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How to test a 'float' or 'double' zero numerically?

Hi,

Suppose T is 'float' or 'double'.

T x;

x < 10 * std::numeric_li mits<T>::epsilo n();

I can use the above comparison to test if 'x' is numerically zero. But
I'm wondering what should be a good multiplicative constant before
epsilon?

Thanks,
Peng
Sep 13 '08 #1
15 27873
I can use the above comparison to test if 'x' is numerically zero.

No, as x can also be negative.
But I'm wondering what should be a good multiplicative constant before
epsilon?
Epsilon is the smallest value such such that 1.0 + epsilon != 1.0. You
need to scale it with the numbers to compare with. Comparing against
zero is always hard. You are probably best of with using abs(x) <
your_own_epsilo n. Set your_own_epsilo n to what ever you want, such as
0.00000001 perhaps.

Regards,
Anders Dalvander
Sep 13 '08 #2
On Sep 13, 11:10 am, Anders Dalvander <goo...@dalvand er.comwrote:
I can use the above comparison to test if 'x' is numerically zero.

No, as x can also be negative.
Right, I meant std::abs(x).
But I'm wondering what should be a good multiplicative constant before
epsilon?

Epsilon is the smallest value such such that 1.0 + epsilon != 1.0. You
need to scale it with the numbers to compare with. Comparing against
zero is always hard. You are probably best of with using abs(x) <
your_own_epsilo n. Set your_own_epsilo n to what ever you want, such as
0.00000001 perhaps.
Therefore, there is no general accept such epsilon?

Thanks,
Peng
Sep 13 '08 #3
Peng Yu <Pe*******@gmai l.comkirjutas:
Hi,

Suppose T is 'float' or 'double'.

T x;

x < 10 * std::numeric_li mits<T>::epsilo n();

I can use the above comparison to test if 'x' is numerically zero. But
Really? What if x is -10000? What if it is equal to std::numeric_li mits
<T>::epsilon( )?
I'm wondering what should be a good multiplicative constant before
epsilon?
To answer your question literally, then comparing to zero is easy, just
use if(x==0). However, this usually does not give you much if x is a
result of some computation, with this expression you can pretty much just
check whether x has been assigned literal zero beforehand.

If you want to compare values appearing in some numeric algorithm then
this all very much depends on the algorithm. What is your actual problem
you are trying to solve?

Paavo


Sep 13 '08 #4
Peng Yu wrote:
x < 10 * std::numeric_li mits<T>::epsilo n();

I can use the above comparison to test if 'x' is numerically zero.
No you can't. A value of x distinct from zero might also test as
"zero" with that.
Sep 13 '08 #5
Hi,

Consider a machine where the smallest number that can be represented is
0.0001

Lets asume I have the following calculation (lets assume the 0.00005 would
be the result of some calculaton).
0.0001 -0.00005 - 0.00005
Now it is obvious that this should result in zero. However the last two
results would be zero since the machine can only have up to four digits
behind the dot. So what should be zero is actually 0.0001 so a correct value
for a multiplier for epsilon would be 0.0002. Reasoning 0.0001 < 0.0002
therefore it is zero?

Consider then the following

The same formula only we also divide by 0.0001 afterwards
( 0.0001 -0.00005 - 0.00005 ) / 0.0001 = 1 However the one actually should
be a zero therefore our first conclusion was incorrect. A correct multiplier
for epsilon should be 10001

Of course one could go on, epsilons multiplier could be anything.

Conclusion there is not a correct multiplier for epsilon. There can be one
per formula but that is probably not very practical.
Regards, Ron AF Greve

http://www.InformationSuperHighway.eu

"Peng Yu" <Pe*******@gmai l.comwrote in message
news:2e******** *************** ***********@w7g 2000hsa.googleg roups.com...
Hi,

Suppose T is 'float' or 'double'.

T x;

x < 10 * std::numeric_li mits<T>::epsilo n();

I can use the above comparison to test if 'x' is numerically zero. But
I'm wondering what should be a good multiplicative constant before
epsilon?

Thanks,
Peng

Sep 13 '08 #6
On Sep 13, 4:48 pm, "Ron AF Greve" <ron@localhostw rote:
Hi,

Consider a machine where the smallest number that can be represented is
0.0001

Lets asume I have the following calculation (lets assume the 0.00005 would
be the result of some calculaton).
0.0001 -0.00005 - 0.00005
Now it is obvious that this should result in zero. However the last two
results would be zero since the machine can only have up to four digits
behind the dot. So what should be zero is actually 0.0001 so a correct value
for a multiplier for epsilon would be 0.0002. Reasoning 0.0001 < 0.0002
therefore it is zero?

Consider then the following

The same formula only we also divide by 0.0001 afterwards
( 0.0001 -0.00005 - 0.00005 ) / 0.0001 = 1 However the one actually should
be a zero therefore our first conclusion was incorrect. A correct multiplier
for epsilon should be 10001

Of course one could go on, epsilons multiplier could be anything.

Conclusion there is not a correct multiplier for epsilon. There can be one
per formula but that is probably not very practical.
I see. Then the problem is how to derive it for a particular formula.

Probably, I need to write down the formula and take the derivatives of
all its arguments, check how much errors there could be for each
arguments. Then I would end up with a bound of the rounding error
(epsilon is equivalent to it). Right?

Thanks,
Peng
Sep 13 '08 #7
On 14 Sep, 01:53, Peng Yu <PengYu...@gmai l.comwrote:
On Sep 13, 4:48 pm, "Ron AF Greve" <ron@localhostw rote:
Of course one could go on, epsilons multiplier could be anything.
Conclusion there is not a correct multiplier for epsilon. There can be one
per formula but that is probably not very practical.

I see. Then the problem is how to derive it for a particular formula.

Probably, I need to write down the formula and take the derivatives of
all its arguments, check how much errors there could be for each
arguments. Then I would end up with a bound of the rounding error
(epsilon is equivalent to it). Right?
Numerical analysis is an art in itself. There are departments
in universities which deal almost exclusively with the analysis
of numerics, which essentially boils down to error analysis.

In my field of work certain analytical solutions were formulated
in the early '50s, but a stable numerical solution wasn't found
until the early/mid '90s.

You might want to check with the math department at your local
university on how to approach whatever problem you work with.

Rune
Sep 14 '08 #8
In my field of work certain analytical solutions were formulated
in the early '50s, but a stable numerical solution wasn't found
until the early/mid '90s.
Would you please give some example references on this?

Thanks,
Peng

Sep 14 '08 #9
On 2008-09-13 18:16, Peng Yu wrote:
On Sep 13, 11:10 am, Anders Dalvander <goo...@dalvand er.comwrote:
I can use the above comparison to test if 'x' is numerically zero.

No, as x can also be negative.

Right, I meant std::abs(x).
But I'm wondering what should be a good multiplicative constant before
epsilon?

Epsilon is the smallest value such such that 1.0 + epsilon != 1.0. You
need to scale it with the numbers to compare with. Comparing against
zero is always hard. You are probably best of with using abs(x) <
your_own_epsil on. Set your_own_epsilo n to what ever you want, such as
0.00000001 perhaps.

Therefore, there is no general accept such epsilon?
No, different applications requires different precision, some would
consider a variable equal to zero if it was 0.0001 from zero while
others might require 0.0000001. You have to analyse your problem to find
a value that suites you.

--
Erik Wikström
Sep 14 '08 #10

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