[CSC 335] Bisection Tolerance Precision

[Andrew J. Pounds]

The eps value is the smallest number that you can add to one and get a 
value different from one.  In IEEE 64 bit precision that is 
$\frac{1}{2}^{52}$.  The other number you mention is the smallest number 
representable in the IEEE double precision format.

Remember -- in machine numbers you have a sign, an exponent, and a 
mantissa.  If you are doing tolerance calculations in some way or 
another you are really only considering values in the mantissa -- they 
may be scaled by the exponent -- but ultimately any tolerance you set 
will have to be equal to the eps (which is problematic) or related to it 
somehow greater.   The common custom is to set a tolerance that is 
greater than the eps.  The problem is how to evaluate if you have 
achieved that tolerance.  I look forward to seeing what you come up with 
for that.




On 09/14/13 19:14, \ wrote:
> Hi Dr. Pounds,
>
> I've been working on implementing the bisection method and for some
> reason the smallest tolerance, using doubles, for (b - a) / 2 that I
> can achieve is 10^ -15. I know that this is the level of precision
> that can usually be achieved using double precision, but isn't the
> minimum value what should be limiting me? And isn't that minimum value
> something like 10^ -304 (i.e. much, much smaller than 10^ -15).
>
> Thanks,
>


-- 
Andrew J. Pounds, Ph.D.  (pounds_aj at mercer.edu)
Professor of Chemistry and Computer Science
Mercer University,  Macon, GA 31207   (478) 301-5627
http://faculty.mercer.edu/pounds_aj

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