(* Content-type: application/mathematica *)

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(* Beginning of Notebook Content *)
Notebook[{
Cell[TextData[StyleBox["In this example we use the package IntroToSymmetry.m \
to work out the point group of the 2-D Compressible Euler equations for \
isentropic flow\n\nUt+U*Ux+VUy+(1/rho)*Px=0,\n\nVt+U*Vx+V*Vy+(1/rho)*Py=0,\n\n\
Pt+U*Px+V*Py+3*P*(Ux+Vy)=0,\n\n(rho)t+U*(rho)x+V*(rho)y+rho(Ux+Vy)=0.",
 FontSize->14,
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["\<\
This next command turns off spurious spelling error warnings.\
\>", "Text"],

Cell[BoxData[
 RowBox[{"Off", "[", 
  RowBox[{"General", "::", "spell"}], "]"}]], "Input"],

Cell["Clear all symbols in the current context.", "Text",
 CellChangeTimes->{{3.4715644485587473`*^9, 3.47156448605101*^9}}],

Cell[BoxData[
 RowBox[{"ClearAll", "[", 
  RowBox[{"Evaluate", "[", 
   RowBox[{
    RowBox[{"Context", "[", "]"}], "<>", "\"\<*\>\""}], "]"}], "]"}]], "Input"],

Cell[TextData[{
 StyleBox["First read in the package which is located in User Home \
Folder/Library/",
  FontWeight->"Plain"],
 StyleBox["Mathematica",
  FontWeight->"Plain",
  FontSlant->"Italic"],
 StyleBox["/Applications/SymmetryAnalysis.",
  FontWeight->"Plain"]
}], "Text",
 CellChangeTimes->{{3.472046215957556*^9, 3.472046255573135*^9}, {
  3.472046289795615*^9, 3.472046308115048*^9}, {3.4720466725423717`*^9, 
  3.472046673357225*^9}},
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["Needs[\"SymmetryAnalysis`IntroToSymmetry`\"]", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[TextData[StyleBox["Enter the list of independent variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"independentvariables", "=", 
   RowBox[{"{", 
    RowBox[{"\"\<x\>\"", ",", "\"\<y\>\"", ",", "\"\<t\>\""}], "}"}]}], 
  ";"}]], "Input"],

Cell[TextData[StyleBox["Enter the list of dependent variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"dependentvariables", "=", 
   RowBox[{"{", 
    RowBox[{
    "\"\<U\>\"", ",", "\"\<V\>\"", ",", "\"\<P\>\"", ",", "\"\<rho\>\""}], 
    "}"}]}], ";"}]], "Input"],

Cell[TextData[StyleBox["Enter the list of function and/or constant names that \
must be preserved when the equations are expressed internally in terms of \
generic variables.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"frozennames", "=", 
   RowBox[{"{", "}"}]}], ";"}]], "Input"],

Cell[TextData[StyleBox["Enter the maximum derivative order of the input \
equation(s).",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"porder", "=", "1"}], ";"}]], "Input"],

Cell[TextData[StyleBox["The maximum derivative order that the infinitesimals \
are assumed to depend on is specified by the input parameter rorder. This \
parameter is only nonzero when the user is looking for Lie contact groups or \
Lie-Backlund groups. For the usual case where one is searching for point \
groups set rorder=0.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"rorder", "=", "0"}], ";"}]], "Input"],

Cell[TextData[StyleBox["When searching for Lie-Backlund groups (rorder=1 or \
greater) one can, without loss of generality, leave the independent variables \
untransformed. The corresponding infinitesimals (the xse's) are set to zero \
by setting xseon=0. If one is searching for point groups then set xseon=1. \
The choice xseon=1 is also an option when looking for Lie-Backlund groups and \
this can be useful when looking for contact symmetries.",
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"xseon", "=", "1"}], ";"}]], "Input"],

