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FYI, Tally is still broken. So are some symbolic eigenvalues

  • To: mathgroup at smc.vnet.net
  • Subject: [mg89599] FYI, Tally is still broken. So are some symbolic eigenvalues
  • From: DrMajorBob <drmajorbob at att.net>
  • Date: Sat, 14 Jun 2008 05:30:09 -0400 (EDT)
  • References: <JNEIICAJLELPIHHIMDJDOEIMCEAA.michael.weyrauch@gmx.de> <200806131010.GAA07718@smc.vnet.net> <3786945.1213366358727.JavaMail.root@m08>
  • Reply-to: drmajorbob at longhorns.com

Some symbolic eigenvalues are still broken as well, in version 6.0.3.  
Daniel Lichtblau said they'd been broken since version 5.1.

http://groups.google.com/group/comp.soft-sys.math.mathematica/browse_thread/thread/6da2b0fcc02f1b04/f1f79e15f20b313a

Like Szabolcs, I consider the Tally bug harder to forgive, since it's  
merely failing to apply a "same" test properly.

Bobby

On Fri, 13 Jun 2008 08:43:32 -0500, Szabolcs Horvát <szhorvat at gmail.com>  
wrote:

> [note: private message, not sent to MathGroup]
>
> That is very unfortunate ... it is not very surprising that Integrate
> does not always return a correct result, but Tally is a very simple
> function.  I think that it is outrageous that a broken version was
> included in several versions ...
>
> What about those nasty symbolic eigenvalue-problems that were reported
> a few months ago?  Do those work correctly now?
>
> Here is an example:
> http://groups.google.com/group/comp.soft-sys.math.mathematica/browse_thread/thread/6da2b0fcc02f1b04/f1f79e15f20b313a
>
> On Fri, Jun 13, 2008 at 13:10, DrMajorBob <drmajorbob at att.net> wrote:
>> For any who might be curious, the Tally failure detailed below (in
>> December 2007) is unchanged in version 6.0.3.
>>
>> $Version
>>
>> "6.0 for Mac OS X x86 (64-bit) (May 21, 2008)"
>>
>> (Here's a repeat of the same code, without all those >> interruptions.)
>>
>> Needs["Combinatorica`"];
>>
>> diagrams::usage = "Calculate diagrams";
>> basicedges::usage;
>> wick[a_, b_] := pair[a, b];
>> wick[a_, b__] :=
>>
>>   Sum[Expand[pair[a, {b}[[i]]] Delete[Unevaluated[wick[b]], i]], {i,
>>     Length[{b}]}];
>> ham[n_Integer] := (ns = 4*n - 3; {c[ns], c[ns + 1], d[ns + 2], d[ns  
>> + 3]});
>> basicedges[n_] :=
>>
>>   Flatten[Table[{{{i, i + 1}, EdgeColor -> Red}}, {i, 1, 2*n, 2}], 1];
>> hamserie[n_Integer] := Flatten[Table[ham[i], {i, n}]];
>> cvertices[n_Integer] := {{n, 1}, {n, 0}};
>> cvertexserie[n_Integer] := Flatten[Table[cvertices[i], {i, n}], 1];
>> pair[c[_], c[_]] := 0;
>> pair[d[_], d[_]] := 0;
>> pair[a_Integer, b_Integer] /; (a > b) := pair[b, a];
>> diagrams[n_] :=
>>
>>  Module[{wickli, rep, i, cgraph, cvertices, congraph, le, un, ta},
>>   wickli = wick[Sequence @@ hamserie[n]] /. Plus -> List;
>>   le = Length[wickli[[1]]];
>>   wickli = wickli /. {pair[c[i_], d[j_]] -> pair[i, j],
>>      pair[d[i_], c[j_]] -> pair[i, j]};
