Help! Numerically handle removable singularities: a more clear example...
Luna Moon
I am helping my friend on this issue.
We have some numerical computation which involves a lot of "log(g)*g"
and we have to evaluate these for g on the interval [0, 1],
numerically... The reason that 0 is included is that because the
singularity at 0 is removable.
In Maple:
> x := 0; log(x)*x;
%;
0
Error, (in ln) numeric exception: division by zero
In Matlab:
>> x=0; log(x)*x
ans =
NaN
Our procedure is that we first derive very complicated symbolic
expressions in Maple and then translate into Matlab to get numerical
calculation.
How do handle the above "NaN" in Matlab? This is just one intermediate
step using "log(x)*x" (this expression is from Maple symbolic
results).
Of course we could take a limit at x=0 and handle that case in Matlab
separately, let say when |x| < 1e-5, we switch to this limiting
result; and when |x|>=1e-5, we still use the numerical approach.
But this creates a discontinuity at |x|=1e-5 in numerical results in
Matlab.
Could anybody shed some lights on us about how to handle such a
situation?
Thanks a lot!
A.L.
<lunamo...@gmail.com> wrote:
>Hi all,
>
>I am helping my friend on this issue.
>
>We have some numerical computation which involves a lot of "log(g)*g"
>and we have to evaluate these for g on the interval [0, 1],
>numerically... The reason that 0 is included is that because the
>singularity at 0 is removable.
http://ideas.repec.org/c/boc/bocode/x031701.html
http://fmwww.bc.edu/repec/bocode/s/slog.ox
A.L.
Jon Slaughter
log(x)*x when x = 0 is undefined because log(x) is undefined
limit(log(x)*x,x=0) = 0
Generally if you are simply doing a numerical computation you could handle
it as a side case such as
if (x == 0) \\ or |x| <= 0.00000001 or something similar
return 0;
or
// hack that is not generally a good way
if (x==0) \\ or |x| <= 0.00000001
x = 0.00001
// do comp
If your doing some advanced numerical method then generally you will find
some numerial appoximation to log(x)*x in the first place where the
discontinuity is removed such as a taylor series
log(x)*x ~= -sum((1 - x)^k/k)*x
(not a good method)
or by using a faster converging series such as some found at
http://en.wikipedia.org/wiki/Logarithm
Gordon Sande
> Hi all,
>
> I am helping my friend on this issue.
>
> We have some numerical computation which involves a lot of "log(g)*g"
> and we have to evaluate these for g on the interval [0, 1],
> numerically... The reason that 0 is included is that because the
> singularity at 0 is removable.
Just as there is a more common name for sin(x)/x as sinc(x) which
is fairly common in signal processing there may be a common name
for x*log(x) which shows up in entropy calculations. There may even
be known approximations for x*log(x) as there is for sin(x)/x. Google
is your friend as I do not know of a ready source.
Try defining a new function xln(x) = x * ln(x) in Maple and let it chew
on the result. At least it will be easier to know where the simplifiction
will occur.
Jon Slaughter
xlog := x->piecewise(x=0, 0, x*log(x));
which should return 0 when x is 0 else x*log(x).
Mate
> On 2009-11-14 17:52:56 -0400, Luna Moon <lunamoonm...@gmail.com> said:
>
> > Hi all,
>
> > I am helping my friend on this issue.
>
> > We have some numerical computation which involves a lot of "log(g)*g"
> > and we have to evaluate these for g on the interval [0, 1],
> > numerically... The reason that 0 is included is that because the
> > singularity at 0 is removable.
>
> Just as there is a more common name for sin(x)/x as sinc(x) which
> is fairly common in signal processing there may be a common name
> for x*log(x) which shows up in entropy calculations. There may even
> be known approximations for x*log(x) as there is for sin(x)/x. Google
> is your friend as I do not know of a ready source.
>
> Try defining a new function xln(x) = x * ln(x) in Maple and let it chew
> on the result. At least it will be easier to know where the simplifiction
> will occur.
>
There is a big difference between sin(x)/x and x*log(x); the first is
analytic
(removable singularity at 0) but the second is not. `xln` is
continuous in
[0,oo) but is not differentiable at 0, so it will generate many
problems.
Mate
Sebastian Nowozin
Hi,
On Nov 15, 2:19 am, Gordon Sande <g.sa...@worldnet.att.net> wrote:
> Just as there is a more common name for sin(x)/x as sinc(x) which
> is fairly common in signal processing there may be a common name
> for x*log(x) which shows up in entropy calculations. There may even
> be known approximations for x*log(x) as there is for sin(x)/x. Google
> is your friend as I do not know of a ready source.
A good idea.
Another idea, not necessarily good and its applicability depends a lot
on the remaining part of your problem.
Do a variable substitution exp(y)=x, hence x*ln(x) becomes exp(y)*ln
(exp(y)) which is exp(y)*y, defined on the entire real line. The
substitution is only really useful in case you can do it on all of
your x variables.
Sebastian