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# SPDX-License-Identifier: GPL-2.0-or-later
# RingsForHomalg: Dictionaries of external rings
#
# Implementations
#
## Implementations for the rings provided by Singular.
####################################
#
# global variables:
#
####################################
BindGlobal( "CommonHomalgTableForSingularTools",
rec(
Zero := HomalgExternalRingElement( R -> homalgSendBlocking( [ "0" ], [ "number" ], R, "Zero" ), "Singular", IsZero ),
One := HomalgExternalRingElement( R -> homalgSendBlocking( [ "1" ], [ "number" ], R, "One" ), "Singular", IsOne ),
MinusOne := HomalgExternalRingElement( R -> homalgSendBlocking( [ "-1" ], [ "number" ], R, "MinusOne" ), "Singular", IsMinusOne ),
RingElement := R -> r -> homalgSendBlocking( [ r ], [ "poly" ], R, "define" ),
IsZero := r -> homalgSendBlocking( [ r, "==0" ] , "need_output", "IsZero" ) = "1",
IsOne := r -> homalgSendBlocking( [ r, "==1" ] , "need_output", "IsOne" ) = "1",
Minus :=
function( a, b )
return homalgSendBlocking( [ a, "-(", b, ")" ], [ "def" ], "Minus" );
end,
DivideByUnit :=
function( a, u )
local e;
if IsHomalgExternalRingElementRep( u ) then
e := homalgPointer( u );
else
e := u;
fi;
if e{[1]} = "-" then
#Info( InfoWarning, 1, "\033[01m\033[5;31;47mdividing by a unit starting with a minus sign:\033[0m ", e );
return homalgSendBlocking( [ "-poly(", a, ")/", e{[ 2..Length( e ) ]} ], [ "def" ], "DivideByUnit" );
else
return homalgSendBlocking( [ "poly(", a, ")/", e ], [ "def" ], "DivideByUnit" );
fi;
end,
IsUnit := #FIXME: just for polynomial rings(?)
function( R, u )
return homalgSendBlocking( [ "deg( ", u, " )" ], "need_output", "IsUnit" ) = "0";
end,
IsUnit_Z := #FIXME: just for polynomial rings(?)
function( R, u )
return homalgSendBlocking( [ "( ", u, " == 1 || ", u, " == -1 )" ], "need_output", "IsUnit" ) = "1";
end,
Sum :=
function( a, b )
return homalgSendBlocking( [ a, "+(", b, ")" ], [ "def" ], "Sum" );
end,
Product :=
function( a, b )
return homalgSendBlocking( [ "(", a, ")*(", b, ")" ], [ "def" ], "Product" );
end,
Gcd :=
function( a, b )
return homalgSendBlocking( [ "gcd(", a, b, ")" ], [ "def" ], "Gcd" );
end,
CancelGcd :=
function( a, b )
local g, a_g, b_g;
g := homalgSendBlocking( [ "gcd(", a, b, ")" ], [ "def" ], "Gcd" );
a_g := homalgSendBlocking( [ "poly(", a, ") / (", g, ")" ], [ "def" ], "CancelGcd" );
b_g := homalgSendBlocking( [ "poly(", b, ") / (", g, ")" ], [ "def" ], "CancelGcd" );
return [ a_g, b_g ];
end,
CopyElement :=
function( r, R )
return homalgSendBlocking( [ "imap(", HomalgRing( r ), r, ")" ], [ "poly" ], R, "CopyElement" );
end,
LaTeXString :=
function( poly )
local l;
l := homalgSendBlocking( [ "texpoly( \"\", ", poly, ")" ], "need_display", "homalgLaTeX" );
RemoveCharacters( l, "$" );
return l;
end,
ShallowCopy := C -> homalgSendBlocking( [ C ], [ "matrix" ], "CopyMatrix" ),
CopyMatrix :=
function( C, R )
return homalgSendBlocking( [ "imap(", HomalgRing( C ), C, ")" ], [ "matrix" ], R, "CopyMatrix" );
end,
ZeroMatrix :=
function( C )
return homalgSendBlocking( [ "0" ] , [ "matrix" ] , [ "[", NumberColumns( C ), "][", NumberRows( C ), "]" ], C, "ZeroMatrix" );
end,
IdentityMatrix :=
function( C )
return homalgSendBlocking( [ "unitmat(", NumberRows(C), ")" ] , [ "matrix" ] , [ "[", NumberRows(C), "][", NumberRows(C), "]"], C, "IdentityMatrix" );
end,
AreEqualMatrices :=
function( A, B )
return homalgSendBlocking( [ A, "==", B ] , "need_output", "AreEqualMatrices" ) = "1";
end,
Involution :=
function( M )
return homalgSendBlocking( [ "Involution(", M, ")" ], [ "matrix" ], "Involution" );
end,
TransposedMatrix :=
function( M )
return homalgSendBlocking( [ "transpose(", M, ")" ], [ "matrix" ], "TransposedMatrix" );
end,
