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# SPDX-License-Identifier: GPL-2.0-or-later
# MatricesForHomalg: Matrices for the homalg project
#
# Implementations
#
####################################
#
# global variables:
#
####################################
HOMALG_MATRICES.colored_info :=
rec(
## Basic Operations:
## Echelon form
RowEchelonForm := [ 4, HOMALG_MATRICES.colors.BOE ],
ColumnEchelonForm := [ 4, HOMALG_MATRICES.colors.BOE ],
## reduced Echelon form
ReducedRowEchelonForm := [ 4, HOMALG_MATRICES.colors.BOE ],
ReducedColumnEchelonForm := [ 4, HOMALG_MATRICES.colors.BOE ],
## Basis
BasisOfRowModule := [ 3, HOMALG_MATRICES.colors.BOB ],
BasisOfColumnModule := [ 3, HOMALG_MATRICES.colors.BOB ],
## BasisCoeff
BasisOfRowsCoeff := [ 3, HOMALG_MATRICES.colors.BOB ],
BasisOfColumnsCoeff := [ 3, HOMALG_MATRICES.colors.BOB ],
## partially reduced Basis
PartiallyReducedBasisOfRowModule := [ 3, HOMALG_MATRICES.colors.BOB ],
PartiallyReducedBasisOfColumnModule := [ 3, HOMALG_MATRICES.colors.BOB ],
## reduced Basis
ReducedBasisOfRowModule := [ 3, HOMALG_MATRICES.colors.BOB ],
ReducedBasisOfColumnModule := [ 3, HOMALG_MATRICES.colors.BOB ],
## Decide zero
DecideZeroRows := [ 2, HOMALG_MATRICES.colors.BOD ],
DecideZeroColumns := [ 2, HOMALG_MATRICES.colors.BOD ],
## Decide zero effectively
DecideZeroRowsEffectively := [ 2, HOMALG_MATRICES.colors.BOD ],
DecideZeroColumnsEffectively := [ 2, HOMALG_MATRICES.colors.BOD ],
## solutions of Homogeneous system
SyzygiesGeneratorsOfRows := [ 2, HOMALG_MATRICES.colors.BOH ],
SyzygiesGeneratorsOfColumns := [ 2, HOMALG_MATRICES.colors.BOH ],
## relative solutions of Homogeneous system
RelativeSyzygiesGeneratorsOfRows := [ 2, HOMALG_MATRICES.colors.BOH ],
RelativeSyzygiesGeneratorsOfColumns := [ 2, HOMALG_MATRICES.colors.BOH ],
## reduced solutions of Homogeneous system
ReducedSyzygiesGeneratorsOfRows := [ 2, HOMALG_MATRICES.colors.BOH ],
ReducedSyzygiesGeneratorsOfColumns := [ 2, HOMALG_MATRICES.colors.BOH ],
);
####################################
#
# global functions:
#
####################################
##
InstallGlobalFunction( ColoredInfoForService,
function( arg )
local nargs, a, string, l, color, s;
nargs := Length( arg );
a := arg[2];
if IsString( a ) then
string := a;
if IsBound( HOMALG_MATRICES.colored_info.(string) ) then
l := HOMALG_MATRICES.colored_info.(string)[1];
color := HOMALG_MATRICES.colored_info.(string)[2];
else
l := 2;
color := "\033[1m";
fi;
elif IsList( a ) and Length( a ) = 2 and IsString( a[1] ) and IsRecord( a[2] ) then
string := a[1];
if IsBound( a[2].(string) ) then
l := a[2].(string)[1];
color := a[2].(string)[2];
else
l := 2;
color := "\033[1m";
fi;
else
string := "unknown";
l := 2;
color := "\033[1m";
fi;
if arg[1] = "busy" then
s := Concatenation( HOMALG_MATRICES.colors.busy, "BUSY>\033[0m ", color );
s := Concatenation( s, string, "\033[0m \033[7m", color );
Append( s, Concatenation( List( arg{[ 3 .. nargs ]}, function( a ) if IsStringRep( a ) then return a; else return String( a ); fi; end ) ) );
s := Concatenation( s, "\033[0m " );
if l < 4 then
Info( InfoHomalgBasicOperations, l , "" );
fi;
Info( InfoHomalgBasicOperations, l, s );
else
s := Concatenation( HOMALG_MATRICES.colors.done, "<DONE\033[0m ", color );
s := Concatenation( s, string, "\033[0m \033[7m", color );
Append( s, Concatenation( List( arg{[ 3 .. nargs ]}, function( a ) if IsStringRep( a ) then return a; else return String( a ); fi; end ) ) );
s := Concatenation( s, "\033[0m ", " in ", homalgTotalRuntimes( arg[1], "" ) );
Info( InfoHomalgBasicOperations, l, s );
if l < 4 then
Info( InfoHomalgBasicOperations, l , "" );
fi;
fi;
end );