Cell[TextData[StyleBox["When searching for Lie-Backlund groups it is \
necessary to differentiate the input equations with respect to all the \
independent variables producing derivatives of order porder+rorder. These \
higher order differential consequences are appended to the set of rules \
applied to the invariance condition. This process is carried out \
automatically when internalrules=1. For point groups the equation or equation \
system is the only rule or set of rules needed and one sets internalrules=0.",
 
 FontWeight->"Plain"]], "Text",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"internalrules", "=", "0"}], ";"}]], "Input"],

Cell["\<\
Construct the array of rules to be applied to the invariance condition.\
\>", "Text"],

Cell["\<\
rulesarray=
{\"D[U[x,y,t],t]->-U[x,y,t]*D[U[x,y,t],x]-
V[x,y,t]*D[U[x,y,t],y]-
(1/rho[x,y,t])*D[P[x,y,t],x]\",
\"D[V[x,y,t],t]->-U[x,y,t]*D[V[x,y,t],x]-
V[x,y,t]*D[V[x,y,t],y]-
(1/rho[x,y,t])*D[P[x,y,t],y]\",
\"D[P[x,y,t],t]->-U[x,y,t]*D[P[x,y,t],x]-
V[x,y,t]*D[P[x,y,t],y]-
2*P[x,y,t]*D[U[x,y,t],x]-2*P[x,y,t]*D[V[x,y,t],y]\",
\"D[rho[x,y,t],t]->-U[x,y,t]*D[rho[x,y,t],x]-
V[x,y,t]*D[rho[x,y,t],y]-
rho[x,y,t]*D[U[x,y,t],x]-rho[x,y,t]*D[V[x,y,t],y]\"};\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["Enter the x-momentum equation as a string. ", "Text"],

Cell["\<\
inputequation1=
\"D[U[x,y,t],t]+U[x,y,t]*D[U[x,y,t],x]+
V[x,y,t]*D[U[x,y,t],y]+
(1/rho[x,y,t])*D[P[x,y,t],x]\";\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["Enter the y-momentum equation as a string. ", "Text"],

Cell["\<\
inputequation2=
\"D[V[x,y,t],t]+U[x,y,t]*D[V[x,y,t],x]+
V[x,y,t]*D[V[x,y,t],y]+
(1/rho[x,y,t])*D[P[x,y,t],y]\";\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["Enter the pressure equation as a string. ", "Text"],

Cell["\<\
inputequation3=
\"D[P[x,y,t],t]+U[x,y,t]*D[P[x,y,t],x]+
V[x,y,t]*D[P[x,y,t],y]+
2*P[x,y,t]*D[U[x,y,t],x]+2*P[x,y,t]*D[V[x,y,t],y]\";\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["Enter the continuity equation as a string. ", "Text"],

Cell["\<\
inputequation4=
\"D[rho[x,y,t],t]+U[x,y,t]*D[rho[x,y,t],x]+
V[x,y,t]*D[rho[x,y,t],y]+
rho[x,y,t]*D[U[x,y,t],x]+rho[x,y,t]*D[V[x,y,t],y]\";\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["\<\
Now work out the determining equations of the Lie point group which leaves \
equation1 invariant. The Mathematica function Timing outputs the time \
required  for the FindDeterminingEquations function to execute.\
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
Timing[FindDeterminingEquations[
independentvariables,dependentvariables,frozennames,porder,rorder,xseon,
inputequation1,rulesarray,internalrules]]\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 InterpretationBox["\<\"The function FindDetermining Equations has begun, the \
memory in use = \\!\\(41770280\\), the time used = \\!\\(69.043429`\\)\"\>",
  StringForm[
  "The function FindDetermining Equations has begun, the memory in use = ``, \
the time used = ``", 41770280, 69.043429],
  Editable->False]], "Print",
 CellChangeTimes->{3.469479559594532*^9, 3.469479618493494*^9, 
  3.471571833255919*^9, 3.48378841611314*^9}],