>>   graph = {}; While[wickli =!= {},
>>    wickli =
>>     rempar[First[wickli], wickli, n,
>>      le]];(*edge reduction and edgelist construction for use by> >
>>   Combinatorica*)
>>
>>   rep = Dispatch[Flatten[Table[{Rule[2*i - 1, i], Rule[2*i, i]}, {i,
>> 2*n}]]];
>>   graph = (Take[#, -le] /. rep /. pair[a__]^_ -> pair[a]) & /@ graph;
>>   be = basicedges[n];
>>   cgraph = Map[List, (graph /. {pair -> List, Times -> List}), {2}];
>>   cvertices = List /@ cvertexserie[n]; cgraph = Join[be, #] & /@ cgraph;
>>   cgraph = Graph[#, cvertices] & /@ cgraph;(*Now I compare Union and
>> Tally*)
>>
>>    saved = cgraph; un = Union[cgraph, SameTest -> IsomorphicQ];
>>   Print["Union: number of elements: ", Length[un]]; Print[GraphicsGrid[
>>     Partition[ShowGraph[#] & /@ un, 3, 3, {1, 1}, {}]]];
>>   ta = Sort@Tally[cgraph, IsomorphicQ][[All, 1]];
>>   Print["Tally: Number of Elements: ", Length[ta]];
>>   Print[GraphicsGrid[Partition[ShowGraph /@ ta, 3, 3, {1, 1}, {}]]];
>>   Print[GraphicsGrid[
>>     Partition[ShowGraph /@ ta[[{2, 4}]], 3, 3, {1, 1}, {}]]];
>>   Print["Are 2 and 4 isomorphic? ", IsomorphicQ[ta[[2]], ta[[4]]]];
>>   Print["Are 4 and 2 isomorphic? ", IsomorphicQ[ta[[4]], ta[[2]]]];
>>   ]
>> rempar[li_, wickli_List, n_Integer, le_] :=
>>  Module[{lis, mult, gem, pre, i}, lis = {Take[li, -le]}; pre = Drop[li,
>> -le];
>>   Do[lis = Join[lis, lis /. {i -> i + 1, i + 1 -> i}], {i, 1, 4*n - 1,  
>> 2}];
>>   lis = Union[lis]; mult = Length[lis];
>>   graph = Join[graph, {li*mult}];
>>   Complement[wickli, pre*lis]]
>> diagrams[3]
>>
>> Union: number of elements: 8
>>
>> Tally: Number of Elements: 11
>>
>> Are 2 and 4 isomorphic? True
>>
>> Are 4 and 2 isomorphic? True
>>
>> saved // Length
>> index = Thread[saved -> Range@Length@saved];
>> (u = Union[saved, SameTest -> IsomorphicQ]) // Length
>> (t = Sort@Tally[saved, IsomorphicQ][[All, 1]]) // Length
>>
>> 21
>>
>> 8
>>
>> 11
>>
>>
>> These are the cgraph indices returned by Union and Tally :
>>
>> u /. index
>> t /. index
>>
>> {1, 11, 4, 21, 8, 18, 15, 6}
>>
>> {1, 11, 4, 2, 8, 14, 3, 18, 15, 7, 6}
>>
>>
>> Comparing cgraph[[5]] and cgraph[[10]] is irrelevant, as you can see.
>>
>> (boo = Boole@Outer[IsomorphicQ, t, t, 1]) // MatrixForm;
>> boo == Transpose@boo
>> Cases[Position[boo, 1], {a_, b_} /; a < b, 1]
>>
>> True
>>
>> {{2, 4}, {5, 7}, {8, 10}}
>>
>> Bobby
>>
>> On Fri, 07 Dec 2007 12:27:32 -0600, Michael Weyrauch
>> <michael.weyrauch at gmx.de> wrote:
>>
>>> Dear Bobby,
>>>
>>>     thanks for asking.
>>>
>>> Yes, indeed, I reported this problem to WRI using official support
>>> channels (thanks to a service contract of my company).
>>>
>>> I got the answer from some WRI support engineer that Tally[] is indeed
>>> broken,
>>> and does not function correctly in more complicated cases. However,
>>> appyling
>>> it
>>> repeatedly until nothing changes any more does give the correct result.