CertainRows :=
function( M, plist )
if Length( plist ) > 1 and IsRangeRep( plist ) and Length( plist ) = AbsInt( plist[Length( plist )] - plist[1] ) + 1 then
return homalgSendBlocking( [ "submat(", M, ",1..", NumberColumns( M ), ",", plist[1] , "..", plist[Length( plist )], ")" ], [ "matrix" ], "CertainRows" );
fi;
return homalgSendBlocking( [ "submat(", M, ",1..", NumberColumns( M ), ",intvec(", plist, "))" ], [ "matrix" ], "CertainRows" );
end,
CertainColumns :=
function( M, plist )
if Length( plist ) > 1 and IsRangeRep( plist ) and Length( plist ) = AbsInt( plist[Length( plist )] - plist[1] ) + 1 then
return homalgSendBlocking( [ "submat(", M, ",", plist[1] , "..", plist[Length( plist )], ",1..", NumberRows( M ), ")" ], [ "matrix" ], "CertainColumns" );
fi;
return homalgSendBlocking( [ "submat(", M, ",intvec(", plist, "),1..", NumberRows( M ), ")" ], [ "matrix" ], "CertainColumns" );
end,
UnionOfRows :=
function( L )
local f;
f := Concatenation( [ "concat(" ], L, [ ")" ] );
return homalgSendBlocking( f, [ "matrix" ], [ "[", NumberColumns(L[1]), "][", Sum( List( L, NumberRows ) ), "]" ], "UnionOfRows" );
end,
UnionOfColumns :=
function( L )
return homalgSendBlocking( L, [ "matrix" ], [ "[", Sum( List( L, NumberColumns ) ), "][", NumberRows(L[1]), "]" ], "UnionOfColumns" );
end,
DiagMat :=
function( e )
local f;
f := Concatenation( [ "dsum(" ], e, [ ")" ] );
return homalgSendBlocking( f, [ "matrix" ], [ "[", Sum( List( e, NumberColumns ) ), "][", Sum( List( e, NumberRows ) ), "]" ], "DiagMat" );
end,
KroneckerMat :=
function( A, B )
return homalgSendBlocking( [ "tensor(", A, B, ")" ], [ "matrix" ], [ "[", NumberColumns( A ) * NumberColumns( B ), "][", NumberRows( A ) * NumberRows( B ), "]" ], "KroneckerMat" );
end,
DualKroneckerMat :=
function( A, B )
return homalgSendBlocking( [ "DualKroneckerMat(", A, B, ")" ], [ "matrix" ], [ "[", NumberColumns( A ) * NumberColumns( B ), "][", NumberRows( A ) * NumberRows( B ), "]" ], "DualKroneckerMat" );
end,
MulMat :=
function( a, A )
return homalgSendBlocking( [ "(", a, ")*", A ], [ "matrix" ], "MulMat" );
end,
MulMatRight :=
function( A, a )
return homalgSendBlocking( [ A, "*(", a, ")" ], [ "matrix" ], "MulMatRight" );
end,
AddMat :=
function( A, B )
return homalgSendBlocking( [ A, "+", B ], [ "matrix" ], "AddMat" );
end,
SubMat :=
function( A, B )
return homalgSendBlocking( [ A, "-", B ], [ "matrix" ], "SubMat" );
end,
Compose :=
function( A, B )
return homalgSendBlocking( [ B, "*", A ], [ "matrix" ], "Compose" ); ## ;-)
end,
NumberRows :=
function( C )
return StringToInt( homalgSendBlocking( [ "ncols(", C, ")" ], "need_output", "NumberRows" ) );
end,
NumberColumns :=
function( C )
return StringToInt( homalgSendBlocking( [ "nrows(", C, ")" ], "need_output", "NumberColumns" ) );
end,
Determinant :=
function( C )
return homalgSendBlocking( [ "det(", C, ")" ], [ "def" ], "Determinant" );
end,
IsZeroMatrix :=
function( M )
return homalgSendBlocking( [ "IsZeroMatrix(", M, ")" ], "need_output", "IsZeroMatrix" ) = "1";
end,
IsIdentityMatrix :=
function( M )
return homalgSendBlocking( [ "IsIdentityMatrix(", M, ")" ], "need_output", "IsIdentityMatrix" ) = "1";
end,
IsDiagonalMatrix :=
function( M )
return homalgSendBlocking( [ "IsDiagonalMatrix(", M, ")" ], "need_output", "IsDiagonalMatrix" ) = "1";
end,
ZeroRows :=
function( C )
local list_string;
list_string := homalgSendBlocking( [ "ZeroRows(", C, ")" ], "need_output", "ZeroRows" );
return StringToIntList( list_string );
end,
ZeroColumns :=
function( C )
local list_string;
list_string := homalgSendBlocking( [ "ZeroColumns(", C, ")" ], "need_output", "ZeroColumns" );
return StringToIntList( list_string );
end,
GetColumnIndependentUnitPositions :=
function( M, pos_list )