####################################
#
# methods for operations:
#
####################################
################################################################
##
## CAUTION: !!!
##
## the following procedures never take ring relations into
## account and hence should never be part of a homalgTable
## of a homalg residue class ring
##
################################################################
if IsBound( _RowEchelonForm ) then
##
InstallOtherMethod( RowEchelonForm,
"for matrices",
[ IsMatrix ],
_RowEchelonForm );
fi;
##
InstallMethod( RowEchelonForm,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, RP, t, B;
R := HomalgRing( M );
RP := homalgTable( R );
ColoredInfoForService( "busy", "RowEchelonForm", NumberRows( M ), " x ", NumberColumns( M ), " : ", RingName( R ) );
t := homalgTotalRuntimes( );
if IsBound(RP!.RowEchelonForm) then
B := RP!.RowEchelonForm( M );
B := CertainRows( B, NonZeroRows( B ) );
ColoredInfoForService( t, "RowEchelonForm", NumberRows( B ) );
return B;
elif IsBound(RP!.ColumnEchelonForm) then
B := Involution( RP!.ColumnEchelonForm( Involution( M ) ) );
B := CertainRows( B, NonZeroRows( B ) );
ColoredInfoForService( t, "RowEchelonForm", NumberRows( B ) );
return B;
fi;
TryNextMethod( );
end );
##
InstallMethod( RowEchelonForm,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix and IsVoidMatrix ],
function( M, T )
local R, RP, t, TI, B;
R := HomalgRing( M );
RP := homalgTable( R );
ColoredInfoForService( "busy", "RowEchelonForm (M,T)", NumberRows( M ), " x ", NumberColumns( M ), " : ", RingName( R ) );
t := homalgTotalRuntimes( );
if IsBound(RP!.RowEchelonForm) then
B := RP!.RowEchelonForm( M, T );
ColoredInfoForService( t, "RowEchelonForm (M,T)", Length( NonZeroRows( B ) ) );
return B;
elif IsBound(RP!.ColumnEchelonForm) then
TI := HomalgVoidMatrix( R );
B := Involution( RP!.ColumnEchelonForm( Involution( M ), TI ) );
ColoredInfoForService( t, "RowEchelonForm (M,T)", Length( NonZeroRows( B ) ) );
SetPreEval( T, Involution( TI ) ); ResetFilterObj( T, IsVoidMatrix );
return B;
fi;
TryNextMethod( );
end );
##
InstallMethod( ColumnEchelonForm,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, RP, t, B;
R := HomalgRing( M );
RP := homalgTable( R );
if IsBound(RP!.ColumnEchelonForm) then
t := homalgTotalRuntimes( );
ColoredInfoForService( "busy", "ColumnEchelonForm", NumberRows( M ), " x ", NumberColumns( M ), " : ", RingName( R ) );
B := RP!.ColumnEchelonForm( M );
B := CertainColumns( B, NonZeroColumns( B ) );
ColoredInfoForService( t, "ColumnEchelonForm", NumberColumns( B ) );
return B;
fi;
B := Involution( RowEchelonForm( Involution( M ) ) );
return B;
end );
##
InstallMethod( ColumnEchelonForm,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix and IsVoidMatrix ],
function( M, T )
local R, RP, t, TI, B;
R := HomalgRing( M );
RP := homalgTable( R );
if IsBound(RP!.ColumnEchelonForm) then
t := homalgTotalRuntimes( );
ColoredInfoForService( "busy", "ColumnEchelonForm (M,T)", NumberRows( M ), " x ", NumberColumns( M ), " : ", RingName( R ) );
B := RP!.ColumnEchelonForm( M, T );
ColoredInfoForService( t, "ColumnEchelonForm (M,T)", Length( NonZeroColumns( B ) ) );
