Re: //N bug, but WHY?

*To*: mathgroup at smc.vnet.net*Subject*: [mg58675] Re: //N bug, but WHY?*From*: Peter Pein <petsie at dordos.net>*Date*: Thu, 14 Jul 2005 02:49:00 -0400 (EDT)*References*: <data3n$mec$1@smc.vnet.net> <db030a$r7o$1@smc.vnet.net> <db2g59$din$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Richard Fateman schrieb: > .... > >>symbio wrote: >> >> >>>Evaluating (using //N) two exact same expressions, gives wrong answer unless >> >>... > > > The same question was asked the previous day, and answered, in sci.math.symbolic. > Symbio knows about email enough to provide a bogus return address, but > doesn't know much about numerical computation, and is unwilling to learn. > So he or she asks and asks. At least Symbio found the mathematica newsgroup. > > It occurs to me that the right answer, for someone like Symbio > who really doesn't want to > be burdened by any computational knowledge, is simple: > > Never EVER use N[...], > but always N[...,<somenumber>], and hope for the best. > > > In Symbio's case he or she should just replace the > the locution ......//N > > with the locution > .....//N[#,16]& > > > and Symbio's answer comes out "right". No need for FullSimplify etc etc. > > So in this regard, there is a "bug" and it even has a "fix". > > More significantly, and the reason I bother to write again, is > the following advice. > (at least if you believe in the validity of the Mathematica > software-float numerics). > > Suggestion to WRI: > > Change the meaning of N to N[#,16]& and alter N so that > the only time the current N[] is used is by explicit > demand, e.g. N[...., DoItFasterAndMaybeWrongUsingMachineArithmetic] > > > Regards > RJF > I tried it this way: Unprotect[N]; N /: Options[N] = Union[Evaluate[Options[N]], {QuickAndDirty -> False}]; HoldPattern[N[x_, opt:(o_ -> v_)..]] := If[(QuickAndDirty /. {opt}) == True, N[x, MachinePrecision, (* NOT $MachinePrecision! *) Sequence @@ DeleteCases[{opt}, QuickAndDirty -> True]], N[x]]; HoldPattern[N[x_]] := N[x, $MachinePrecision]; N::usage = "N[expr] gives the numerical value of expr.\ N[expr, n] attempts to give a result with n-digit precision.\ N[expr,QuickAndDirty->True] will most likely ignore problems\ arising from a poorly conditioned expr."; QuickAndDirty::usage = "QuickAndDirty is a boolean option for N.\ If set to True, N will not recognize ill conditioned expressions"; Protect[N]; In[8]:= expr = Cosh[(43*Pi)/Sqrt[2]] + (1 - Cosh[43*Sqrt[2]*Pi])*Csch[43*Sqrt[2]*Pi]*Sinh[(43*Pi)/Sqrt[2]]; In[9]:=N[expr, QuickAndDirty -> True] Out[9]=-1.9342813113834067*^25 In[10]:=N[expr] From In[10]:= N::meprec: Internal precision limit $MaxExtraPrecision = 49.99999999999999` reached while evaluating Cosh[(43*Pi)/Sqrt[2]] + (1 - Cosh[43*Sqrt[2]*Pi])*Csch[43*Sqrt[2]*Pi]*Sinh[(43*Pi)/Sqrt[2]]. More... Out[10]= 0``21.45463265565461 so we see at least, that something has gone wrong. In[11]:= Block[{$MaxExtraPrecision = 86}, N[expr]] Out[11]= 6.551787517854344014050058`15.954589770191*^-42 -- Peter Pein Berlin http://people.freenet.de/Peter_Berlin/