local list;
if pos_list = [ ] then
list := [ 0 ];
else
Error( "a non-empty second argument is not supported in Singular yet: ", pos_list, "\n" );
list := pos_list;
fi;
return StringToDoubleIntList( homalgSendBlocking( [ "GetColumnIndependentUnitPositions(", M, ", list (", list, "))" ], "need_output", "GetColumnIndependentUnitPositions" ) );
end,
GetColumnIndependentUnitPositions_Z :=
function( M, pos_list )
local list;
if pos_list = [ ] then
list := [ 0 ];
else
Error( "a non-empty second argument is not supported in Singular yet: ", pos_list, "\n" );
list := pos_list;
fi;
return StringToDoubleIntList( homalgSendBlocking( [ "GetColumnIndependentUnitPositions_Z(", M, ", list (", list, "))" ], "need_output", "GetColumnIndependentUnitPositions" ) );
end,
GetRowIndependentUnitPositions :=
function( M, pos_list )
local list;
if pos_list = [ ] then
list := [ 0 ];
else
Error( "a non-empty second argument is not supported in Singular yet: ", pos_list, "\n" );
list := pos_list;
fi;
return StringToDoubleIntList( homalgSendBlocking( [ "GetRowIndependentUnitPositions(", M, ", list (", list, "))" ], "need_output", "GetRowIndependentUnitPositions" ) );
end,
GetRowIndependentUnitPositions_Z :=
function( M, pos_list )
local list;
if pos_list = [ ] then
list := [ 0 ];
else
Error( "a non-empty second argument is not supported in Singular yet: ", pos_list, "\n" );
list := pos_list;
fi;
return StringToDoubleIntList( homalgSendBlocking( [ "GetRowIndependentUnitPositions_Z(", M, ", list (", list, "))" ], "need_output", "GetRowIndependentUnitPositions" ) );
end,
GetUnitPosition :=
function( M, pos_list )
local list, list_string;
if pos_list = [ ] then
list := [ 0 ];
else
list := pos_list;
fi;
list_string := homalgSendBlocking( [ "GetUnitPosition(", M, ", list (", list, "))" ], "need_output", "GetUnitPosition" );
if list_string = "fail" then
return fail;
else
return StringToIntList( list_string );
fi;
end,
GetUnitPosition_Z :=
function( M, pos_list )
local list, list_string;
if pos_list = [ ] then
list := [ 0 ];
else
list := pos_list;
fi;
list_string := homalgSendBlocking( [ "GetUnitPosition_Z(", M, ", list (", list, "))" ], "need_output", "GetUnitPosition" );
if list_string = "fail" then
return fail;
else
return StringToIntList( list_string );
fi;
end,
PositionOfFirstNonZeroEntryPerRow :=
function( M )
local L;
L := homalgSendBlocking( [ "PositionOfFirstNonZeroEntryPerRow( ", M, " )" ], "need_output", "PositionOfFirstNonZeroEntryPerRow" );
L := StringToIntList( L );
if Length( L ) = 1 then
return ListWithIdenticalEntries( NumberRows( M ), L[1] );
fi;
return L;
end,
PositionOfFirstNonZeroEntryPerColumn :=
function( M )
local L;
L := homalgSendBlocking( [ "PositionOfFirstNonZeroEntryPerColumn( ", M, " )" ], "need_output", "PositionOfFirstNonZeroEntryPerColumn" );
L := StringToIntList( L );
if Length( L ) = 1 then
return ListWithIdenticalEntries( NumberColumns( M ), L[1] );
fi;
return L;
end,
DivideEntryByUnit :=
function( M, i, j, u )
homalgSendBlocking( [ M, "[", j, i, "] =", M, "[", j, i, "]/", u ], "need_command", "DivideEntryByUnit" );
end,
DivideRowByUnit :=
function( M, i, u, j )
local v;
v := homalgStream( HomalgRing( M ) )!.variable_name;
homalgSendBlocking( [ v, "i=", i, ";", v, "j=", j, ";for(", v, "k=1;", v, "k<=", NumberColumns( M ), ";", v, "k=", v, "k+1){", M, "[", v, "k,", v, "i]=", M, "[", v, "k,", v, "i]/", u, ";};if(", v, "j>0){", M, "[", v, "j,", v, "i]=1;}" ], "need_command", "DivideRowByUnit" );
end,
DivideColumnByUnit :=
function( M, j, u, i )
local v;
v := homalgStream( HomalgRing( M ) )!.variable_name;
homalgSendBlocking( [ v, "j=", j, ";", v, "i=", i, ";for(", v, "k=1;", v, "k<=", NumberRows( M ), ";", v, "k=", v, "k+1){", M, "[", v, "j,", v, "k]=", M, "[", v, "j,", v, "k]/", u, ";};if(", v, "i>0){", M, "[", v, "j,", v, "i]=1;}" ], "need_command", "DivideColumnByUnit" );