return B;
fi;
TI := HomalgVoidMatrix( R );
B := Involution( RowEchelonForm( Involution( M ), TI ) );
SetPreEval( T, Involution( TI ) ); ResetFilterObj( T, IsVoidMatrix );
return B;
end );
##
InstallMethod( ReducedRowEchelonForm,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, RP, t, B;
R := HomalgRing( M );
RP := homalgTable( R );
ColoredInfoForService( "busy", "ReducedRowEchelonForm", NumberRows( M ), " x ", NumberColumns( M ), " : ", RingName( R ) );
t := homalgTotalRuntimes( );
if IsBound(RP!.ReducedRowEchelonForm) then
B := RP!.ReducedRowEchelonForm( M );
ColoredInfoForService( t, "ReducedRowEchelonForm", Length( NonZeroRows( B ) ) );
return B;
elif IsBound(RP!.ReducedColumnEchelonForm) then
B := Involution( RP!.ReducedColumnEchelonForm( Involution( M ) ) );
ColoredInfoForService( t, "ReducedRowEchelonForm", Length( NonZeroRows( B ) ) );
return B;
fi;
TryNextMethod( );
end );
##
InstallMethod( ReducedRowEchelonForm,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix and IsVoidMatrix ],
function( M, T )
local R, RP, t, TI, B;
R := HomalgRing( M );
RP := homalgTable( R );
ColoredInfoForService( "busy", "ReducedRowEchelonForm (M,T)", NumberRows( M ), " x ", NumberColumns( M ), " : ", RingName( R ) );
t := homalgTotalRuntimes( );
if IsBound(RP!.ReducedRowEchelonForm) then
B := RP!.ReducedRowEchelonForm( M, T );
ColoredInfoForService( t, "ReducedRowEchelonForm (M,T)", Length( NonZeroRows( B ) ) );
return B;
elif IsBound(RP!.ReducedColumnEchelonForm) then
TI := HomalgVoidMatrix( R );
B := Involution( RP!.ReducedColumnEchelonForm( Involution( M ), TI ) );
ColoredInfoForService( t, "ReducedRowEchelonForm (M,T)", Length( NonZeroRows( B ) ) );
SetPreEval( T, Involution( TI ) ); ResetFilterObj( T, IsVoidMatrix );
return B;
fi;
TryNextMethod( );
end );
##
InstallMethod( ReducedColumnEchelonForm,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, RP, t, B;
R := HomalgRing( M );
RP := homalgTable( R );
if IsBound(RP!.ReducedColumnEchelonForm) then
t := homalgTotalRuntimes( );
ColoredInfoForService( "busy", "ReducedColumnEchelonForm", NumberRows( M ), " x ", NumberColumns( M ), " : ", RingName( R ) );
B := RP!.ReducedColumnEchelonForm( M );
ColoredInfoForService( t, "ReducedColumnEchelonForm", Length( NonZeroColumns( B ) ) );
return B;
fi;
B := Involution( ReducedRowEchelonForm( Involution( M ) ) );
return B;
end );
##
InstallMethod( ReducedColumnEchelonForm,
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix and IsVoidMatrix ],
function( M, T )
local R, RP, t, TI, B;
R := HomalgRing( M );
RP := homalgTable( R );
if IsBound(RP!.ReducedColumnEchelonForm) then
t := homalgTotalRuntimes( );
ColoredInfoForService( "busy", "ReducedColumnEchelonForm (M,T)", NumberRows( M ), " x ", NumberColumns( M ), " : ", RingName( R ) );
B := RP!.ReducedColumnEchelonForm( M, T );
ColoredInfoForService( t, "ReducedColumnEchelonForm (M,T)", Length( NonZeroColumns( B ) ) );
return B;
fi;
TI := HomalgVoidMatrix( R );
B := Involution( ReducedRowEchelonForm( Involution( M ), TI ) );
SetPreEval( T, Involution( TI ) ); ResetFilterObj( T, IsVoidMatrix );
return B;
end );
