20Â¥: Originally posted by loujing at 2016-01-08 18:15:38
×·ÎÊһϣ¬Ôڵõ½ÕâÀïµÄÁªÁ¢·½³Ì×éºó£¬·ûºÅÍƵ¼µÄ¹¤×÷ÊDz»ÊǾͿÉÒÔËã½áÊøÁË£¿

Èç¹û½ÓÏÂÀ´£¬ÎÒ¸ø³ö¾ßÌåµÄÒ»×é(x_i,y_i)£¬ÈçºÎÔÙÀûÓÃÕâÀïµÄÍƵ¼£¬¼ÆËã³ö¾ßÌåµÄw_j£¿...

(*֮ǰµÄ´úÂëÎҾͲ»¸´ÖÆÁË*)
n = 3; m = 2;
points = RandomReal[1, {n, 2}];

{x[i_], y[i_]} := {points[[i, 1]], points[[i, 2]]};

var = w /@ Range[0, m];
Solve[Table[eqn, {j, 0, m}], var][[1]]
(var /. %).x^Range[0, m]
Show[Plot[%, {x, Min[points[[All, 1]]], Max[points[[All, 1]]]}], ListPlot[points],
PlotRange -> All]
Clear[x, y]

{xlist, ylist} = points\[Transpose];
With[{xm = #^Range[0, m] & /@ xlist}, Solve[ylist.xm == var.xm\[Transpose].xm, var]]

f[i_, m_, x_] := Sum[w@j*x[[i]]^j, {j, 0, m}]
l[m_, n_, x_, y_] := 1/2 Sum[(f[i, m, x] - y[[i]])^2, {i, n}];
n = 4; m = 2;
var = w /@ Range[0, m];
{xlist, ylist} = Transpose@RandomReal[1, {n, 2}];
NSolve[D[l[m, n, xlist, ylist], {var}], var]
Show[ListPlot[Transpose[{xlist, ylist}]],
Plot[Evaluate[var.x^Range[0, m] /. %], {x, 0, 1}], PlotRange -> All]

21Â¥: Originally posted by xzczd at 2016-01-08 21:36:58
˵½áÊøÒ²Ëã½áÊøÁË£¬ËµÃ»½áÊøҲû½áÊø£¬4Â¥µÄÍƵ¼ÔÚÕâÖ®ºóÓÖ¶àÁËÒ»²½£¬ÊÇ°Ñ֮ǰµÄ½á¹ûд³ÉÁ˾ØÕó³Ë·¨µÄÐÎʽ£¬Õâ¸öÆäʵÊǾØÕóµã³Ë¹æÔòµÄÖ±½ÓÓ¦Ó㬣¬µ«ÊÇÈç¹ûÄãÏë°ÑÕâ²½Ò²½»¸øÈí¼þÍƵ¼µÄ»°£¬ÎÒ¿ÉÒÔ¸æËßÄãÕâ¸ö¹æÔòÓÖÊÇ ...

eqn = (D[l /. rulej1[w@j], w[j]] /. {rulej2, rulej3} //. {ruleExpand, ruleDistribute,
      ruleExtract} // Expand) == 0

18Â¥: Originally posted by xzczd at 2016-01-08 17:59:09
¡­¡­¸Õ·¢ÏÖÎÒΪÁË´Õ³öºÍÄãµÄ¶¥Â¥ÏàËƵĽá¹û·¸Á˸ö´íÎ󡣶¢×ÅËÄÂ¥µÄͼ¿´ÁË°ëÌ죨ÕæÉËÑÛ£©£¬ÏÖÔÚÎÒ»ù±¾È·¶¨£¬ÕýÈ·µÄÍƵ¼ÊÇ£º

f := Sum
l = 1/2 Sum;

rulej1 := Sum /; ! FreeQ :> sum
rulej2 = sum' :> ...