end,
CopyRowToIdentityMatrix :=
function( M, i, L, j )
local v, l;
v := homalgStream( HomalgRing( M ) )!.variable_name;
l := Length( L );
if l > 1 and ForAll( L, IsHomalgMatrix ) then
homalgSendBlocking( [ v, "i=", i, ";", v, "j=", j, ";for(", v, "k=1;", v, "k<=", NumberColumns( M ), ";", v, "k=", v, "k+1){", L[1], "[", v, "k,", v, "j]=-", M, "[", v, "k,", v, "i];", L[2], "[", v, "k,", v, "j]=", M, "[", v, "k,", v, "i];", "};", L[1], "[", v, "j,", v, "j]=1;", L[2], "[", v, "j,", v, "j]=1" ], "need_command", "CopyRowToIdentityMatrix" );
elif l > 0 and IsHomalgMatrix( L[1] ) then
homalgSendBlocking( [ v, "i=", i, ";", v, "j=", j, ";for(", v, "k=1;", v, "k<=", NumberColumns( M ), ";", v, "k=", v, "k+1){", L[1], "[", v, "k,", v, "j]=-", M, "[", v, "k,", v, "i];};", L[1], "[", v, "j,", v, "j]=1;" ], "need_command", "CopyRowToIdentityMatrix" );
elif l > 1 and IsHomalgMatrix( L[2] ) then
homalgSendBlocking( [ v, "i=", i, ";", v, "j=", j, ";for(", v, "k=1;", v, "k<=", NumberColumns( M ), ";", v, "k=", v, "k+1){", L[2], "[", v, "k,", v, "j]=", M, "[", v, "k,", v, "i];", "};", L[2], "[", v, "j,", v, "j]=1" ], "need_command", "CopyRowToIdentityMatrix" );
fi;
end,
CopyColumnToIdentityMatrix :=
function( M, j, L, i )
local v, l;
v := homalgStream( HomalgRing( M ) )!.variable_name;
l := Length( L );
if l > 1 and ForAll( L, IsHomalgMatrix ) then
homalgSendBlocking( [ v, "j=", j, ";", v, "i=", i, ";for(", v, "k=1;", v, "k<=", NumberRows( M ), ";", v, "k=", v, "k+1){", L[1], "[", v, "i,", v, "k]=-", M, "[", v, "j,", v, "k];", L[2], "[", v, "i,", v, "k]=", M, "[", v, "j,", v, "k];", "};", L[1], "[", v, "i,", v, "i]=1;", L[2], "[", v, "i,", v, "i]=1" ], "need_command", "CopyColumnToIdentityMatrix" );
elif l > 0 and IsHomalgMatrix( L[1] ) then
homalgSendBlocking( [ v, "j=", j, ";", v, "i=", i, ";for(", v, "k=1;", v, "k<=", NumberRows( M ), ";", v, "k=", v, "k+1){", L[1], "[", v, "i,", v, "k]=-", M, "[", v, "j,", v, "k];};", L[1], "[", v, "i,", v, "i]=1;" ], "need_command", "CopyColumnToIdentityMatrix" );
elif l > 1 and IsHomalgMatrix( L[2] ) then
homalgSendBlocking( [ v, "j=", j, ";", v, "i=", i, ";for(", v, "k=1;", v, "k<=", NumberRows( M ), ";", v, "k=", v, "k+1){", L[2], "[", v, "i,", v, "k]=", M, "[", v, "j,", v, "k];", "};", L[2], "[", v, "i,", v, "i]=1" ], "need_command", "CopyColumnToIdentityMatrix" );
fi;
end,
SetColumnToZero :=
function( M, i, j )
local v;
v := homalgStream( HomalgRing( M ) )!.variable_name;
homalgSendBlocking( [ v, "i=", i, ";", v, "j=", j, ";for(", v, "k=1;", v, "k<", v, "i;", v, "k=", v, "k+1){", M, "[", v, "j,", v, "k]=0;};for(", v, "k=", v, "i+1;", v, "k<=", NumberRows( M ), ";", v, "k=", v, "k+1){", M, "[", v, "j,", v, "k]=0;}" ], "need_command", "SetColumnToZero" );
end,
GetCleanRowsPositions :=
function( M, clean_columns )
local list_string;
list_string := homalgSendBlocking( [ "GetCleanRowsPositions(", M, ", list (", clean_columns, "))" ], "need_output", "GetCleanRowsPositions" );
return StringToIntList( list_string );
end,
ConvertMatrixToRow :=
function( M )
return homalgSendBlocking( [ "ConvertMatrixToRow(", M, ")" ], [ "matrix" ], "ConvertMatrixToRow" );
end,
ConvertRowToMatrix :=
function( M, r, c )
return homalgSendBlocking( [ "ConvertRowToMatrix(", M, r, c, ")" ], [ "matrix" ], "ConvertRowToMatrix" );
end,
## determined by CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries
AffineDimension :=
function( mat )
if ZeroColumns( mat ) <> [ ] then
if not ( IsZero( mat ) and NumberRows( mat ) = 1 and NumberColumns( mat ) = 1 ) then
Error( "Singular (<= 3-1-3) does not handle nontrivial free direct summands correctly\n" );
fi;