#### Basis, DecideZero, Syzygies:
## <#GAPDoc Label="BasisOfRowModule">
## <ManSection>
## <Oper Arg="M" Name="BasisOfRowModule" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Let <M>R</M> be the ring over which <A>M</A> is defined (<M>R:=</M><C>HomalgRing</C>( <A>M</A> )) and
## <M>S</M> be the row span of <A>M</A>, i.e. the <M>R</M>-submodule of the free left module
## <M>R^{(1 \times NumberColumns( <A>M</A> ))}</M> spanned by the rows of <A>M</A>. A solution to the
## <Q>submodule membership problem</Q> is an algorithm which can decide if an element <M>m</M> in
## <M>R^{(1 \times NumberColumns( <A>M</A> ))}</M> is contained in <M>S</M> or not. And exactly like
## the Gaussian (resp. Hermite) normal form when <M>R</M> is a field (resp. principal ideal ring), the row span of
## the resulting matrix <M>B</M> coincides with the row span <M>S</M> of <A>M</A>, and computing <M>B</M> is typically
## the first step of such an algorithm. (&see; Appendix <Ref Chap="Basic_Operations"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( BasisOfRowModule, ### defines: BasisOfRowModule (BasisOfModule (low-level))
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, RP, t, nr_cols, nr_rows, MI, B, nz;
## this is an attribute
R := HomalgRing( M );
RP := homalgTable( R );
t := homalgTotalRuntimes( );
nr_cols := NumberColumns( M );
ColoredInfoForService( "busy", "BasisOfRowModule", NumberRows( M ), " x ", nr_cols, " : ", RingName( R ) );
if IsBound(RP!.BasisOfRowModule) then
B := RP!.BasisOfRowModule( M );
if HasRowRankOfMatrix( B ) then
SetRowRankOfMatrix( M, RowRankOfMatrix( B ) );
fi;
SetNumberColumns( B, nr_cols );
nr_rows := NumberRows( B );
if HasIsZero( M ) and not IsZero( M ) then
## check assertion
Assert( 6, IsZero( B ) = IsZero( M ) );
SetIsZero( B, false );
elif nr_rows = 1 and IsZero( B ) then
## most computer algebra systems do not support empty matrices
## and return for a trivial BasisOfRowModule a one-row zero matrix
B := HomalgZeroMatrix( 0, nr_cols, R );
SetIsZero( M, true );
else
## check assertion
Assert( 6, R!.asserts.BasisOfRowModule( B ) );
SetIsZero( B, nr_rows = 0 );
SetIsZero( M, nr_rows = 0 );
fi;
SetIsBasisOfRowsMatrix( B, true );
if HasIsReducedBasisOfRowsMatrix( M ) and IsReducedBasisOfRowsMatrix( M ) then
SetReducedBasisOfRowModule( B, M );
fi;
ColoredInfoForService( t, "BasisOfRowModule", NumberRows( B ) );
IncreaseRingStatistics( R, "BasisOfRowModule" );
return B;
elif IsBound(RP!.BasisOfColumnModule) then
MI := Involution( M );
B := RP!.BasisOfColumnModule( MI );
if HasColumnRankOfMatrix( B ) then
SetRowRankOfMatrix( M, ColumnRankOfMatrix( B ) );
fi;
SetNumberRows( B, nr_cols );
nr_rows := NumberColumns( B ); ## this is not a mistake
if HasIsZero( M ) and not IsZero( M ) then
## check assertion
Assert( 6, IsZero( B ) = IsZero( M ) );
SetIsZero( B, false );
elif nr_rows = 1 and IsZero( B ) then
## most computer algebra systems do not support empty matrices
## and return for a trivial BasisOfColumnModule a one-column zero matrix
B := HomalgZeroMatrix( nr_cols, 0, R );
SetIsZero( M, true );
else
## check assertion
Assert( 6, R!.asserts.BasisOfColumnModule( B ) );
