813032