Calculating numerical integral using integral or quadgk - matlab

Question

I am using MATLAB to calculate the numerical integral of a complex function including natural exponent.
I get a warning:
Infinite or Not-a-Number value encountered
if I use the function integral, while another error is thrown:
Output of the function must be the same size as the input
if I use the function quadgk.
I think the reason could be that the integrand is infinite when the variable ep is near zero.
Code shown below. Hope you guys can help me figure it out.
close all
clear
clc
%%
N = 10^5;
edot = 10^8;
yita = N/edot;
kB = 8.6173324*10^(-5);
T = 300;
gamainf = 0.115;
dTol = 3;
K0 = 180;
K = K0/160.21766208;
nu = 3*10^12;
i = 1;
data = [];
%% lambda = ec/ef < 1
for ef = 0.01:0.01:0.1
for lambda = 0.01:0.01:0.08
ec = lambda*ef;
f = #(ep) exp(-((32/3)*pi*gamainf^3*(0.5+0.5*sqrt(1+2*dTol*K*(ep-ec)/gamainf)-dTol*K*(ep-ec)/gamainf).^3/(K*(ep-ec)).^2-16*pi*gamainf^3*(0.5+0.5*sqrt(1+2*dTol*K*(ep-ec)/gamainf)-dTol*K*(ep-ec)/gamainf).^2/((1+dTol*K*(ep-ec)/(gamainf*(0.5+0.5*sqrt(1+2*dTol*K*(ep-ec)/gamainf)-dTol*K*(ep-ec)/gamainf)))*(K*(ep-ec)).^2))/(kB*T));
q = integral(f,0,ef,'ArrayValued',true);
% q = quadgk(f,0,ef);
prob = 1-exp(-yita*nu*q);
data(i,1) = ef;
data(i,2) = lambda;
data(i,3) = q;
i = i+1;
end
end

Answer 1

I've rewritten your equations so that a human can actually understand it:
function integration
N = 1e5;
edot = 1e8;
yita = N/edot;
kB = 8.6173324e-5;
T = 300;
gamainf = 0.115;
dTol = 3;
K0 = 180;
K = K0/160.21766208;
nu = 3e12;
i = 1;
data = [];
%% lambda = ec/ef < 1
for ef = 0.01:0.01:0.1
for lambda = 0.01:0.01:0.08
ec = lambda*ef;
q = integral(#f,0,ef,'ArrayValued',true);
% q = quadgk(f,0,ef);
prob = 1 - exp(-yita*nu*q);
data(i,:) = [ef lambda q];
i = i+1;
end
end
function y = f(ep)
G = K*(ep - ec);
r = dTol*G/gamainf;
S = sqrt(1 + 2*r);
x = (1 + S)/2 - r;
Z = 16*pi*gamainf^3;
y = exp( -Z*x.^2.*( 2*x/(3*G.^2) - 1/(G.^2*(1 + r/x))) ) /...
(kB*T));
end
end
Now, for the first iteration, ep = 0.01, the value of the argument of the exp() function inside f is huge. In fact, if I rework the function to return the argument to the exponent (not the value):
function y = f(ep)
% ... all of the above
% NOTE: removed the exp() to return the argument
y = -Z*x.^2.*( 2*x/(3*G.^2) - 1/(G.^2*(1 + r/x))) ) /...
(kB*T);
end
and print its value at some example nodes like so:
for ef = 0.01 : 0.01 : 0.1
for lambda = 0.01 : 0.01 : 0.08
ec = lambda*ef;
zzz(i,:) = [f(0) f(ef/4) f(ef)];
i = i+1;
end
end
zzz
I get this:
% f(0) f(ef/4) f(ef)
zzz =
7.878426438111721e+07 1.093627454284284e+05 3.091140080273912e+03
1.986962280947140e+07 1.201698288371587e+05 3.187767404903769e+03
8.908646053687230e+06 1.325435523124976e+05 3.288027743119838e+03
5.055141696747510e+06 1.467952125661714e+05 3.392088351112798e+03
...
3.601790797707676e+04 2.897200140791236e+02 2.577170427480841e+01
2.869829209254144e+04 3.673888685004256e+02 2.404148067956737e+01
2.381082059148755e+04 4.671147785149462e+02 2.238181495716831e+01
So, integral() has to deal with things like exp(10^7). This may not be a problem per se if the argument would fall off quickly enough, but as shown above, it doesn't.
So basically you're asking for the integral of a function that ranges in value between exp(~10^7) and exp(~10^3). Needless to say, The d(ef) in the order of 0.01 isn't going to compensate for that, and it'll be non-finite in floating point arithmetic.
I suspect you have a scaling problem. Judging from the names of your variables as well as the equations, I would think that this has something to do with thermodynamics; a reworked form of Planck's law? In that case, I'd check if you're working in nanometers; a few factors of 10^(-9) will creep in, rescaling your integrand to the compfortably computable range.
In any case, it'll be wise to check all your units, because it's something like that that's messing up the numbers.
NB: the maximum exp() you can compute is around exp(709.7827128933840)