>>> (The last bit I did not yet check for myself.)
>>>
>>> In practice I wrote my own Tally[], which works but  is probably much
>>> much
>>> slower
>>> than a (correctly working) built-in Tally[].
>>>
>>> I hope that in a future version Tally will work correctly, because I
>>> find it
>>> very useful in principle.
>>>
>>>
>>> Regards   Michael
>>>
>>> "DrMajorBob" <drmajorbob at bigfoot.com> schrieb im Newsbeitrag
>>> news:<fjav9f$rjb$1 at smc.vnet.net>...
>>>> Did we get an answer on whether this is a bug in Tally?
>>>>
>>>> Bobby
>>>>
>>>> On Sun, 18 Nov 2007 16:09:01 -0600, DrMajorBob  
>>>> <drmajorbob at bigfoot.com>
>>>> wrote:
>>>>
>>>> > There is, apparently, something wrong with Tally, but your test  
>>>> wasn't
>>>> > the right one, since cgraph's 5th and 10th elements were not  
>>>> returned
>>>> in
>>>
>>>>
>>>> > the Tally results. Here's a modification of your code and some  
>>>> tests:
>>>> >
>>>> > Needs["Combinatorica`"];
>>>> > diagrams::usage = "Calculate diagrams";
>>>> > basicedges::usage;
>>>> >
>>>> > wick[a_, b_] := pair[a, b];
>>>> > wick[a_, b__] :=
>>>> >    Sum[Expand[pair[a, {b}[[i]]] Delete[Unevaluated[wick[b]], i]],  
>>>> {i,
>>>> >      Length[{b}]}];
>>>> > ham[n_Integer] := (ns = 4*n - 3; {c[ns], c[ns + 1], d[ns + 2],
>>>> >      d[ns + 3]});
>>>> > basicedges[n_] :=
>>>> >    Flatten[Table[{{{i, i + 1}, EdgeColor -> Red}}, {i, 1, 2*n, 2}],
>>>> >     1];
>>>> > hamserie[n_Integer] := Flatten[Table[ham[i], {i, n}]];
>>>> > cvertices[n_Integer] := {{n, 1}, {n, 0}};
>>>> > cvertexserie[n_Integer] := Flatten[Table[cvertices[i], {i, n}], 1];
>>>> > pair[c[_], c[_]] := 0;
>>>> > pair[d[_], d[_]] := 0;
>>>> > pair[a_Integer, b_Integer] /; (a > b) := pair[b, a];
>>>> >
>>>> > diagrams[n_] :=
>>>> >    Module[{wickli, rep, i, cgraph, cvertices, congraph, le, un, ta}  
>>>> ,
>>>> >     wickli = wick[Sequence @@ hamserie[n]] /. Plus -> List;
>>>> >     le = Length[wickli[[1]]];
>>>> >     wickli =
>>>> >      wickli /. {pair[c[i_], d[j_]] -> pair[i, j],
>>>> >        pair[d[i_], c[j_]] -> pair[i, j]};
>>>> >     graph = {};
>>>> >     While[wickli =!= {},
>>>> >      wickli = rempar[First[wickli], wickli, n, le]];
>>>> >     (*edge reduction and edgelist construction for use by \
>>>> > Combinatorica*)
>>>> >     rep = Dispatch[
>>>> >       Flatten[Table[{Rule[2*i - 1, i], Rule[2*i, i]}, {i, 2*n}]]];
>>>> >     graph = (Take[#, -le] /. rep /. pair[a__]^_ -> pair[a]) & /@
>>>> >       graph;
>>>> >     be = basicedges[n];
>>>> >     cgraph = Map[List, (graph /. {pair -> List, Times -> List}),  
>>>> {2} >> ];
>>>> >     cvertices = List /@ cvertexserie[n];
>>>> >     cgraph = Join[be, #] & /@ cgraph;
>>>> >     cgraph = Graph[#, cvertices] & /@ cgraph;
>>>> >     (*Now I compare Union and Tally*)