## the only case of a free direct summand we are allowed to send to Singular (<= 3-1-3)
return Int( homalgSendBlocking( [ "dim(std(", mat, "))" ], "need_output", "AffineDimension" ) );
fi;
return Int( homalgSendBlocking( [ "dim(", mat, ")" ], "need_output", "AffineDimension" ) );
end,
CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries :=
function( mat )
local hilb;
if ZeroColumns( mat ) <> [ ] then
if not ( IsZero( mat ) and NumberRows( mat ) = 1 and NumberColumns( mat ) = 1 ) then
Error( "Singular (<= 3-1-3) does not handle nontrivial free direct summands correctly\n" );
fi;
## the only case of a free direct summand we allowed to send to Singular (<= 3-1-3)
hilb := homalgSendBlocking( [ "string(hilb(std(", mat, "),1))" ], "need_output", "HilbertPoincareSeries" );
else
hilb := homalgSendBlocking( [ "string(hilb(", mat, ",1))" ], "need_output", "HilbertPoincareSeries" );
fi;
hilb := StringToIntList( hilb );
if hilb = [ 0, 0 ] then
return [ ];
fi;
return hilb{[ 1 .. Length( hilb ) - 1 ]};
end,
### commented out since Singular (<= 3-1-3) does not handle nontrivial free direct summands correctly;
## determined by CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries
#XCoefficientsOfNumeratorOfHilbertPoincareSeries :=
# function( mat )
# local hilb;
#
# if ZeroColumns( mat ) <> [ ] and
# ## the only case of a free direct summand we can to send to Singular (<= 3-1-3)
# not ( IsZero( mat ) and NumberRows( mat ) = 1 and NumberColumns( mat ) = 1 ) then
# Error( "Singular (<= 3-1-3) does not handle nontrivial free direct summands correctly\n" );
# fi;
#
# hilb := homalgSendBlocking( [ "hilb(std(", mat, "),2)" ], "need_output", "HilbertPoincareSeries" );
#
# hilb := StringToIntList( hilb );
#
# if hilb = [ 0, 0 ] then
# return [ ];
# fi;
#
# return hilb{[ 1 .. Length( hilb ) - 1 ]};
#
# end,
CoefficientsOfUnreducedNumeratorOfWeightedHilbertPoincareSeries :=
function( mat, weights, degrees )
local R, v, hilb;
R := HomalgRing( mat );
v := homalgStream( R )!.variable_name;
if ZeroColumns( mat ) <> [ ] and
## the only case of a free direct summand we allowed to send to Singular (<= 3-1-3)
not ( IsZero( mat ) and NumberRows( mat ) = 1 and NumberColumns( mat ) = 1 ) then
Error( "Singular (<= 3-1-3) does not handle nontrivial free direct summands correctly\n" );
fi;
hilb := homalgSendBlocking( [ "CoefficientsOfUnreducedNumeratorOfWeightedHilbertPoincareSeries(", mat, ", intvec(", weights,"), intvec(", degrees, "))" ], "need_output", "HilbertPoincareSeries" );
hilb := StringToIntList( hilb );
if hilb = [ 0, 0 ] then
return [ 0 ];
fi;
return hilb;
end,
MaxDimensionalRadicalSubobject :=
function( mat )
if not NumberColumns( mat ) = 1 then
Error( "only maximal dimensional radical subobjects of one-column matrices is supported\n" );
fi;
return homalgSendBlocking( [ "matrix(equiRadical(", mat, "))" ], [ "matrix" ], "MaxDimensionalRadicalSubobject" );
end,
RadicalSubobject :=
function( mat )
if not NumberColumns( mat ) = 1 then
Error( "only radical of one-column matrices is supported\n" );
fi;
return homalgSendBlocking( [ "RadicalSubobject(", mat, ")" ], [ "matrix" ], "RadicalSubobject" );
end,
RadicalSubobject_Z :=
function( mat )
if not NumberColumns( mat ) = 1 then
Error( "only radical of one-column matrices is supported\n" );
fi;
return homalgSendBlocking( [ "RadicalSubobject_Z(", mat, ")" ], [ "matrix" ], "RadicalSubobject" );
end,
RadicalDecomposition :=
function( mat )
local R, v, c;
if not NumberColumns( mat ) = 1 then
Error( "only primary decomposition of one-column matrices is supported\n" );
fi;
R := HomalgRing( mat );
v := homalgStream( R )!.variable_name;
homalgSendBlocking( [ "list ", v, "l=RadicalDecomposition(", mat, ")" ], "need_command", "RadicalDecomposition" );