Cell[BoxData["\<\"The function FindDeterminingEquations is nearly complete. \
The invariance condition has been created with all rules applied. The final \
step in the generation of the determining equations is to sum together terms \
in the table of invariance condition terms (called infinitesimaltable) that \
are multiplied by the same combination of products of free y derivatives. The \
result is the table infinitesimaltablesums corresponding to matching \
y-derivative expressions. If the invariance condition is long as it often is \
this process could take a long time since it requires sorting through the \
table infinitesimaltable once for each possible combination of y derivative \
products. This is the rate limiting step in the function \
FindDeterminingEquations.Virtually all other steps are quite fast including \
the generation of the extended derivatives of the infinitesimals.\"\>"], \
"Print",
 CellChangeTimes->{3.469479559594532*^9, 3.469479618493494*^9, 
  3.471571833255919*^9, 3.483788416367415*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The determining equations have been expressed in \
terms of z-variables, the length of zdeterminingequations = \\!\\(22\\), the \
byte count of zdeterminingequations = \\!\\(6224\\), the memory in use = \
\\!\\(42193848\\), the time used = \\!\\(69.353154`\\)\"\>",
  StringForm[
  "The determining equations have been expressed in terms of z-variables, the \
length of zdeterminingequations = ``, the byte count of zdeterminingequations \
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Cell[BoxData[
 InterpretationBox["\<\"FindDeterminingEquations is done. The memory in use = \
\\!\\(42196520\\), the time used = \\!\\(69.35351299999999`\\)\"\>",
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  "FindDeterminingEquations is done. The memory in use = ``, the time used = \
``", 42196520, 69.35351299999999],
  Editable->False]], "Print",
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  3.471571833255919*^9, 3.483788416403483*^9}],

Cell[BoxData["\<\"FindDeterminingEquations has finished executing. You can \
look at the output in the table zdeterminingequations. Each entry in this \
table is a determining equation in string format expressed in terms of \
z-variables. Rules for converting between z-variables and conventional \
variables are contained in the table ztableofrules. To view the determining \
equations in terms of conventional variables use the command \
ToExpression[zdeterminingequations]/.ztableofrules. There are two other items \
the user may wish to look at; the equation converted to generic \
(x1,x2,...,y1,y2,...) variables is designated equationgenericvariables and \
the various derivatives of the equation that appear in the invariance \
condition can be viewed in the table invarconditiontable. Rules for \
converting between z-variables and generic variables are contained in the \
table ztableofrulesxy.\"\>"], "Print",
 CellChangeTimes->{3.469479559594532*^9, 3.469479618493494*^9, 
  3.471571833255919*^9, 3.483788416405478*^9}],

Cell[BoxData[
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}, Open  ]],

Cell["The determiningequations of equation1.", "Text"],

Cell["zdeterminingequations1=zdeterminingequations;", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["\<\
Now work out the determining equations of the Lie point group which leaves \
equation2 invariant.\
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
Timing[FindDeterminingEquations[
independentvariables,dependentvariables,frozennames,porder,rorder,xseon,
inputequation2,rulesarray,internalrules]]\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 InterpretationBox["\<\"The function FindDetermining Equations has begun, the \
memory in use = \\!\\(42148184\\), the time used = \\!\\(69.355423`\\)\"\>",
  StringForm[
  "The function FindDetermining Equations has begun, the memory in use = ``, \
the time used = ``", 42148184, 69.355423],
  Editable->False]], "Print",
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Cell[BoxData["\<\"The function FindDeterminingEquations is nearly complete. \
The invariance condition has been created with all rules applied. The final \
step in the generation of the determining equations is to sum together terms \
in the table of invariance condition terms (called infinitesimaltable) that \
are multiplied by the same combination of products of free y derivatives. The \
result is the table infinitesimaltablesums corresponding to matching \
y-derivative expressions. If the invariance condition is long as it often is \
this process could take a long time since it requires sorting through the \
table infinitesimaltable once for each possible combination of y derivative \
products. This is the rate limiting step in the function \
FindDeterminingEquations.Virtually all other steps are quite fast including \
the generation of the extended derivatives of the infinitesimals.\"\>"], \
"Print",
 CellChangeTimes->{3.469479560011949*^9, 3.46947961894445*^9, 
  3.4715718335883417`*^9, 3.483788416740519*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The determining equations have been expressed in \
terms of z-variables, the length of zdeterminingequations = \\!\\(22\\), the \
byte count of zdeterminingequations = \\!\\(6224\\), the memory in use = \
\\!\\(42223632\\), the time used = \\!\\(69.665385`\\)\"\>",
  StringForm[
  "The determining equations have been expressed in terms of z-variables, the \
length of zdeterminingequations = ``, the byte count of zdeterminingequations \
= ``, the memory in use = ``, the time used = ``", 22, 6224, 42223632, 
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  3.4715718335883417`*^9, 3.4837884167730417`*^9}],