SetIsZero( B, nr_rows = 0 );
SetIsZero( M, nr_rows = 0 );
fi;
SetIsBasisOfColumnsMatrix( B, true );
SetBasisOfColumnModule( MI, B );
B := Involution( B );
SetIsBasisOfRowsMatrix( B, true );
if HasIsReducedBasisOfRowsMatrix( M ) and IsReducedBasisOfRowsMatrix( M ) then
SetReducedBasisOfRowModule( B, M );
fi;
ColoredInfoForService( t, "BasisOfRowModule", NumberRows( B ) );
DecreaseRingStatistics( R, "BasisOfRowModule" );
IncreaseRingStatistics( R, "BasisOfColumnModule" );
return B;
fi;
#=====# begin of the core procedure #=====#
B := ReducedRowEchelonForm( M );
nz := Length( NonZeroRows( B ) );
if nz = 0 then
B := HomalgZeroMatrix( 0, NumberColumns( B ), R );
else
B := CertainRows( B, [ 1 .. nz ] );
fi;
## check assertion
Assert( 6, R!.asserts.BasisOfRowModule( B ) );
nr_rows := NumberRows( B );
SetIsZero( B, nr_rows = 0 );
SetIsZero( M, nr_rows = 0 );
SetIsBasisOfRowsMatrix( B, true );
if HasIsReducedBasisOfRowsMatrix( M ) and IsReducedBasisOfRowsMatrix( M ) then
SetReducedBasisOfRowModule( B, M );
fi;
ColoredInfoForService( t, "BasisOfRowModule", NumberRows( B ) );
IncreaseRingStatistics( R, "BasisOfRowModule" );
return B;
end );
## <#GAPDoc Label="BasisOfColumnModule">
## <ManSection>
## <Oper Arg="M" Name="BasisOfColumnModule" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Let <M>R</M> be the ring over which <A>M</A> is defined (<M>R:=</M><C>HomalgRing</C>( <A>M</A> )) and
## <M>S</M> be the column span of <A>M</A>, i.e. the <M>R</M>-submodule of the free right module
## <M>R^{(NumberRows( <A>M</A> ) \times 1)}</M> spanned by the columns of <A>M</A>. A solution to the
## <Q>submodule membership problem</Q> is an algorithm which can decide if an element <M>m</M> in
## <M>R^{(NumberRows( <A>M</A> ) \times 1)}</M> is contained in <M>S</M> or not. And exactly like
## the Gaussian (resp. Hermite) normal form when <M>R</M> is a field (resp. principal ideal ring), the column span of
## the resulting matrix <M>B</M> coincides with the column span <M>S</M> of <A>M</A>, and computing <M>B</M> is typically
## the first step of such an algorithm. (&see; Appendix <Ref Chap="Basic_Operations"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( BasisOfColumnModule, ### defines: BasisOfColumnModule (BasisOfModule (low-level))
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, RP, t, nr_rows, nr_cols, MI, B, nz;
## this is an attribute
R := HomalgRing( M );
RP := homalgTable( R );
t := homalgTotalRuntimes( );
nr_rows := NumberRows( M );
ColoredInfoForService( "busy", "BasisOfColumnModule", nr_rows, " x ", NumberColumns( M ), " : ", RingName( R ) );
if IsBound(RP!.BasisOfColumnModule) then
B := RP!.BasisOfColumnModule( M );
if HasColumnRankOfMatrix( B ) then
SetColumnRankOfMatrix( M, ColumnRankOfMatrix( B ) );
fi;
SetNumberRows( B, nr_rows );
nr_cols := NumberColumns( B );
if HasIsZero( M ) and not IsZero( M ) then
## check assertion
Assert( 6, IsZero( B ) = IsZero( M ) );
SetIsZero( B, false );
elif nr_cols = 1 and IsZero( B ) then
## most computer algebra systems do not support empty matrices
## and return for a trivial BasisOfColumnModule a one-column zero matrix
B := HomalgZeroMatrix( nr_rows, 0, R );
