Re: locating overlow/underflow (and the issue of accuracy)

*To*: mathgroup at smc.vnet.net*Subject*: [mg112277] Re: locating overlow/underflow (and the issue of accuracy)*From*: Leslaw Bieniasz <nbbienia at cyf-kr.edu.pl>*Date*: Tue, 7 Sep 2010 06:07:51 -0400 (EDT)

Hi, Well, I am afraid you do not answer my question. Assuming that I eliminated the underflow/overflow error messages by restricting down the range of y, for which I tabulate my expression, what is the accuracy of the results I obtain? Can I trust that it is 70 significant digits? This is what I need to know in the first place. If there are limitations of N[] in this respect, then how can I know if the limitations enter into play? Does the fact that I don't get error messages or warnings mean that there are no errors, or not? Apart from that, I believe that the issues of accuracy and maximum/minimum numbers are closely interrelated, since they both depend on the floating point representation used. What I don't know is whether this representation is adaptively chosen by Mathematica in order to achieve a particular goal (accuracy), or it is fixed. In the latter case, I would argue that the feature of "arbitrary precision" is a fiction. Leslaw On Tue, 7 Sep 2010, Bill Rowe wrote: > On 9/6/10 at 6:36 AM, nbbienia at cyf-kr.edu.pl (Leslaw Bieniasz) wrote: > >> Thanks for your comments. I have reduced the range of y, and I am >> now obtaining results without any error messages. However, this >> exercise raises my doubts whether the desired "arbitrary precision" >> is indeed reached, especially when y approaches 100. The program >> output yields infinity for the number of valid significant digits, >> but how many digits can I really trust if I declare 70 digits? >> Shouldn't the output contain warnings if the number of valid digits >> is smaller than 70? I don't perhaps need as many as 70, something >> like 30 would be enough, but I need to be sure that I really have >> that accuracy. I don't know how all that works, but I can imagine >> MATHEMATICA should automatically allocate appropriately more bytes >> to the floating point representations used, if the current number of >> bytes is insufficient. There are algorithms that can do that, I >> believe, like interval arithmetics? How can I check whether the >> accuracy is indeed as I want it to be? > > You are conflating two separate issues. Your comments above > center on arbitrary precision. But the issue you are having is > related to underflow/overflow. There are limits to how > large/small of a number can be represented in Mathematica's > arbitrary precision format. Specifically, these are given by > $MinNumber and $MaxNumber. On my system, I get > > In[1]:= Log[10, $MaxNumber] > > Out[1]= 3.232284580911158146609071*10^8 > > In[2]:= Log[10, $MinNumber] > > Out[2]= -3.232284580911158146609071*10^8 > > In[3]:= Log[10, Erfc[10^4]] // N > > Out[3]= -4.34295*10^7 > > That is long before you get to a value of 5 10^19 for y, > Erfc[Sqrt[y]] is outside the range of values that can be > represented in Mathematica's arbitrary precision format. > > It might be possible to obtain the number of accurate digits you > want. But this cannot be done using a simplistic approach with > Mathematica's built-in functions such as N due to the > limitations of what can be represented. > > Note, I am not saying Mathematica cannot be used to obtain what > you want. Since Mathematica can be seen as a general purpose > programming language, any mathematical problem that can be > solved on your system can be solved with the tools available in > Mathematica. But to obtain accurate digits for values that lie > outside the range of $MinNumber to $MaxNumber means you will > need to write your own code in Mathematica to deal with such > large/small numbers. Moreover, since you are using functions > like Erfc, you will need to write code to replace these as well. > This is clearly not a simple thing to do. It will not be easy to > do even if you use something other than Mathematica.