c := Int( homalgSendBlocking( [ "size(", v, "l)" ], "need_output", R, "RadicalDecomposition" ) );
return
List( [ 1 .. c ],
function( i )
local prime;
prime := HomalgVoidMatrix( "unkown_number_of_rows", 1, R );
homalgSendBlocking( [ "matrix ", prime, "[1][size(", v, "l[", i, "])]=", v, "l[", i, "]" ], "need_command", "RadicalDecomposition" );
return prime;
end
);
end,
RadicalDecomposition_Z :=
function( mat )
local R, v, c;
if not NumberColumns( mat ) = 1 then
Error( "only primary decomposition of one-column matrices is supported\n" );
fi;
R := HomalgRing( mat );
v := homalgStream( R )!.variable_name;
homalgSendBlocking( [ "list ", v, "l=RadicalDecomposition_Z(", mat, ")" ], "need_command", "RadicalDecomposition" );
c := Int( homalgSendBlocking( [ "size(", v, "l)" ], "need_output", R, "RadicalDecomposition" ) );
return
List( [ 1 .. c ],
function( i )
local prime;
prime := HomalgVoidMatrix( "unkown_number_of_rows", 1, R );
homalgSendBlocking( [ "matrix ", prime, "[1][size(", v, "l[", i, "])]=", v, "l[", i, "]" ], "need_command", "RadicalDecomposition" );
return prime;
end
);
end,
MaxDimensionalSubobject :=
function( mat )
if not NumberColumns( mat ) = 1 then
Error( "only maximal dimensional subobjects of one-column matrices is supported\n" );
fi;
return homalgSendBlocking( [ "matrix(equidimMax(", mat, "))" ], [ "matrix" ], "MaxDimensionalSubobject" );
end,
EquiDimensionalDecomposition :=
function( mat )
local R, v, c;
if not NumberColumns( mat ) = 1 then
Error( "only primary decomposition of one-column matrices is supported\n" );
fi;
R := HomalgRing( mat );
v := homalgStream( R )!.variable_name;
homalgSendBlocking( [ "list ", v, "l=equidim(", mat, ")" ], "need_command", "RadicalDecomposition" );
c := Int( homalgSendBlocking( [ "size(", v, "l)" ], "need_output", R, "RadicalDecomposition" ) );
return
List( [ 1 .. c ],
function( i )
local prime;
prime := HomalgVoidMatrix( "unkown_number_of_rows", 1, R );
homalgSendBlocking( [ "matrix ", prime, "[1][size(", v, "l[", i, "])]=", v, "l[", i, "]" ], "need_command", "RadicalDecomposition" );
return prime;
end
);
end,
PrimaryDecomposition :=
function( mat )
local R, v, c;
if not NumberColumns( mat ) = 1 then
Error( "only primary decomposition of one-column matrices is supported\n" );
fi;
R := HomalgRing( mat );
v := homalgStream( R )!.variable_name;
homalgSendBlocking( [ "list ", v, "l=PrimaryDecomposition(", mat, ")" ], "need_command", "PrimaryDecomposition" );
c := Int( homalgSendBlocking( [ "size(", v, "l)" ], "need_output", R, "PrimaryDecomposition" ) );
return
List( [ 1 .. c ],
function( i )
local primary, prime;
primary := HomalgVoidMatrix( "unkown_number_of_rows", 1, R );
prime := HomalgVoidMatrix( "unkown_number_of_rows", 1, R );
homalgSendBlocking( [ "matrix ", primary, "[1][size(", v, "l[", i, "][1])]=", v, "l[", i, "][1]" ], "need_command", "PrimaryDecomposition" );
homalgSendBlocking( [ "matrix ", prime, "[1][size(", v, "l[", i, "][2])]=", v, "l[", i, "][2]" ], "need_command", "PrimaryDecomposition" );
return [ primary, prime ];
end
);
end,
PrimaryDecomposition_Z :=
function( mat )
local R, v, c;
if not NumberColumns( mat ) = 1 then
Error( "only primary decomposition of one-column matrices is supported\n" );
fi;
R := HomalgRing( mat );
v := homalgStream( R )!.variable_name;
homalgSendBlocking( [ "list ", v, "l=PrimaryDecomposition_Z(", mat, ")" ], "need_command", "PrimaryDecomposition" );
c := Int( homalgSendBlocking( [ "size(", v, "l)" ], "need_output", R, "PrimaryDecomposition" ) );
return
List( [ 1 .. c ],
function( i )
local primary, prime;
primary := HomalgVoidMatrix( "unkown_number_of_rows", 1, R );
prime := HomalgVoidMatrix( "unkown_number_of_rows", 1, R );