Cell[BoxData[
 InterpretationBox["\<\"FindDeterminingEquations is done. The memory in use = \
\\!\\(42225096\\), the time used = \\!\\(69.665749`\\)\"\>",
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  3.4715718335883417`*^9, 3.483788416775042*^9}],

Cell[BoxData["\<\"FindDeterminingEquations has finished executing. You can \
look at the output in the table zdeterminingequations. Each entry in this \
table is a determining equation in string format expressed in terms of \
z-variables. Rules for converting between z-variables and conventional \
variables are contained in the table ztableofrules. To view the determining \
equations in terms of conventional variables use the command \
ToExpression[zdeterminingequations]/.ztableofrules. There are two other items \
the user may wish to look at; the equation converted to generic \
(x1,x2,...,y1,y2,...) variables is designated equationgenericvariables and \
the various derivatives of the equation that appear in the invariance \
condition can be viewed in the table invarconditiontable. Rules for \
converting between z-variables and generic variables are contained in the \
table ztableofrulesxy.\"\>"], "Print",
 CellChangeTimes->{3.469479560011949*^9, 3.46947961894445*^9, 
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Cell[BoxData[
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}, Open  ]],

Cell["The determiningequations of equation2.", "Text"],

Cell["zdeterminingequations2=zdeterminingequations;", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["\<\
Now work out the determining equations of the Lie point group which leaves \
equation3 invariant.\
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
Timing[FindDeterminingEquations[
independentvariables,dependentvariables,frozennames,porder,rorder,xseon,
inputequation3,rulesarray,internalrules]]\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 InterpretationBox["\<\"The function FindDetermining Equations has begun, the \
memory in use = \\!\\(42176344\\), the time used = \\!\\(69.672392`\\)\"\>",
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the time used = ``", 42176344, 69.672392],
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Cell[BoxData["\<\"The function FindDeterminingEquations is nearly complete. \
The invariance condition has been created with all rules applied. The final \
step in the generation of the determining equations is to sum together terms \
in the table of invariance condition terms (called infinitesimaltable) that \
are multiplied by the same combination of products of free y derivatives. The \
result is the table infinitesimaltablesums corresponding to matching \
y-derivative expressions. If the invariance condition is long as it often is \
this process could take a long time since it requires sorting through the \
table infinitesimaltable once for each possible combination of y derivative \
products. This is the rate limiting step in the function \
FindDeterminingEquations.Virtually all other steps are quite fast including \
the generation of the extended derivatives of the infinitesimals.\"\>"], \
"Print",
 CellChangeTimes->{3.4694795605142612`*^9, 3.469479619580062*^9, 
  3.471571834022444*^9, 3.48378841727641*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The determining equations have been expressed in \
terms of z-variables, the length of zdeterminingequations = \\!\\(29\\), the \
byte count of zdeterminingequations = \\!\\(8896\\), the memory in use = \
\\!\\(42425528\\), the time used = \\!\\(70.086466`\\)\"\>",
  StringForm[
  "The determining equations have been expressed in terms of z-variables, the \
length of zdeterminingequations = ``, the byte count of zdeterminingequations \
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  3.471571834022444*^9, 3.4837884173149633`*^9}],