SetIsZero( M, true );
else
## check assertion
Assert( 6, R!.asserts.BasisOfColumnModule( B ) );
SetIsZero( B, nr_cols = 0 );
SetIsZero( M, nr_cols = 0 );
fi;
SetIsBasisOfColumnsMatrix( B, true );
if HasIsReducedBasisOfColumnsMatrix( M ) and IsReducedBasisOfColumnsMatrix( M ) then
SetReducedBasisOfColumnModule( B, M );
fi;
ColoredInfoForService( t, "BasisOfColumnModule", NumberColumns( B ) );
IncreaseRingStatistics( R, "BasisOfColumnModule" );
return B;
elif IsBound(RP!.BasisOfRowModule) then
MI := Involution( M );
B := RP!.BasisOfRowModule( MI );
if HasRowRankOfMatrix( B ) then
SetColumnRankOfMatrix( M, RowRankOfMatrix( B ) );
fi;
SetNumberColumns( B, nr_rows );
nr_cols := NumberRows( B ); ## this is not a mistake
if HasIsZero( M ) and not IsZero( M ) then
## check assertion
Assert( 6, IsZero( B ) = IsZero( M ) );
SetIsZero( B, false );
elif nr_cols = 1 and IsZero( B ) then
## most computer algebra systems do not support empty matrices
## and return for a trivial BasisOfRowModule a one-row zero matrix
B := HomalgZeroMatrix( 0, nr_rows, R );
SetIsZero( M, true );
else
## check assertion
Assert( 6, R!.asserts.BasisOfRowModule( B ) );
SetIsZero( B, nr_cols = 0 );
SetIsZero( M, nr_cols = 0 );
fi;
SetIsBasisOfRowsMatrix( B, true );
SetBasisOfRowModule( MI, B );
B := Involution( B );
SetIsBasisOfColumnsMatrix( B, true );
if HasIsReducedBasisOfColumnsMatrix( M ) and IsReducedBasisOfColumnsMatrix( M ) then
SetReducedBasisOfColumnModule( B, M );
fi;
ColoredInfoForService( t, "BasisOfColumnModule", NumberColumns( B ) );
DecreaseRingStatistics( R, "BasisOfColumnModule" );
IncreaseRingStatistics( R, "BasisOfRowModule" );
return B;
fi;
#=====# begin of the core procedure #=====#
B := ReducedColumnEchelonForm( M );
nz := Length( NonZeroColumns( B ) );
if nz = 0 then
B := HomalgZeroMatrix( NumberRows( B ), 0, R );
else
B := CertainColumns( B, [ 1 .. nz ] );
fi;
## check assertion
Assert( 6, R!.asserts.BasisOfColumnModule( B ) );
nr_cols := NumberColumns( B );
SetIsZero( B, nr_cols = 0 );
SetIsZero( M, nr_cols = 0 );
SetIsBasisOfColumnsMatrix( B, true );
if HasIsReducedBasisOfColumnsMatrix( M ) and IsReducedBasisOfColumnsMatrix( M ) then
SetReducedBasisOfColumnModule( B, M );
fi;
ColoredInfoForService( t, "BasisOfColumnModule", NumberColumns( B ) );
IncreaseRingStatistics( R, "BasisOfColumnModule" );
return B;
end );
## <#GAPDoc Label="DecideZeroRows">
## <ManSection>
## <Oper Arg="A, B" Name="DecideZeroRows" Label="for pairs of matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Let <A>A</A> and <A>B</A> be matrices having the same number of columns and defined over the same ring <M>R</M>
## (<M>:=</M><C>HomalgRing</C>( <A>A</A> )) and <M>S</M> be the row span of <A>B</A>,
## i.e. the <M>R</M>-submodule of the free left module <M>R^{(1 \times NumberColumns( <A>B</A> ))}</M>
## spanned by the rows of <A>B</A>. The result is a matrix <M>C</M> having the same shape as <A>A</A>,
## for which the <M>i</M>-th row <M><A>C</A>^i</M> is equivalent to the <M>i</M>-th row <M><A>A</A>^i</M> of <A>A</A> modulo <M>S</M>,
## i.e. <M><A>C</A>^i-<A>A</A>^i</M> is an element of the row span <M>S</M> of <A>B</A>. Moreover, the row <M><A>C</A>^i</M> is zero,
## if and only if the row <M><A>A</A>^i</M> is an element of <M>S</M>. So <C>DecideZeroRows</C> decides which rows of <A>A</A>
## are zero modulo the rows of <A>B</A>. (&see; Appendix <Ref Chap="Basic_Operations"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( DecideZeroRows, ### defines: DecideZeroRows (Reduce)
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ],
