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critical points of a third order polynomial fit (simplification)

• To: mathgroup at smc.vnet.net
• Subject: [mg68466] critical points of a third order polynomial fit (simplification)
• From: "Chris Chiasson" <chris at chiasson.name>
• Date: Sun, 6 Aug 2006 02:56:47 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

Here are some commands that find the (two) critical locations of a
cubic polynomial fitted to four points. The four points are given as
bracket={y1,x1,y2,x2,y3,x3,y4,x4}, where y is the range and x is the
domain location. The code produces two sets of critical points. Both
solution sets involve substituting the coefficients obtained from the
four point "fit" into the general solution of the critical points
(from calculus D[poly,x]==0). The only difference is that one of them
calls a Simplify command that the other doesn't (compare rep[3] with
rep[4]).

Why are the answers for the second critical point so different? Do you
think the simplification made in rep[3] is invalid? Does your computer
give the same results that mine gives?

In[1]:=
$Version

Out[1]=
5.2 for Microsoft Windows (June 20, 2005)

In[2]:=
bracket={0.040911501042171394,9.797734083340343,0.003643048928782312,
      9.939642325021731,0.0007478278717134088,10.027346441664564,
      0.028647147834538127,10.169254683345951};

In[3]:=
eqn[1]=Plus@@Table[Times[c[i],Power[x,i-1]],{i,4}]\[Equal]y;
eqn[2]=eqn[1]/.{x\[Rule]x[#],y\[Rule]y[#]}&/@Range[4];
rep[1]=FullSimplify@Solve[eqn[2],Array[c,{4}]];
rep[2]=FullSimplify@Solve[D[eqn[1],x],x];
rep[3]=Simplify[rep[2]/.rep[1][[1]]];
rep[4]=rep[2]/.rep[1][[1]];
rep[5]={x[num_]\[RuleDelayed]bracket[[2 num]],
      y[num_]\[RuleDelayed]bracket[[2 num-1]]};

In[10]:=
x/.rep[3]/.rep[5]//InputForm

Out[10]//InputForm=
{-1.4838127878952383*^10,
 17.999998818597952}

In[11]:=
x/.rep[4]/.rep[5]//InputForm

Out[11]//InputForm=
{-1.4838127878952383*^10,
 9.999999069105009}

-- 
http://chris.chiasson.name/

• Follow-Ups:
  • Re: critical points of a third order polynomial fit (simplification)
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>