Cell[BoxData[
 InterpretationBox["\<\"FindDeterminingEquations is done. The memory in use = \
\\!\\(42427192\\), the time used = \\!\\(70.08684300000002`\\)\"\>",
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Cell[BoxData["\<\"FindDeterminingEquations has finished executing. You can \
look at the output in the table zdeterminingequations. Each entry in this \
table is a determining equation in string format expressed in terms of \
z-variables. Rules for converting between z-variables and conventional \
variables are contained in the table ztableofrules. To view the determining \
equations in terms of conventional variables use the command \
ToExpression[zdeterminingequations]/.ztableofrules. There are two other items \
the user may wish to look at; the equation converted to generic \
(x1,x2,...,y1,y2,...) variables is designated equationgenericvariables and \
the various derivatives of the equation that appear in the invariance \
condition can be viewed in the table invarconditiontable. Rules for \
converting between z-variables and generic variables are contained in the \
table ztableofrulesxy.\"\>"], "Print",
 CellChangeTimes->{3.4694795605142612`*^9, 3.469479619580062*^9, 
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Cell[BoxData[
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Cell["The determiningequations of equation3.", "Text"],

Cell["zdeterminingequations3=zdeterminingequations;", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell["\<\
Now work out the determining equations of the Lie point group which leaves \
equation4 invariant.\
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
Timing[FindDeterminingEquations[
independentvariables,dependentvariables,frozennames,porder,rorder,xseon,
inputequation4,rulesarray,internalrules]]\
\>", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[BoxData[
 InterpretationBox["\<\"The function FindDetermining Equations has begun, the \
memory in use = \\!\\(42359560\\), the time used = \
\\!\\(70.08848200000001`\\)\"\>",
  StringForm[
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the time used = ``", 42359560, 70.08848200000001],
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  3.471571834456932*^9, 3.483788417366346*^9}],

Cell[BoxData["\<\"The function FindDeterminingEquations is nearly complete. \
The invariance condition has been created with all rules applied. The final \
step in the generation of the determining equations is to sum together terms \
in the table of invariance condition terms (called infinitesimaltable) that \
are multiplied by the same combination of products of free y derivatives. The \
result is the table infinitesimaltablesums corresponding to matching \
y-derivative expressions. If the invariance condition is long as it often is \
this process could take a long time since it requires sorting through the \
table infinitesimaltable once for each possible combination of y derivative \
products. This is the rate limiting step in the function \
FindDeterminingEquations.Virtually all other steps are quite fast including \
the generation of the extended derivatives of the infinitesimals.\"\>"], \
"Print",
 CellChangeTimes->{3.469479561015979*^9, 3.4694796202664413`*^9, 
  3.471571834456932*^9, 3.483788417730722*^9}],

Cell[BoxData[
 InterpretationBox["\<\"The determining equations have been expressed in \
terms of z-variables, the length of zdeterminingequations = \\!\\(32\\), the \
byte count of zdeterminingequations = \\!\\(8944\\), the memory in use = \
\\!\\(42419960\\), the time used = \\!\\(70.494883`\\)\"\>",
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Cell[BoxData[
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Cell[BoxData["\<\"FindDeterminingEquations has finished executing. You can \
look at the output in the table zdeterminingequations. Each entry in this \
table is a determining equation in string format expressed in terms of \
z-variables. Rules for converting between z-variables and conventional \
variables are contained in the table ztableofrules. To view the determining \
equations in terms of conventional variables use the command \
ToExpression[zdeterminingequations]/.ztableofrules. There are two other items \
the user may wish to look at; the equation converted to generic \
(x1,x2,...,y1,y2,...) variables is designated equationgenericvariables and \
the various derivatives of the equation that appear in the invariance \
condition can be viewed in the table invarconditiontable. Rules for \
converting between z-variables and generic variables are contained in the \
table ztableofrulesxy.\"\>"], "Print",
 CellChangeTimes->{3.469479561015979*^9, 3.4694796202664413`*^9, 
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Cell["The determiningequations of equation3.", "Text"],

