Re: bug ?

*To*: mathgroup at smc.vnet.net*Subject*: [mg122363] Re: bug ?*From*: Richard Fateman <fateman at cs.berkeley.edu>*Date*: Wed, 26 Oct 2011 17:40:25 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <j862b6$5ls$1@smc.vnet.net>

On 10/25/2011 3:17 AM, swiss wrote: > Following rational function has a double pole matched by a double root at (4+\sqrt{17})/4. > The first result is correct, the other ones are false, but come without warning. > >Can somebody explain this to me? As you know, you are dividing by zero. That's why you take a limit. If you are not doing arithmetic NOT with the exact values you think you have, but some other numbers more or less nearby, you get different results. Consider a slight variation on your input 54.. -(((s (4 s - 5)^2 - (5 + 3 Sqrt[17])/8 (4 s + 1)) ((4 s - 5)^2 - 4 (5 + 3 Sqrt[17])/8 (4 s - 7)))/(16 s^2 - 32 s - 1)^2) /. s -> 1/4 (4 + Sqrt[17.0000000000000000]) which returns Indeterminate. Although -(((s (4 s - 5)^2 - (5 + 3 Sqrt[17])/8 (4 s + 1)) ((4 s - 5)^2 - 4 (5 + 3 Sqrt[17])/8 (4 s - 7)))/(16 s^2 - 32 s - 1)^2) /. s -> 1/4 (4 + Sqrt[17.000000000000000]) (one fewer "0") in there, returns without comment, 0.09375. Is this a bug? I am sure that some people will call it a feature. It is a result of compromises made in numerics. Your example on line 52 is more interesting. Let q = -(((s (4 s - 5)^2 - (5 + 3 Sqrt[17])/ 8 (4 s + 1)) ((4 s - 5)^2 - 4 (5 + 3 Sqrt[17])/8 (4 s - 7)))/(16 s^2 - 32 s - 1)^2) /. s -> 1/4 (4 + Sqrt[17]) Then Simplify[q] returns, as you show, 1/16* (1+2*Sqrt[17]) FullSimplify[q] returns the same N[q] returns 0. N[q,1] returns Indeterminate (with warning) Expand[q] returns 0 Map[N,Distribute[q]] returns -1.49737* 10^14 [A HUGE number!] Oh, for fun, take that q value and do this: qq= (q/. 17 -> r) That is, replace all the instances of 17 with r. Limit[qq, r->17] is 1/16*(1+2*Sqrt[17]) which is of course not the same as output 50, but is the result of Simplify. Some of the results about can be explained by a feature/bug that is likely to be in Mathematica. That is essentially a rule that says if one is computing a product like A * B, and one is able to simplify A to zero, there is no need to look at B. The product is zero. Probably other programs like Mathematica do the same thing, by the way. In this case B = 1/0, but B wasn't examined. So that is an explanation. It is a result of a heuristic in the Simplify program. Is it a bug or a feature? Well, it wouldn't be the first time that something in Mathematica is simultaneously declared to be a feature and produces um, mathematically questionable results. I guess that a fan of Mathematica could argue that it must be a feature because this behavior is well known to WRI and has not been changed, and WRI would change it if it were a bug, so it must be a feature. :) > In[50]:= Limit[-(((s (4 s - 5)^2 - (5 + 3 Sqrt[17])/ > 8 (4 s + 1)) ((4 s - 5)^2 - > 4 (5 + 3 Sqrt[17])/8 (4 s - 7)))/(16 s^2 - 32 s - 1)^2), > s -> 1/4 (4 + Sqrt[17])] > > Out[50]= 1/544 (331 - 19 Sqrt[17]) > > In[51]:= 1/544 (331 - 19 Sqrt[17]) // N > > Out[51]= 0.46445 > > In[52]:= -(((s (4 s - 5)^2 - (5 + 3 Sqrt[17])/ > 8 (4 s + 1)) ((4 s - 5)^2 - > 4 (5 + 3 Sqrt[17])/8 (4 s - 7)))/(16 s^2 - 32 s - 1)^2) /. > s -> 1/4 (4 + Sqrt[17]) // Simplify > > Out[52]= 1/16 (1 + 2 Sqrt[17]) > > In[53]:= 1/16 (1 + 2 Sqrt[17]) // N > > Out[53]= 0.577888 > > In[54]:= -(((s (4 s - 5)^2 - (5 + 3 Sqrt[17])/ > 8 (4 s + 1)) ((4 s - 5)^2 - > 4 (5 + 3 Sqrt[17])/8 (4 s - 7)))/(16 s^2 - 32 s - 1)^2) /. > s -> 1/4 (4 + Sqrt[17.]) > > Out[54]= 0.09375 >

**Follow-Ups**:**Re: bug ?***From:*DrMajorBob <btreat1@austin.rr.com>