function( A, B )
local R, RP, t, l, m, n, id, zz, M, C;
## we can only reduce with respect to a distinguished basis
B := BasisOfRowModule( B );
if IsBound( A!.DecideZeroRows ) and
IsIdenticalObj( A!.DecideZeroRows, B ) then
return A;
fi;
if not ( IsMutable( A ) or IsMutable( B ) ) then
if IsBound( B!.DecideZeroRows ) then
C := _ElmWPObj_ForHomalg( B!.DecideZeroRows, A, fail );
if C <> fail then
return C;
fi;
else
B!.DecideZeroRows :=
ContainerForWeakPointers(
TheTypeContainerForWeakPointersOnComputedValues,
[ "operation", "DecideZeroRows" ] );
fi;
fi;
### causes many IsZero external calls without
### reducing the effective number of DecideZero calls;
### observed with Purity.g in ExamplesForHomalg
## if IsZero( A ) or IsZero( B ) then
## ## redispatch to the specialized methods
## return DecideZeroRows( A, B );
## fi;
if HasIsZero( B ) and IsZero( B ) then
## redispatch to the specialized methods
return DecideZeroRows( A, B );
fi;
R := HomalgRing( B );
R!.asserts.DecideZeroRowsWRTNonBasis( B );
RP := homalgTable( R );
t := homalgTotalRuntimes( );
l := NumberRows( A );
m := NumberColumns( A );
n := NumberRows( B );
ColoredInfoForService( "busy", "DecideZeroRows", "( ", l, " + ", n, " ) x ", m, " : ", RingName( R ) );
if IsBound(RP!.DecideZeroRows) then
C := RP!.DecideZeroRows( A, B );
SetNumberRows( C, l ); SetNumberColumns( C, m );
if not IsZero( C ) then
Assert( 6, not IsZero( A ) );
SetIsZero( A, false );
fi;
C!.DecideZeroRows := B;
if not ( IsMutable( A ) or IsMutable( B ) ) then
_AddTwoElmWPObj_ForHomalg( B!.DecideZeroRows, A, C );
fi;
ColoredInfoForService( t, "DecideZeroRows" );
IncreaseRingStatistics( R, "DecideZeroRows" );
return C;
elif IsBound(RP!.DecideZeroColumns) then
C := RP!.DecideZeroColumns( Involution( A ), Involution( B ) );
SetNumberRows( C, m ); SetNumberColumns( C, l );
if not IsZero( C ) then
Assert( 6, not IsZero( A ) );
SetIsZero( A, false );
fi;
C := Involution( C );
C!.DecideZeroRows := B;
if not ( IsMutable( A ) or IsMutable( B ) ) then
_AddTwoElmWPObj_ForHomalg( B!.DecideZeroRows, A, C );
fi;
ColoredInfoForService( t, "DecideZeroRows" );
DecreaseRingStatistics( R, "DecideZeroRows" );
IncreaseRingStatistics( R, "DecideZeroColumns" );
return C;
fi;
#=====# begin of the core procedure #=====#
if HasIsOne( A ) and IsOne( A ) then ## save as much new definitions as possible
id := A;
else
id := HomalgIdentityMatrix( l, R );
fi;
zz := HomalgZeroMatrix( n, l, R );
M := UnionOfRows( UnionOfColumns( id, A ), UnionOfColumns( zz, B ) );
M := ReducedRowEchelonForm( M );
C := CertainRows( CertainColumns( M, [ l + 1 .. l + m ] ), [ 1 .. l ] );
if not IsZero( C ) then
Assert( 6, not IsZero( A ) );