>>>> >     saved = cgraph;
>>>> >     un = Union[cgraph, SameTest -> IsomorphicQ];
>>>> >     Print["Union: number of elements: ", Length[un]];
>>>> >     Print[GraphicsGrid[
>>>> >       Partition[ShowGraph[#] & /@ un, 3, 3, {1, 1}, {}]]];
>>>> >     ta = Sort@Tally[cgraph, IsomorphicQ][[All, 1]];
>>>> >     Print["Tally: Number of Elements: ", Length[ta]];
>>>> >     Print[GraphicsGrid[Partition[ShowGraph /@ ta, 3, 3, {1, 1},
>>
>>>> {}]]];=
>>>>
>>>> >     Print[GraphicsGrid[
>>>> >       Partition[ShowGraph /@ ta[[{2, 4}]], 3, 3, {1, 1}, {}]]];
>>>> >     Print["Are 2 and 4 isomorphic? ", IsomorphicQ[ta[[2]],  
>>>> ta[[4]]]];
>>>> >     Print["Are 4 and 2 isomorphic? ", IsomorphicQ[ta[[4]],  
>>>> ta[[2]]]];
>>>> >     ];
>>>> >
>>>> > rempar[li_, wickli_List, n_Integer, le_] :=
>>>> >    Module[{lis, mult, gem, pre, i}, lis = {Take[li, -le]};
>>>> >     pre = Drop[li, -le];
>>>> >     Do[lis = Join[lis, lis /. {i -> i + 1, i + 1 -> i}], {i, 1,
>>>> >       4*n - 1, 2}];
>>>> >     lis = Union[lis];
>>>> >     mult = Length[lis];
>>>> >     graph = Join[graph, {li*mult}];
>>>> >     Complement[wickli, pre*lis]];
>>>> >
>>>> > diagrams[3]
>>>> > Union: number of elements: 8
>>>> > Tally: Number of Elements: 11
>>>> > Are 2 and 4 isomorphic? True
>>>> > Are 4 and 2 isomorphic? True
>>>> >
>>>> > saved // Length
>>>> > index = Thread[saved -> Range@Length@saved];
>>>> > (u = Union[saved, SameTest -> IsomorphicQ]) // Length
>>>> > (t = Sort@Tally[saved, IsomorphicQ][[All, 1]]) // Length
>>>> >
>>>> > 21
>>>> >
>>>> > 8
>>>> >
>>>> > 11
>>>> >
>>>> > These are the cgraph indices returned by Union and Tally:
>>>> >
>>>> > u /. index
>>>> > t /. index
>>>> >
>>>> > {1, 11, 4, 21, 8, 18, 15, 6}
>>>> >
>>>> > {1, 11, 4, 2, 8, 14, 3, 18, 15, 7, 6}
>>>> >
>>>> > Comparing cgraph[[5]] and cgraph[[10]] is irrelevant, as you can  
>>>> see.
>>>> >
>>>> > (boo = Boole@Outer[IsomorphicQ, t, t, 1]) // MatrixForm;
>>>> > boo == Transpose@boo
>>>> > Cases[Position[boo, 1], {a_, b_} /; a < b, 1]
>>>> >
>>>> > True
>>>> >
>>>> > {{2, 4}, {5, 7}, {8, 10}}
>>>> >
>>>> > boo is NOT an identity matrix, so Tally did something very odd.
>>>> >
>>>> > Bobby
>>>> >
>>>> > On Sun, 18 Nov 2007 03:53:16 -0600, Michael Weyrauch  >>
>>>> > <michael.weyrauch at gmx.de> wrote:
>>>> >
>>>> >> Hello,
>>>> >>
>>>> >> in the  Mathematica 6.0 documentation it says in the entry for
>>>> Tally:=
>>>>   =
>>>>
>>>> >> Properties and Relations:
>>>> >>
>>>> >> "A sorted  Tally is equivalent to a list of counts for the  Union:  
>>>> "
>>>> >>
>>>> >> This is what I indeed expect of Tally and Union, in particular then
>>>> i=
>>>> t  =
>>>>
>>>> >> holds for any list:
>>>> >> Length[Tally[list]]    is equal to     Length[Union[list]].
