The discussion revolves around a potential issue with Mathematica 6.0 related to precision in calculations involving a small value for x. Participants explore whether the behavior observed is due to a bug or a misunderstanding of how Mathematica handles precision in numerical computations.
Participants express differing views on whether the observed behavior is a bug or a result of how precision is handled in Mathematica. There is no consensus on the nature of the issue, but multiple suggestions for addressing it are presented.
The discussion highlights limitations in understanding how Mathematica manages precision, particularly when transitioning between machine and arbitrary precision. Specific assumptions about input values and their impact on calculations are noted but remain unresolved.
In[46]:= x = 0.00004
Out[46]= 0.00004
In[45]:= N[(1 + x)^(1/3) - 1 - x/3 + (x^2)/9 - 5 (x^3)/81, 100]
Out[45]= 1.18826*10^-16
In[44]:= FullSimplify[(1 + y)^(1/3) - 1 - y/3 + y^2/9 < 5 y^3/81,
Assumptions -> y > 0]
Out[44]= True
When you do calculations with arbitrary‐precision numbers, as discussed in the previous section, Mathematica always keeps track of the precision of your results, and gives only those digits which are known to be correct, given the precision of your input. When you do calculations with machine‐precision numbers, however, Mathematica always gives you a machine‐precision result, whether or not all the digits in the result can, in fact, be determined to be correct on the basis of your input.
alphysicist said:Hi ehrenfest,
I believe the problem occurs when you give the value to x; at that point machine precision is used for x which carries over to the rest of the calculation. If you want 100 digits in arbitrary precision, you could use:
x=0.00004`100
and then calculate
N[(1 + x)^(1/3) - 1 - x/3 + (x^2)/9 - 5 (x^3)/81, 100]