Sorry for the delay, I took some time to put the code to work. However,
there's something strange.
Below is a "concise" version of my code. The strange thing that I noticed
is that the *find_local_maximum* function returns a polynomial as the
maximum value, and changing the tolerance (*tol*) argument of the function
do not change the result. You can see in the graph that the maximum point
is slightly away from the calculated point.
Thank you for the help!
n=6 # polynomial order
listOmega=[]
for i in range(n+1):
listOmega.append(i/n)
listPhases=[0]
for i in range(1,n+1):
listPhases.append(listOmega[i]^2/2 + 9/2*listOmega[i])
Alpha0=listOmega[1]/tan(listPhases[1])
listAlpha=[Alpha0]
for i in range(1,n):
firstline=listOmega[i+1]^2-listOmega[i]^2
lastline=Alpha0-listOmega[i+1]/tan(listPhases[i+1])
ic=1
while ic<i:
lastline=listAlpha[ic]-((listOmega[i+1]^2 -
listOmega[ic]^2)/lastline)
ic+=1
listAlpha.append(firstline/lastline)
var('s,omega')
H=[1,s + listAlpha[0]]
for i in range(2,n+1):
H.append(listAlpha[i-1]*H[i-1] + (s^2 + listOmega[i-1]^2)*H[i-2])
f=sqrt(H[6](s=I*omega)*H[6](s=-I*omega));
g=[]
for i in range(n+1):
g.append(f*chebyshev_T(i,omega)/sqrt(1-omega^2))
up=0
for i in range(n+1):
up=up + c[i]*chebyshev_T(i,omega)
hmod(omega)=(H[6](s=I*omega)).polynomial(CC)
den=lambda omega: up/abs(hmod(omega))
norm_factor=find_local_maximum(den,-1,1,tol=1e-18); norm_factor
print("Point where local maximum occurs: " + str(norm_factor[1]))
print("Local maximum: " + str(norm_factor[0](omega=norm_factor[1]).n()))
plot(den(omega),-1,1)
On Friday, January 12, 2018 at 4:51:32 PM UTC+1, Vegard Lima wrote:
>
> Hi
>
> On Fri, Jan 12, 2018 at 4:10 PM, João Alberto de França Ferreira
> <[email protected] <javascript:>> wrote:
> > Is there a way to compute the local maximum of the absolute value of a
> > polynomial with complex coefficients? the below snippet should exemplify
> my
> > problem.
> ...
>
> Maybe a trick like this works:
> f(x) = x^2 + 2*I*x + 2
> g = lambda x: abs(f(x))
> find_local_maximum(g,-1,1)
>
> this gives
> (3.6055511727769516, 0.99999996297566163)
>
> while
> find_local_maximum(abs(f(x)),-1,1)
> gives
> TypeError: unable to coerce to a real number
>
> The difference is that one is a python function while
> the other one is a SymbolicExpression, not sure
> why one works and the other doesn't.
>
>
> Thanks,
> --
> Vegard
>
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