>>>> >>
>>>> >> Now, I have an example, where Mathematica 6.0 produces a result
>>
>>>> where=
>>>>
>>>> >> Tally[list] and Union[list] are different in length, which surpris=
>> es  =
>>
>>>> =
>>>> me.
>>>> >> And in fact,  the result of Tally[ ] seems wrong to me.
>>>> >>
>>>> >> You can reproduce this result using the small Mathematica package =
>>  =
>>>>
>>>> >> enclosed, which
>>>> >> uses Combinatorica.  (Sorry for the somewhat complicated example, =
>>  =
>>
>>>> but=
>>>>  I  =
>>>>
>>>> >> did not find
>>>> >> a simpler case showing the effect.)
>>>> >>
>>>> >> If you load this package into a notebook and then execute
>>>> >>
>>>> >> diagrams[2]
>>>> >>
>>>> >> Tally and Union produce the expected result: both lists have equal=
>>  =
>>>>
>>>> >> length.
>>>> >> (The list elements are diagrams.)
>>>> >>
>>>> >> However, if you execute
>>>> >>
>>>> >> diagrams[3]
>>>> >>
>>>> >> Tally and Union produce lists of different length.
>>>> >>
>>>> >> To my opinion,  it really should never happen that Tally and Union=
>>
>>>> >> produce lists of different length. I just expect of Tally to tell =
>> me  =
>>
>>>> =
>>>>  =
>>>>
>>>> >> the multpilicities in the equivalence classes, in addition to
>>>> >> the information produced by Union.  (The two list to be compared a=
>> re  =
>>
>>>> =
>>>>  =
>>>>
>>>> >> called "ta" and "un" in the package  enclosed.)
>>>> >>
>>>> >> Strangely enough, the program compares list elements 5 and 10  ==
>>
>>>>
>>>> >> explicitly, and comes to the
>>>> >> conclusion that element 5 and 10 belong to the same equivalence  =
>>
>>>> class=
>>>> ,  =
>>>>
>>>> >> nevertheless they are
>>>> >> both listed seperately in the Tally, but - correctly - lumped up  =
>> in  =
>>
>>>> =
>>>>  =
>>>>
>>>> >> the Union.
>>>> >>
>>>> >> Do I misinterpret something here or is there a bug in Tally?  (Tal=
>> ly  =
>>
>>>> =
>>>> is  =
>>>>
>>>> >> new in Mathematica 6, and I
>>>> >> would find it extremely useful, if it would do what I expect it to=
>>  =
>>
>>>> do=
>>>> .)
>>>> >>
>>>> >> Michael
>>>> >>
>>>> >> Here comes my little package in order to reproduce the effect....
