Re: Would you post this Mathematica question to news group for me, please?

*To*: MathGroup at yoda.physics.unc.edu*Subject*: Re: Would you post this Mathematica question to news group for me, please?*From*: jacobson at cello.hpl.hp.com*Date*: Mon, 06 Jul 92 09:20:16 -0700

andrew at ll.mit.edu (Andrew Hecht) writes: > I have a problem with mathematica, version 2.0 (on NeXT): When I > perform the following iteration using numbers with high precision, I > get erroneous results. For example, with > > a = > 3.750000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 > > and > > t = a/2 > > and using the following iteration technique: > > x = t; Do[Print[x]; x = a x - x^2 ,{140}] ; N[x,80] > > the results for the 135th through 140th iteration are > > 3.0, 2.0, 0.0, 0.0 > You have encounted the infamous ratcheting precision problem. It is instructive to note what is happening to the accuracy at each step. In[1]:= a=SetAccuracy[3.5,100] Out[1]= 3.5 In[10]:= x=SetAccuracy[2.5,30];Do[Print[{x,Accuracy[x]}];x=a x - x^2,{50}] {2.5, 30} {2.5, 29} {2.5, 28} {2.5, 27} {2.5, 26} {2.5, 25} {2.5, 24} {2.5, 23} {2.5, 23} {2.5, 22} {2.5, 21} {2.5, 20} {2.5, 19} {2.5, 18} {2.5, 17} {2.5, 16} {2.5, 15} {2.5, 14} {2.5, 13} {2.5, 12} {2.5, 11} {2.5, 10} {2.5, 10} {2.5, 9} {2.5, 8} {2.5, 7} {2.5, 6} {2.5, 5} {2.5, 4} {2.5, 3} {2.5, 2} {2.5, 1} {3., 0} {0., 0} {0., -1} {0., -2} {0., -5} {0., -10} {0., -20} {0., -39} {0., -78} {0., -157} {0., -313} {0., -626} {0., -1252} {0., -2505} {0., -5009} {0., -10019} {0., -20038} {0., -40075} Mathematica's problem is that it has entangled two subtlely different ideas all under the name of precision. One is an estimate of the error that might be in a calculation. The other is the number of digits to represent the data. For the first notion, it assumes that at every addition/subtraction, the errors in each operand are independent, and might combine in a worst-case way. It then uses this to estimate the precision of the result. However, in many calculations, both terms derive from the same varibles (x, in the example), and if one is off a little, the other is off a little in the same direction, so the errors in fact cancel rather than reinforce each other. In fact, in interative calculations, we don't care about error based on the original value, as the result is getting more and more accurate, not less and less accurate with each iteration. But Mathematica does not know this and lowers the precision (first definition) with each iteration. This is not a bug, but just a natural result of such a system. But, and this is a fundamental mistake, I believe, Mathematica links these two notions of precision, so when it sees a lower precision (first definition) it then throws away digits of representation (making the second notion of precision match the first). Fortunately, you can keep Mathematica from doing this by setting the variable $MinPrecision: In[12]:= x=SetAccuracy[2.5,30];\ Block[{$MinPrecision=50},Do[Print[{x,Accuracy[x]}];x=a x - x^2,{50}]] {2.5, 50} {2.5, 50} {2.5, 50} ... I am writing an article on this phenomona for the Mathematica Journal. The galleys just arrived Friday, so it should be off the presses soon. -- David Jacobson

**Follow-Ups**:**Re: Would you post this Mathematica question to news group for me, please?***From:*jacobson@cello.hpl.hp.com