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Mathematica Wolfram Language

FullSimplify issue in Mathematica 8.0.1

Mario Weitzer

Posted 11 years ago

Input: FullSimplify[5916 + 5 Sqrt[606] >= Sqrt[6 (5835701 + 9860 Sqrt[606])]] Output: {} <= 5916 + 5 Sqrt[606]

POSTED BY: Mario Weitzer

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Szabolcs Horvát

Posted 11 years ago

It is generally a good idea to upgrade to the latest bugfix release of whichever version you have ... with Mathematica 8.0.4 I get `True` with the warning that N::meprec: "Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating 5916+5\ Sqrt[606]-Sqrt[6\(5835701+9860\Sqrt[606])]." `5916 + 5 Sqrt[606] - Sqrt[6 (5835701 + 9860 Sqrt[606])] // FullSimplify` gives `0`.

POSTED BY: Szabolcs Horvát

Jim Baldwin

Jim Baldwin, Retired

Posted 11 years ago

I get the correct answer (True) in both 9.0.1.0 and 10.0.0.0 (Windows 7) with the same warning as above.

POSTED BY: Jim Baldwin

Mario Weitzer

Posted 11 years ago

That's interesting. I restarted the kernel and now I also get the correct result. I will try to reproduce the error.

POSTED BY: Mario Weitzer

Mario Weitzer

Posted 11 years ago

I was able to reproduce the bug. Just restart your kernel and evaluate the following commands exactly as they are (will take a minute or two). The problem seems to be related to my usage of the protected symbol E as a local variable in a Block because if I replace all ocurrences of E in the definition of fSailIntersection with, say F, I get correct results. FullSimplify seems to have some sort of cache where it stores previous results and this gets corrupted by the evaluation of the code below. Still a serious bug imo. fOmega[d_] := If[MemberQ[{1, 2}, d], {0, Sqrt[d]}, {1/2, Sqrt[d]/2}]; fSailTestLessEqual[d_,S_,T_,{x_,y_}]:=Block[{},Off[N::meprec];Return[FullSimplify[(x((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2)+(S[[1]]+S[[2]]fOmega[d][[1]])(T[[1]]+T[[2]]fOmega[d][[1]])+S[[2]]T[[2]]fOmega[d][[2]]^2)^2+(y((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2)+(S[[1]]+S[[2]]fOmega[d][[1]])T[[2]]fOmega[d][[2]]-(T[[1]]+T[[2]]fOmega[d][[1]])S[[2]]fOmega[d][[2]])^2<=(S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2<=(x((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2)+(S[[1]]+S[[2]]fOmega[d][[1]])(T[[1]]+T[[2]]fOmega[d][[1]]+1)+S[[2]]T[[2]]fOmega[d][[2]]^2)^2+(y((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2)+(S[[1]]+S[[2]]fOmega[d][[1]])T[[2]]fOmega[d][[2]]-(T[[1]]+T[[2]]fOmega[d][[1]]+1)S[[2]]fOmega[d][[2]])^2&&-fOmega[d][[2]]/2<=S[[2]]fOmega[d][[2]]x+(S[[1]]+S[[2]]fOmega[d][[1]])y+T[[2]]fOmega[d][[2]]<=fOmega[d][[2]]/2]];On[N::meprec];]; fSailEdges[d_,S_,T_]:={(x-(-(S[[1]]+S[[2]]fOmega[d][[1]])(T[[1]]+T[[2]]fOmega[d][[1]])-S[[2]]T[[2]]fOmega[d][[2]]^2)/((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2))^2+(y-(-(S[[1]]+S[[2]]fOmega[d][[1]])T[[2]]fOmega[d][[2]]+(T[[1]]+T[[2]]fOmega[d][[