How to avoid overflow or underflow in mathematica?

[quantumfang]

It's well known that Mathematica can do evaluation to arbitrary precision. But I don't know how to avoid the following trouble in computing:

Ef = 0.;
mu=0.;
KBT = 1.*10^-10;
FL[omega_Real] := 1./(1. + E^((omega - mu - Ef)/KBT));

the error message shows

General::ovfl: Overflow occurred in computation. >>
General::unfl: Underflow occurred in computation. >>

[W_Craig Carter]

Hello Quantum Fang,

You could use arbitrary precision and do something like this (although
I don't advise this approach):

fermiseries =
Normal[Series[1/(1 + E^((muShift) beta)), {beta, 0, 14}]];

colist = Simplify[CoefficientList[fermiseries, beta]]

and then write a function that calls N[beta, bigNumber] and N[mushift,
bigNumber] on the product of the two list until you have the accuracy
that you want.... This is a bad idea.

However your function, except for ((omega - mu - Ef)/KBT)
approximately 1, is the unit step function (0 if x<0, 1/2 x=0, 1 if
x>0). If you are interested in the physical aspects of this problem
then expand (omega - mu - ef) near 1/(kb T). In other words, scale
your energies with the thermal energy.

WCC

--
W. Craig Carter

[Szabolcs Horvát]

You've practically written E^(10^10.) here. I hope that you do realize
how great the magnitude of this double exponential is.

Mathematica has its limits too: the biggest number it can handle is
$MaxNumber, which is 1.920224672692357*10^646456887 on my computer. The
magnitude of your expression is greater than this.

[Carl Woll]

quantumfang wrote:

>It's well known that Mathematica can do evaluation to arbitrary precision. But I don't know how to avoid the following trouble in computing:
>
>Ef = 0.;
>mu=0.;
>KBT = 1.*10^-10;
>FL[omega_Real] := 1./(1. + E^((omega - mu - Ef)/KBT));
>
>the error message shows
>
>General::ovfl: Overflow occurred in computation. >>
>General::unfl: Underflow occurred in computation. >>
>
>

You don't tell us what value of omega you used. However, keep in mind
that there is a maximum number in Mathematica. On my computer this
number is:

In[34]:= $MaxNumber

Out[34]= 1.920224672692357*10^646456887

The log of this number is:

In[35]:= Log[$MaxNumber]

Out[35]= 1.488521991921978478647057*10^9

So, if (omega-mu-Ef)/KBT is larger than the above number you will get
overflow/underflow problem. With your parameters, this means that an
order 1 value of omega will cause overflow/underflow problems.

Carl Woll
Wolfram Research