doubt regarding floating point arithmetic Rounding schemes/ Truncation schemes

doubt regarding floating point arithmetic Rounding schemes/ Truncation schemes

Post by Nick Maclar » Fri, 21 Nov 2003 07:49:59





>Rounding schemes/ Truncation schemes

>The choice of schemes decides the max error and bias.

Correct.

Quote:>Zero bias schemes on an average cause little error but there is a large
>chance htat they might hav a bias too and not end up with an error of
>zero. Am I right or wrong

I am not quite sure what you mean, but I think that you are roughly
right.  Most zero bias schemes have no bias, assuming that the errors
are uniformly distributed.  True probabilistic (stochastic) rounding
has no bias, whatever the distribution of errors, but has twice the
mean square error and makes debugging slightly (!) harder.

Quote:>What are the rounding schemes / truncation schemes in use nowadays

Almost always IEEE 754, or minor variants of it.

Regards,
Nick Maclaren.

 
 
 

doubt regarding floating point arithmetic Rounding schemes/ Truncation schemes

Post by Robert Myer » Fri, 21 Nov 2003 09:21:39





>>Rounding schemes/ Truncation schemes

>>The choice of schemes decides the max error and bias.

>Correct.

>>Zero bias schemes on an average cause little error but there is a large
>>chance htat they might hav a bias too and not end up with an error of
>>zero. Am I right or wrong

>I am not quite sure what you mean, but I think that you are roughly
>right.  Most zero bias schemes have no bias, assuming that the errors
>are uniformly distributed.  True probabilistic (stochastic) rounding
>has no bias, whatever the distribution of errors, but has twice the
>mean square error and makes debugging slightly (!) harder.

When I saw the subject line, I thought, now that will get a response
from Nick.

For extra credit, you might want to study

http://www.cs.ucla.edu/~stott/mca/CSD-970014.ps.gz

and the links to the literature it provides.  In any case, roundoff
bias isn't likely to matter much one way or the other for
poorly-conditioned problems:

http://www4.ncsu.edu/~mtchu/Teaching/Lectures/MA529/chapter1.pdf

http://www.math.cmu.edu/~shlomo/VKI-Lectures/lecture1/node5.html

http://www.math.princeton.edu/~ellenber/Math204Lects/Week12.pdf

but that probably isn't what the homework problem asked about. ;-).

RM

 
 
 

doubt regarding floating point arithmetic Rounding schemes/ Truncation schemes

Post by Dragan Cvetkovi » Sat, 22 Nov 2003 05:16:54



> http://www.math.cmu.edu/~shlomo/VKI-Lectures/lecture1/node5.html
> and http://www4.ncsu.edu/~mtchu/Teaching/Lectures/MA529/chapter1.pdf are
> too mathematical

chapter1 too mathematical?! No way. It is just an intro.

Bye, Dragan

--
Dragan Cvetkovic,

To be or not to be is true. G. Boole      No it isn't.  L. E. J. Brouwer

!!! Sender/From address is bogus. Use reply-to one !!!

 
 
 

doubt regarding floating point arithmetic Rounding schemes/ Truncation schemes

Post by Nick Maclar » Sat, 22 Nov 2003 05:33:01




>CAN u please explain this part i dont understand
>True probabilistic (stochastic) rounding

Start by performing the operation to infinite precision.  Then
consider how to truncate it to fit in your representation.
Consider the bits that won't fit as a fixed point number in the
rage [0,1).  Add 1 to the bottom bit of the representation with
probability that number.

Regards,
Nick Maclaren.

 
 
 

doubt regarding floating point arithmetic Rounding schemes/ Truncation schemes

Post by Robert Myer » Sat, 22 Nov 2003 06:25:50




Quote:>http://www.math.cmu.edu/~shlomo/VKI-Lectures/lecture1/node5.html
>and http://www4.ncsu.edu/~mtchu/Teaching/Lectures/MA529/chapter1.pdf are
>too mathematical

If you skip the intimidating mathematical notation of section 1.1,
Chapter 1 is actually a pretty down-to-earth introduction of some of
the issues involved in error analysis.  It may not be a quick read for
you, but I'll bet you can understand more than you think if you give
it a chance.

The link from cmu puts you right into the middle of a real numerical
method and shows how the condition number plays a critical role in
determining whether and how a given iterative method converges.  It
isn't important that you understand the details.  For an important
class of matrices that arises often in practice, the condition number
is the ratio of the largest eigenvalue of the matrix to the smallest
and usually determines how much numerical precision you need.  The
Princeton link explains the connection between precision and condition
number.  It also shows you that the eigenvalue recipe doesn't apply to
all matrices.

If you took the trouble to follow the links, you're halfway home.

RM