>>>> >>
>>>> >> BeginPackage["wick`"]
>>>> >>
>>>> >> Needs["Combinatorica`"];
>>>> >> diagrams::usage="Calculate diagrams";
>>>> >> basicedges::usage;
>>>> >>
>>>> >> Begin["`Private`"]
>>>> >>
>>>> >> wick[a_, b_] := pair[a, b];
>>>> >> wick[a_, b__]:= Sum[Expand[pair[a, {b}[[i]]]  =
>>>>
>>>> >> Delete[Unevaluated[wick[b]], i]], {i, Length[{b}]}];
>>>> >> ham[n_Integer]:=(ns=4*n-3;{c[ns],c[ns+1],d[ns+2],d[ns+3]});
>>>> >> basicedges[n_]:=Flatten[Table[{{{i,i+1},  EdgeColor->Red}},  ==
>>
>>>>
>>>> >> {i,1,2*n,2}],1];
>>>> >> hamserie[n_Integer]:=Flatten[Table[ham[i],{i,n}]];
>>>> >> cvertices[n_Integer]:={{n,1},{n,0}};
>>>> >> cvertexserie[n_Integer]:=Flatten[Table[cvertices[i],{i,n}],1];
>>>> >> pair[c[_],c[_]]:=0;
>>>> >> pair[d[_],d[_]]:=0;
>>>> >> pair[a_Integer,b_Integer]/;(a>b):=pair[b,a];
>>>> >>
>>>> >> diagrams[n_]:=Module[{wickli, rep, i, cgraph, cvertices, congrap=
>> h, =
>>>> le,  =
>>>>
>>>> >> un, ta},
>>>> >>
>>>> >>   wickli=wick[Sequence@@hamserie[n]]/.Plus->List;
>>>> >>   le=Length[wickli[[1]]];
>>>> >>   wickli=wickli/.{pair[c[i_],d[j_]]->pair[i,j],
>>>> >>                   pair[d[i_],c[j_]]->pair[i,j]};
>>>> >>   graph={};
>>>> >>   While[wickli=!={},
>>>> >>         wickli=rempar[First[wickli],wickli,n, le]];
>>>> >>
>>>> >>   (*edge reduction and edgelist construction for use by  =
>>
>>>> Combinatorica=
>>>> *)
>>>> >>   rep=Dispatch[Flatten[Table[{Rule[2*i-1,i],Rule[2*i,i]},{i,2*n}=
>> ]]]=
>>>> ;
>>>> >>   graph=(Take[#,-le]/.rep/.pair[a__]^_->pair[a])&/@graph;
>>>> >>
>>>> >>   be=basicedges[n];
>>>> >>   cgraph=Map[List,(graph/.{pair->List, Times->List}),{2}];
>>>> >>   cvertices=List/@cvertexserie[n];
>>>> >>   cgraph=Join[be,#]&/@cgraph;
>>>> >>   cgraph=Graph[#,cvertices]&/@cgraph;
>>>> >>
>>>> >>   (* Now I compare Union and Tally *)
>>>> >>   un=Union[cgraph,SameTest->IsomorphicQ];
>>>> >>   Print["Union: number of elements: ", Length[un]];
>>>> >>   Print[GraphicsGrid[Partition[ShowGraph[#]&/@un, 3,3,{1,1},{}]]];=
>>
>>>> >>
>>>> >>   ta=Tally[cgraph,IsomorphicQ];
>>>> >>   ta=Sort[ta];
>>>> >>   Print["Tally: Number of Elements: ", Length[ta]];
>>>> >>   Print[GraphicsGrid[Partition[ShowGraph[#]&/@(First/@ta),  =
>>>>
>>>> >> 3,3,{1,1},{}]]];
>>>> >>
>>>> >>   Print["Are 5 and 10 isomorphic? ", IsomorphicQ[cgraph[[5]],  ==
>>
>>>>
>>>> >> cgraph[[10]]]];
>>>> >>
>>>> >>   ];
>>>> >>
>>>> >> rempar[li_,wickli_List,n_Integer,le_]:=Module[{lis, mult, gem,  =
>> pre=
>>>> , i},
>>>> >>   lis={Take[li,-le]}; pre=Drop[li,-le];
>>>> >>   Do[lis=Join[lis,lis /. {i->i+1, i+1->i}], {i,1,4*n-1,2}];
>>>> >>   lis =Union[lis];
>>>> >>   mult=Length[lis];
>>>> >>   graph=Join[graph,{li*mult}];
>>>> >>   Complement[wickli,pre*lis]
>>>> >> ];
>>>> >>
>>>> >> End[];
>>>> >> EndPackage[];
>>>> >>
>>>> >>
>>>> >>
>>>> >>
>>>> >>
>>>> >>
>>>> >>
>>>> >
>>>> >
>>>> >
>>>>
>>>>
>>>>
>>>> -- =
>>>>
>>>> DrMajorBob at bigfoot.com
>>>>
>>>
>>>
>>
>>
>>
>> -- =
>>
>> DrMajorBob at longhorns.com
>>
>>
>



-- 
DrMajorBob at longhorns.com


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