Cell["zdeterminingequations4=zdeterminingequations;", "Input",
 ImageRegion->{{0, 1}, {0, 1}}],

Cell[CellGroupData[{

Cell["\<\
zdeterminingequations5=
Join[zdeterminingequations1,
zdeterminingequations2,zdeterminingequations3,zdeterminingequations4];\
\>", "Input",
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Cell["\<\
Here is the correspondence between z-variables and conventional variables.\
\>", "Text"],

Cell[CellGroupData[{

Cell["ztableofrules", "Input",
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Cell["\<\
Now solve the determining equations in terms of multivariable polynomials of \
some selected order. The Mathematica function Solve uses Gaussian elimination \
to solve a large number of linear equations for the polynomial coefficients. \
The time roughly follows

time/timeref=((number of equations)/(number of equationsref))^n

where the exponent is between 2.4 and 2.7. The Mathematica function Timing \
outputs the time required for the SolveDeterminingEquations function to \
execute.\
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
Timing[SolveDeterminingEquations[
independentvariables,dependentvariables,rorder,xseon,zdeterminingequations5,\
order=2]]\
\>", "Input",
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Cell[BoxData[
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required for each multivariate polynomial used for the infinitesimals. This \
number is determined by the choice of polynomial order and the number of \
zvariables. The time needed to solve the determining equations increases as \
powertable increases. powertablelength = \\!\\(36\\)\"\>",
  StringForm[
  "The variable powertablelength is the number of terms required for each \
multivariate polynomial used for the infinitesimals. This number is \
determined by the choice of polynomial order and the number of zvariables. \
The time needed to solve the determining equations increases as powertable \
increases. powertablelength = ``", 36],
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Cell[BoxData["\<\"The polynomial expansions have been substituted into the \
determining equations. It is now time to collect the coefficients of various \
powers of zvariables into a table called table of coefficientsall. This step \
uses the function CoefficientList and is a fairly time consuming procedure.\"\
\>"], "Print",
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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coefficients = \\!\\(803\\)\"\>",
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Cell[BoxData["\<\"Now it we are ready to use the function Solve to find the \
nonzero polynomial coefficients corresponding to the symmetries of the input \
equation(s). This can take a while.\"\>"], "Print",
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Cell[BoxData[
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Cell[BoxData["\<\"The function SolveDeterminingEquations is finished \
executing.\"\>"], "Print",
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Cell[BoxData[
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\\(70.850784`\\)\"\>",
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Cell[BoxData["\<\"You can look at the output in the tables xsefunctions and \
etafunctions. Each entry in these tables is an infinitesimal function in \
string format expressed in terms of z-variables and the group parameters. The \
output can also be viewed with the group parameters stripped away by looking \
at the table infinitesimalgroups. In either case you may wish to convert the \
z-variables to conventional variables using the table ztableofrules. \
\\n\\nKeep in mind that this function only finds solutions of the determining \
equations that are of polynomial form. The determining equations may admit \
solutions that involve transcendental functions and/or integrals. Note that \
arbitrary functions may appear in the infinitesimals and that these can be \
detected by running the package function SolveDeterminingEquations for \
several polynomial orders. If terms of ever increasing order appear, then an \
arbitrary function is indicated.\"\>"], "Print",
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Cell["\<\
Here are the infinitesimal transformation functions for the independent \
variables.\
\>", "Text"],

Cell[CellGroupData[{

Cell["xsefunctions", "Input",
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Cell[BoxData[
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Cell["and for the dependent variables.", "Text"],

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b210*z4 + (b321/2 - b428/2)*z5 + (b431*z3*z5)/2\"\>", 
   ",", "\<\"eta3[z1_, z2_, z3_, z4_, z5_, z6_, z7_]=b321*z6 + 2*b431*z3*z6\"\
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Cell["\<\
Check the groups. xse's are on the left, eta's are on the right.\
\>", "Text"],

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Cell["Check the infinitesimals in the determining equations.", "Text",
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Cell["Create the commutator table.", "Text"],

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will have terms that cancel. To get rid of these terms use the function \
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