homalgSendBlocking( [ "matrix ", primary, "[1][size(", v, "l[", i, "][1])]=", v, "l[", i, "][1]" ], "need_command", "PrimaryDecomposition" );
homalgSendBlocking( [ "matrix ", prime, "[1][size(", v, "l[", i, "][2])]=", v, "l[", i, "][2]" ], "need_command", "PrimaryDecomposition" );
return [ primary, prime ];
end
);
end,
Eliminate :=
function( rel, indets, R )
local elim;
elim := Product( indets );
return homalgSendBlocking( [ "matrix(eliminate(ideal(", rel, "),", elim, "))" ], [ "matrix" ], R, "Eliminate" );
end,
Coefficients :=
function( poly, var )
local R, v, vars, coeffs;
R := HomalgRing( poly );
v := homalgStream( R )!.variable_name;
var := Product( var );
homalgSendBlocking( [ "matrix ", v, "m = coef(", poly, var, ")" ], "need_command", "Coefficients" );
vars :=homalgSendBlocking( [ "submat(", v, "m,1..1,1..ncols(", v, "m))" ], [ "matrix" ], R, "Coefficients" );
coeffs := homalgSendBlocking( [ "submat(", v, "m,2..2,1..ncols(", v, "m))" ], [ "matrix" ], R, "Coefficients" );
return [ vars, coeffs ];
end,
CoefficientsMatrix :=
function( matrix, var )
local R, v, vars, coeffs;
R := HomalgRing( matrix );
v := homalgStream( R )!.variable_name;
var := Product( var );
homalgSendBlocking( [ "matrix ", v, "m = coef(ideal(", matrix, "),", var, ")" ], "need_command", "Coefficients" );
vars := homalgSendBlocking( [ "submat(", v, "m,1..1,1..ncols(", v, "m))" ], [ "matrix" ], R, "Coefficients" );
coeffs := homalgSendBlocking( [ "submat(", v, "m,2..nrows(", v,"m),1..ncols(", v, "m))" ], [ "matrix" ], R, "Coefficients" );
return [ vars, coeffs ];
end,
CoefficientsWithGivenMonomials :=
function( M, monomials )
return homalgSendBlocking( [ "coeffs(", M, ", ", monomials, ")" ], [ "matrix" ], "Coefficients" );
end,
IndicatorMatrixOfNonZeroEntries :=
function( mat )
local l;
l := StringToIntList( homalgSendBlocking( [ "IndicatorMatrixOfNonZeroEntries(", mat, ")" ], "need_output", "IndicatorMatrixOfNonZeroEntries" ) );
return ListToListList( l, NumberRows( mat ), NumberColumns( mat ) );
end,
DegreeOfRingElement :=
function( r, R )
return Int( homalgSendBlocking( [ "deg( ", r, " )" ], "need_output", "DegreeOfRingElement" ) );
end,
CoefficientsOfUnivariatePolynomial :=
function( r, var )
return homalgSendBlocking( [ "coeffs( ", r, var, " )" ], [ "matrix" ], "Coefficients" );
end,
LeadingModule :=
function( mat )
return homalgSendBlocking( [ "matrix(lead(module(", mat, ")))" ], [ "matrix" ], "LeadingModule" );
end,
MaximalDegreePart :=
function( r, weights )
return homalgSendBlocking( [ "MaximalDegreePart( ", r, ", intvec(", weights,") )" ], "need_output", "MaximalDegreePart" );
end,
MonomialMatrix :=
function( i, vars, R )
return homalgSendBlocking( [ "matrix(ideal(", vars, ")^", i, ")" ], [ "matrix" ], R, "MonomialMatrix" );
end,
MatrixOfSymbols :=
function( mat )
return homalgSendBlocking( [ "MatrixOfSymbols(", mat, ")" ], [ "matrix" ], "MatrixOfSymbols" );
end,
MatrixOfSymbols_workaround :=
function( mat )
return homalgSendBlocking( [ "MatrixOfSymbols_workaround(", mat, ")" ], [ "matrix" ], "MatrixOfSymbols" );
end,
Diff :=
function( D, N )
return homalgSendBlocking( [ "Diff(", D, N, ")" ], [ "matrix" ], "Diff" );
end,
Pullback :=
function( phi, M )
if not IsBound( phi!.RingMap ) then
phi!.RingMap :=
homalgSendBlocking( [ Source( phi ), ImagesOfRingMap( phi ) ], [ "map" ], Range( phi ), "define" );
fi;
Eval( M ); ## the following line might reset Range( phi ) to HomalgRing( M ) in order to evaluate the (eventually lazy) matrix M -> error
return homalgSendBlocking( [ phi!.RingMap, "(", M, ")" ], [ "matrix" ], Range( phi ), "Pullback" );
end,
RandomPol :=
function( R, args... )
return homalgSendBlocking( [ "sparsepoly(", args, ")" ], [ "def" ], R, "RandomPol" );
end,
RandomMat :=
function( R, r, c, args... )
local tmp;