1]])S[[2]]fOmega[d][[2]])/((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2))^2==1/((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2),(x-(-(S[[1]]+S[[2]]fOmega[d][[1]])(T[[1]]+T[[2]]fOmega[d][[1]]+1)-S[[2]]T[[2]]fOmega[d][[2]]^2)/((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2))^2+(y-(-(S[[1]]+S[[2]]fOmega[d][[1]])T[[2]]fOmega[d][[2]]+(T[[1]]+T[[2]]fOmega[d][[1]]+1)S[[2]]fOmega[d][[2]])/((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2))^2==1/((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2),-S[[2]]fOmega[d][[2]]x-(S[[1]]+S[[2]]fOmega[d][[1]])y-T[[2]]fOmega[d][[2]]==fOmega[d][[2]]/2,S[[2]]fOmega[d][[2]]x+(S[[1]]+S[[2]]fOmega[d][[1]])y+T[[2]]fOmega[d][[2]]==fOmega[d][[2]]/2}; fEqualities[d_,C_]:=Flatten[Table[Map[{{#[[1]],#[[1]]/.{Equal->Less}},{#[[2]],#[[2]]/.{Equal->GreaterEqual}},{#[[3]],#[[3]]/.{Equal->LessEqual}},{#[[4]],#[[4]]/.{Equal->Less}}}&,{fSailEdges[d,C[[i]],C[[Mod[i,Length[C]]+1]]]}][[1]],{i,1,Length[C]}],1]; fVerticesSelected[d_,C_]:=Block[{V,VT,VS,a,b},V=Select[fVertices[fEqualities[d,C]],Apply[And,Table[fSailTestLessEqual[d,C[[i]],C[[Mod[i,Length[C]]+1]],#[[1]]],{i,1,Length[C]}]]&];VT=DeleteDuplicates[Union[Map[#[[1]]&,V]],FullSimplify[#1==#2]&];VS=Table[{},{a,1,Length[VT]}];For[a=1,a<=Length[VT],{For[b=1,b<=Length[V],{If[FullSimplify[VT[[a]]==V[[b]][[1]]],VS[[a]]=Union[VS[[a]],V[[b]][[2;;3]]]],b++}],VS[[a]]=Sort[VS[[a]],MemberQ[{Less,Greater},Head[#1[[2]]]]&&MemberQ[{LessEqual,GreaterEqual},Head[#2[[2]]]]&],VS[[a]]=DeleteDuplicates[VS[[a]],FindInstance[Not[Implies[#1[[2]],#2[[2]]]],{x,y},Reals]==={}\|\|FindInstance[Not[Implies[#2[[2]],#1[[2]]]],{x,y},Reals]==={}&],VS[[a]]={VT[[a]],VS[[a]]},a++}];Return[VS];]; fSailIntersection[d_,C_]:=Block[{a=0,V={},E={},P={},PF={},A={},S={},T=0,S2={},R={}},V=fVerticesSelected[d,C];E=Table[If[i!=j,Intersection[V[[i]][[2]],V[[j]][[2]]],{}],{i,1,Length[V]},{j,1,Length[V]}];If[Complement[Union[Flatten[Map[Length[#]&,E,{2}]]],{1,0}]==={},{P=Select[Position[Map[Length[#]==1&,E,{2}],True],#[[1]]<#[[2]]&];R=Table[{V[[P[[i]][[1]]]][[1]],V[[P[[i]][[2]]]][[1]],E[[P[[i]][[1]]]][[P[[i]][[2]]]][[1]][[2]]},{i,1,Length[P]}];}];Return[R];]; Z={{-5,0},{4,2},{-1,-3},{-2,3},{5,-1},{-4,-2},{1,3},{2,-3},{-4,1},{4,1},{-2,-3},{-1,3},{4,-2}}; ZI = fSailIntersection[2, Z] ZI = fSailIntersection[2, Z] fSailTestLessEqual[2,{-4,-2},{1,3},{5065/6232-(17 Sqrt[101/6])/1558,(12 Sqrt[2])/19-1/19 Sqrt[-206+570 (5065/6232-(17 Sqrt[101/6])/1558)-361 (5065/6232-(17 Sqrt[101/6])/1558)^2]}] FullSimplify[24<=(-20+24 (5065/6232-(17 Sqrt[101/6])/1558))^2+(-8 Sqrt[2]+24 ((12 Sqrt[2])/19-1/19 Sqrt[-206+570 (5065/6232-(17 Sqrt[101/6])/1558)-361 (5065/6232-(17 Sqrt[101/6])/1558)^2]))^2] Simplify[24<=(-20+24 (5065/6232-(17 Sqrt[101/6])/1558))^2+(-8 Sqrt[2]+24 ((12 Sqrt[2])/19-1/19 Sqrt[-206+570 (5065/6232-(17 Sqrt[101/6])/1558)-361 (5065/6232-(17 Sqrt[101/6])/1558)^2]))^2] FullSimplify[5916+5 Sqrt[606]>=Sqrt[6 (5835701+9860 Sqrt[606])]] Attachments:

I was able to reproduce the bug. Just restart your kernel and evaluate the following commands exactly as they are (will take a minute or two). The problem seems to be related to my usage of the protected symbol E as a local variable in a Block because if I replace all ocurrences of E in the definition of fSailIntersection with, say F, I get correct results. FullSimplify seems to have some sort of cache where it stores previous results and this gets corrupted by the evaluation of the code below. Still a serious bug imo.

fOmega[d_] := If[MemberQ[{1, 2}, d], {0, Sqrt[d]}, {1/2, Sqrt[d]/2}];
fSailTestLessEqual[d_,S_,T_,{x_,y_}]:=Block[{},Off[N::meprec];Return[FullSimplify[(x((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2)+(S[[1]]+S[[2]]fOmega[d][[1]])(T[[1]]+T[[2]]fOmega[d][[1]])+S[[2]]T[[2]]fOmega[d][[2]]^2)^2+(y((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2)+(S[[1]]+S[[2]]fOmega[d][[1]])T[[2]]fOmega[d][[2]]-(T[[1]]+T[[2]]fOmega[d][[1]])S[[2]]fOmega[d][[2]])^2<=(S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2<=(x((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2)+(S[[1]]+S[[2]]fOmega[d][[1]])(T[[1]]+T[[2]]fOmega[d][[1]]+1)+S[[2]]T[[2]]fOmega[d][[2]]^2)^2+(y((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2)+(S[[1]]+S[[2]]fOmega[d][[1]])T[[2]]fOmega[d][[2]]-(T[[1]]+T[[2]]fOmega[d][[1]]+1)S[[2]]fOmega[d][[2]])^2&&-fOmega[d][[2]]/2<=S[[2]]fOmega[d][[2]]x+(S[[1]]+S[[2]]fOmega[d][[1]])y+T[[2]]fOmega[d][[2]]<=fOmega[d][[2]]/2]];On[N::meprec];];
fSailEdges[d_,S_,T_]:={(x-(-(S[[1]]+S[[2]]fOmega[d][[1]])(T[[1]]+T[[2]]fOmega[d][[1]])-S[[2]]T[[2]]fOmega[d][[2]]^2)/((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2))^2+(y-(-(S[[1]]+S[[2]]fOmega[d][[1]])T[[2]]fOmega[d][[2]]+(T[[1]]+T[[2]]fOmega[d][[1]])S[[2]]fOmega[d][[2]])/((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2))^2==1/((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2),(x-(-(S[[1]]+S[[2]]fOmega[d][[1]])(T[[1]]+T[[2]]fOmega[d][[1]]+1)-S[[2]]T[[2]]fOmega[d][[2]]^2)/((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2))^2+(y-(-(S[[1]]+S[[2]]fOmega[d][[1]])T[[2]]fOmega[d][[2]]+(T[[1]]+T[[2]]fOmega[d][[1]]+1)S[[2]]fOmega[d][[2]])/((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2))^2==1/((S[[1]]+S[[2]]fOmega[d][[1]])^2+(S[[2]]fOmega[d][[2]])^2),-S[[2]]fOmega[d][[2]]x-(S[[1]]+S[[2]]fOmega[d][[1]])y-T[[2]]fOmega[d][[2]]==fOmega[d][[2]]/2,S[[2]]fOmega[d][[2]]x+(S[[1]]+S[[2]]fOmega[d][[1]])y+T[[2]]fOmega[d][[2]]==fOmega[d][[2]]/2};
fEqualities[d_,C_]:=Flatten[Table[Map[{{#[[1]],#[[1]]/.{Equal->Less}},{#[[2]],#[[2]]/.{Equal->GreaterEqual}},{#[[3]],#[[3]]/.{Equal->LessEqual}},{#[[4]],#[[4]]/.{Equal->Less}}}&,{fSailEdges[d,C[[i]],C[[Mod[i,Length[C]]+1]]]}][[1]],{i,1,Length[C]}],1];
fVerticesSelected[d_,C_]:=Block[{V,VT,VS,a,b},V=Select[fVertices[fEqualities[d,C]],Apply[And,Table[fSailTestLessEqual[d,C[[i]],C[[Mod[i,Length[C]]+1]],#[[1]]],{i,1,Length[C]}]]&];VT=DeleteDuplicates[Union[Map[#[[1]]&,V]],FullSimplify[#1==#2]&];VS=Table[{},{a,1,Length[VT]}];For[a=1,a<=Length[VT],{For[b=1,b<=Length[V],{If[FullSimplify[VT[[a]]==V[[b]][[1]]],VS[[a]]=Union[VS[[a]],V[[b]][[2;;3]]]],b++}],VS[[a]]=Sort[VS[[a]],MemberQ[{Less,Greater},Head[#1[[2]]]]&&MemberQ[{LessEqual,GreaterEqual},Head[#2[[2]]]]&],VS[[a]]=DeleteDuplicates[VS[[a]],FindInstance[Not[Implies[#1[[2]],#2[[2]]]],{x,y},Reals]==={}||FindInstance[Not[Implies[#2[[2]],#1[[2]]]],{x,y},Reals]==={}&],VS[[a]]={VT[[a]],VS[[a]]},a++}];Return[VS];];
fSailIntersection[d_,C_]:=Block[{a=0,V={},E={},P={},PF={},A={},S={},T=0,S2={},R={}},V=fVerticesSelected[d,C];E=Table[If[i!=j,Intersection[V[[i]][[2]],V[[j]][[2]]],{}],{i,1,Length[V]},{j,1,Length[V]}];If[Complement[Union[Flatten[Map[Length[#]&,E,{2}]]],{1,0}]==={},{P=Select[Position[Map[Length[#]==1&,E,{2}],True],#[[1]]<#[[2]]&];R=Table[{V[[P[[i]][[1]]]][[1]],V[[P[[i]][[2]]]][[1]],E[[P[[i]][[1]]]][[P[[i]][[2]]]][[1]][[2]]},{i,1,Length[P]}];}];Return[R];];

