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Question: Root Finding again

Question:Root Finding again

longrob 1409 Maple
rootfinding
July 25 2011
2 
I came across this problem while helping another user find the maxima of an expression for 
various values of a parameter here:
http://www.mapleprimes.com/questions/124104-Maximum-Points--Of-Function-With-More

For various values of ga, it was required to find the maxima in a range of 0<delta<2. 
Plots of the expression indicate that one such maxima exists for each value of the parameter.

My approach was to find the zeros of the first derivative of the expression, but this failed both 
with fsolve and RootFinding:-NextZero. I was quite surprised that the latter did not work.

I was able find the zeros by wrapping an eval in a procedure and running fsolve against the 
procedure. This, however, was very slow.

I'd be grateful for any comments on this and suggestions for other/better ways to do it. 
I also tried findindg the maxima with the DirectSearch package, which also did not work.
> 

f:=int((50-50*delta)*(50+50*delta)*(0.3e-1+(.36*(1-exp(-.39*t)))/(12.00000000*exp(-.39*t)+1))*(1-(1-exp(-.39*t))/(12.00000000*exp(-.39*t)+1))/1.08^t, t = 0 .. -2.564102564*ln((1.+5.*delta)/(53.+5.*delta)))+(500.*(-1.-5.*delta))*(-(1.*(1.+4.*ga))/(-53.+48.*ga))^.197336002905/((-1.+1.*ga)*(1.+delta)*((1.+5.*delta)/(53.+5.*delta))^.197336002905);

(1)
> 

plots:-display(seq(plot(eval(f,ga=m),delta=0..2),m=0.1..0.8,0.1));

 
> 

df:=diff(eval(f,ga=0.1),delta):
> 

 

> 

#find zero of df
> 

 

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fsolve(df=0);
> 

 

Error, (in fsolve) number of equations, 1, does not match number of variables, 2

  
> 

fdf:=m->eval(df,delta=m):
> 

RootFinding:-NextZero(fdf,0.1);

(2)
> 

# Interestingly, RootFinding:-NextZero does work for some values of ga (eg 0.8)
> 

  
> 

pq:=proc(x) global df;return eval(df,delta=x); end proc:
> 

fsolve(pq(z)=0,z);
> 

 

(3)
> 

 

> 

 

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