MaplePrimes - answers and comments on Question, Increased precision for complex integration

Reasonably fast

Preben Alsholm — Tue, 29 Jan 2013 04:25:16 Z

If Digits = 15 is OK with you then method = _d01amc is very fast (as fast as Digits=10 without specifying a method):

evalf[15](Int((phi(g_line(t,mu0,c0))/(g_line(t,mu0,c0)-mu0)*exp(-l*g_line(t,mu0,c0))
*diff(g_line(t,mu0,c0),t)),t=0..infinity,method = _d01amc));

#Result 0.999887009423573e-1-.605301458903814*I

This agrees with the result 0.9998870094e-1-.6053014589*I from

evalf(Int((phi(g_line(t,mu0,c0))/(g_line(t,mu0,c0)-mu0)*exp(-l*g_line(t,mu0,c0))
*diff(g_line(t,mu0,c0),t)),t=0..infinity));

This was done in Maple 16.

decay

acer — Tue, 29 Jan 2013 12:29:15 Z

Perhaps Axel might have some clever insight.

I was suspicious of the last 3 digits or so from _d01amc. So here is _d01ajc on a finite interval. I doubt that I understand this integral well yet...

restart;

phi:=z->(z-1)^(9/10)*(z-2)^(6/10)*(z+1)^(8/10)
        /(z*(z-(21/10))^(9/10)*(z+2)^(4/10)):
g_line:=(t,mu,c)->mu+t+c*t*I:
igrand:=(phi(g_line(t,mu0,c0))/(g_line(t,mu0,c0)-mu0)
        *exp(-l*g_line(t,mu0,c0))*diff(g_line(t,mu0,c0),t)):

igrand:=eval(igrand,[mu0=2,c0=0.1,l=1]):

# Various kinds of simplification (while integrand is exact) do not
# seem to improve significantly the accuracy of the integrand itself.
# (But maybe someone can do that better...)
# For now, reverting to floats throughout.
igrand:=subsindets(igrand,rational,evalf):
T:=Int(igrand, t=0..30, digits=15, epsilon=0.5e-14, method=_d01ajc):

CodeTools:-Usage( evalf[15](T) );
memory used=4.20MiB, alloc change=10.06MiB, cpu time=514.00ms, real time=530.00ms
            0.0999887009423848 - 0.605301458903795 I

acer

Thank you

Rune - math — Wed, 30 Jan 2013 14:06:02 Z

Thank you very much for suggesting the option,
method = _d01amc

It did the job perfectly. I needed to compute the integral for 9 different phi functions similar to the one described, but earlier it didn't get any result after 10 minutes.

The suggestion to write phi exact instead of with floats increased the speed of an integral where the integration curve was finite.

Thank you both very much :D