Z={{-5,0},{4,2},{-1,-3},{-2,3},{5,-1},{-4,-2},{1,3},{2,-3},{-4,1},{4,1},{-2,-3},{-1,3},{4,-2}};
ZI = fSailIntersection[2, Z]
ZI = fSailIntersection[2, Z]

fSailTestLessEqual[2,{-4,-2},{1,3},{5065/6232-(17 Sqrt[101/6])/1558,(12 Sqrt[2])/19-1/19 Sqrt[-206+570 (5065/6232-(17 Sqrt[101/6])/1558)-361 (5065/6232-(17 Sqrt[101/6])/1558)^2]}]

FullSimplify[24<=(-20+24 (5065/6232-(17 Sqrt[101/6])/1558))^2+(-8 Sqrt[2]+24 ((12 Sqrt[2])/19-1/19 Sqrt[-206+570 (5065/6232-(17 Sqrt[101/6])/1558)-361 (5065/6232-(17 Sqrt[101/6])/1558)^2]))^2]

Simplify[24<=(-20+24 (5065/6232-(17 Sqrt[101/6])/1558))^2+(-8 Sqrt[2]+24 ((12 Sqrt[2])/19-1/19 Sqrt[-206+570 (5065/6232-(17 Sqrt[101/6])/1558)-361 (5065/6232-(17 Sqrt[101/6])/1558)^2]))^2]

FullSimplify[5916+5 Sqrt[606]>=Sqrt[6 (5835701+9860 Sqrt[606])]]

POSTED BY: Mario Weitzer

Szabolcs Horvát

Posted 11 years ago

It looks like you are running several symbolic processing functions, such as `FullSimplify` and `FindInstance`, within `Block[{E = ...}, ...]`. These functions could easily use `E` internally. They also cache partial symbolic results. So ultimately it is not surprising that doing this breaks the system. Just don't try to use builtins such as E or C as `Block` variables. Also, use `Module` unless you know you need `Block`. I would not call this a bug.

POSTED BY: Szabolcs Horvát

Frank Kampas

Frank Kampas, Physicist at Large Consulting

Posted 11 years ago

I thought Block was recommended unless you need Module.

POSTED BY: Frank Kampas

David Reiss

David Reiss, Scientific Arts

Posted 11 years ago

Nope. The opposite. Though in most cases there is no practical difference.

POSTED BY: David Reiss

Frank Kampas

Frank Kampas, Physicist at Large Consulting

Posted 11 years ago

Isn't Block faster because it doesn't create a lot of new variables?

POSTED BY: Frank Kampas

Szabolcs Horvát

Posted 11 years ago

Consider In[4]:= f[x_] := D[x a, a] In[5]:= f[b] Out[5]= b In[6]:= f[a] Out[6]= 2 a which is just one simple example of why `Block` doesn't localize the "usual way" (it uses dynamic scoping). `Block` does have its uses. If we were to implement something similar to `Table` and localize the iterator, `Block` would be the choice. Choose between `Module` and `Block` depending on what they do, not performance. They are quite different. Reading: What are the use cases for different scoping constructs?

POSTED BY: Szabolcs Horvát

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