Re: minmum of a function

*To*: mathgroup at smc.vnet.net*Subject*: [mg75349] Re: minmum of a function*From*: Peter Pein <petsie at dordos.net>*Date*: Thu, 26 Apr 2007 03:30:06 -0400 (EDT)*References*: <f014ar$8qr$1@smc.vnet.net> <f0ejkm$pad$1@smc.vnet.net>

skruege at gwdg.de schrieb: > Hi Jean-Marc, > > Thank for your help. > I think (Head /@ t1)[[All,1]] is exactly what I need. > Probably I could simplify my definitions of function even before. > In fact, I defined > > amp := fmax/f5 > > with fmax := FindMaximum[Sum[Integrate[f1[x], {x, i, i + a}], {i, 0, 2}], > {p, 0}] whereas f1[x_] := c + (Cos[Pi*x + p])^2 > and > f5 := Integrate[f4[x], {x, 0, 2 + a}] whereas f4[x_] := c + (Cos[Pi*x + > pmax])^2 and pmax := p /. fmax[[2]]. > > With (Head /@ t1)[[All,1]] I'm now able to create a list amp[a] and to > plot it. > Before your help, I managed to get the list by using 'Block' like f6 := > Block[{a = 0.1}, amp], but it isn't elegant at all. > > Regards, > Sven > Hi Sven, I think you should try to find an explicit expression for amp. Then each call to amp will be faster. If amp := fmax/f5 then am will be of the form {number1/number3,{(p->number2)/number3}}, which does not make much sense to me. amp:=fmax[[1]]/f5 would be more meaningful, if the handling of the parameters was sufficient (sorry). I tried the following and had to do some assumptions: I'll assume 0<a<1 and pmax<0 In[1]:= f1and4[x_, p_] := c + Cos[Pi*x + p]^2 In[2]:= targetToMaximize = FullSimplify /@ Apart[Sum[Integrate[f1and4[x, p], {x, i, i + a}], {i, 0, 2}], a] Out[2]= (3/2)*a*(1 + 2*c) + (3*Cos[2*p + a*Pi]*Sin[a*Pi])/(2*Pi) In[3]:= redu = Module[{t1 = Simplify[D[targetToMaximize, p]], t2}, t2 = Simplify[D[t1, p]]; Assuming[0 < a < 1, FullSimplify[ Reduce[t1 == 0 > t2], p, Reals]]]] Out[3]= (C[1] | C[2]) \[Element] Integers && ((p + (a*Pi)/2 == Pi*C[1] && 0 < a - 2*C[2] < 1) || (-1 < a - 2*C[2] < 0 && p == ArcTan[Cot[(a*Pi)/2]] + Pi*C[1])) In[4]:= sol = Solve[FullSimplify[redu /. {C[1 | 2] -> 0}, 0 < a < 1], p] Out[4]= {{p -> -((a*Pi)/2)}} In[5]:= pmax = p /. sol[[1]] Out[5]= -((a*Pi)/2) In[6]:= fmax = targetToMaximize /. p -> pmax Out[6]= (3/2)*a*(1 + 2*c) + (3*Sin[a*Pi])/(2*Pi) In[7]:= f5[a_] = Integrate[f1and4[x, pmax], {x, 0, 2 + a}] Out[7]= ((2 + a)*(1 + 2*c)*Pi + Sin[a*Pi])/(2*Pi) In[8]:= amp[a_] = FullSimplify[fmax/f5[a]] Out[8]= (3*(a*(1 + 2*c)*Pi + Sin[a*Pi]))/((2 + a)*(1 + 2*c)*Pi + Sin[a*Pi]) In[9]:= Plot[amp[a] /. c -> 0.3, {a, 0, 1}] hth, Peter