if Length( args ) >= 2 then
# The 3rd and 4th argument of sparsematrix correspond to the 2nd and 1st argument of sparsepoly above, i.e. are swapped.
# To keep the interface consistent, we swap the corresponding entries of args here.
args := ShallowCopy( args );
tmp := args[2];
args[2] := args[1];
args[1] := tmp;
fi;
return homalgSendBlocking( [ "sparsematrix(", Concatenation( [ c, r ], args ), ")" ], [ "def" ], R, "RandomMat" );
end,
Evaluate :=
function( p, L )
# Remember here the list L is of the form var1, val1, var2, val2, ...
return homalgSendBlocking( [ "subst(", p, L, ")" ], [ "poly" ], "Evaluate" );
end,
EvaluateMatrix :=
function( M, L )
# Remember here the list L is of the form var1, val1, var2, val2, ...
return homalgSendBlocking( [ "EvaluateMatrix(", M, L, ")" ], [ "matrix" ], "Evaluate" );
end,
NumeratorAndDenominatorOfPolynomial :=
function( p )
local R, v, numer, denom;
R := HomalgRing( p );
v := homalgStream( R )!.variable_name;
homalgSendBlocking( [ "list ", v, "l=NumeratorAndDenominatorOfPolynomial(", p, ")" ], "need_command", "Numerator" );
numer := homalgSendBlocking( [ v, "l[1]" ], [ "poly" ], R, "Numerator" );
denom := homalgSendBlocking( [ v, "l[2]" ], [ "poly" ], R, "Numerator" );
numer := HomalgExternalRingElement( numer, R );
denom := HomalgExternalRingElement( denom, R );
return [ numer, denom ];
end,
NumeratorAndDenominatorOfRational :=
function( p )
local R, v, numer, denom;
R := HomalgRing( p );
v := homalgStream( R )!.variable_name;
homalgSendBlocking( [ "list ", v, "l=NumeratorAndDenominatorOfRational(", p, ")" ], "need_command", "Numerator" );
numer := homalgSendBlocking( [ v, "l[1]" ], [ "poly" ], R, "Numerator" );
denom := homalgSendBlocking( [ v, "l[2]" ], [ "poly" ], R, "Numerator" );
numer := HomalgExternalRingElement( numer, R );
denom := HomalgExternalRingElement( denom, R );
return [ numer, denom ];
end,
Inequalities :=
function( R )
local v, l;
v := homalgStream( R )!.variable_name;
homalgSendBlocking( [ "list ", v, "l=system(\"denom_list\")" ], R, "need_command", "Inequalities" );
l := Int( homalgSendBlocking( [ "size(", v, "l)" ], R, "need_output", "Inequalities" ) );
l := List( [ 1 .. l ], i -> homalgSendBlocking( [ v, "l[", i, "]" ], [ "poly" ], R, "Inequalities" ) );
return List( l, a -> HomalgExternalRingElement( a, R ) );
end,
MaximalIndependentSet :=
function( I )
local R, indets, indep;
R := HomalgRing( I );
indets := Indeterminates( R );
indep := homalgSendBlocking( [ "indepSet(ideal(", I, "))" ], "need_output", "MaximalIndependentSet" );
indep := StringToIntList( indep );
return indets{Positions( indep, 1 )};
end,
PolynomialExponents :=
function( poly )
local ring, return_string, nr_indets;
ring := HomalgRing( poly );
return_string := homalgSendBlocking( [ "string(PolynomialExponentsAndCoefficients(", poly, ")[1])" ], ring, "need_output", "Numerator" );
nr_indets := Length( Indeterminates( ring ) );
return_string := StringToIntList( return_string );
return ListToListList( return_string, Length( return_string )/nr_indets, nr_indets );
end,
PolynomialCoefficients :=
function( poly )
local ring;
ring := HomalgRing( poly );
return homalgSendBlocking( [ "string(PolynomialExponentsAndCoefficients(", poly, ")[2])" ], ring, "need_output", "Numerator" );
end,
)
);
[ Original von:0.64Diese Quellcodebibliothek enthält Beispiele in vielen Programmiersprachen.
Man kann per Verzeichnistruktur darin navigieren.
Der Code wird farblich markiert angezeigt.
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]
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