SetIsZero( A, false );
fi;
C!.DecideZeroRows := B;
if not ( IsMutable( A ) or IsMutable( B ) ) then
_AddTwoElmWPObj_ForHomalg( B!.DecideZeroRows, A, C );
fi;
ColoredInfoForService( t, "DecideZeroRows" );
IncreaseRingStatistics( R, "DecideZeroRows" );
return C;
end );
## <#GAPDoc Label="DecideZeroColumns">
## <ManSection>
## <Oper Arg="A, B" Name="DecideZeroColumns" Label="for pairs of matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Let <A>A</A> and <A>B</A> be matrices having the same number of rows and defined over the same ring <M>R</M>
## (<M>:=</M><C>HomalgRing</C>( <A>A</A> )) and <M>S</M> be the column span of <A>B</A>,
## i.e. the <M>R</M>-submodule of the free right module <M>R^{(NumberRows( <A>B</A> ) \times 1)}</M>
## spanned by the columns of <A>B</A>. The result is a matrix <M>C</M> having the same shape as <A>A</A>,
## for which the <M>i</M>-th column <M><A>C</A>_i</M> is equivalent to the <M>i</M>-th column <M><A>A</A>_i</M> of <A>A</A>
## modulo <M>S</M>, i.e. <M><A>C</A>_i-<A>A</A>_i</M> is an element of the column span <M>S</M> of <A>B</A>. Moreover,
## the column <M><A>C</A>_i</M> is zero, if and only if the column <M><A>A</A>_i</M> is an element of <M>S</M>.
## So <C>DecideZeroColumns</C> decides which columns of <A>A</A> are zero modulo the columns of <A>B</A>.
## (&see; Appendix <Ref Chap="Basic_Operations"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( DecideZeroColumns, ### defines: DecideZeroColumns (Reduce)
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ],
function( A, B )
local R, RP, t, l, m, n, id, zz, M, C;
## we can only reduce with respect to a distinguished basis
B := BasisOfColumnModule( B );
if IsBound( A!.DecideZeroColumns ) and
IsIdenticalObj( A!.DecideZeroColumns, B ) then
return A;
fi;
if not ( IsMutable( A ) or IsMutable( B ) ) then
if IsBound( B!.DecideZeroColumns ) then
C := _ElmWPObj_ForHomalg( B!.DecideZeroColumns, A, fail );
if C <> fail then
return C;
fi;
else
B!.DecideZeroColumns :=
ContainerForWeakPointers(
TheTypeContainerForWeakPointersOnComputedValues,
[ "operation", "DecideZeroColumns" ] );
fi;
fi;
### causes many IsZero external calls without
### reducing the effective number of DecideZero calls;
### observed with Purity.g in ExamplesForHomalg
## if IsZero( A ) or IsZero( B ) then
## ## redispatch to the specialized methods
## return DecideZeroColumns( A, B );
## fi;
if HasIsZero( B ) and IsZero( B ) then
## redispatch to the specialized methods
return DecideZeroColumns( A, B );
fi;
R := HomalgRing( B );
R!.asserts.DecideZeroColumnsWRTNonBasis( B );
RP := homalgTable( R );
t := homalgTotalRuntimes( );
l := NumberColumns( A );
m := NumberRows( A );
n := NumberColumns( B );
ColoredInfoForService( "busy", "DecideZeroColumns", m, " x ( ", l, " + ", n, " ) : ", RingName( R ) );
if IsBound(RP!.DecideZeroColumns) then
C := RP!.DecideZeroColumns( A, B );
SetNumberRows( C, m ); SetNumberColumns( C, l );
if not IsZero( C ) then
Assert( 6, not IsZero( A ) );
SetIsZero( A, false );
fi;
C!.DecideZeroColumns := B;
if not ( IsMutable( A ) or IsMutable( B ) ) then
_AddTwoElmWPObj_ForHomalg( B!.DecideZeroColumns, A, C );
fi;
ColoredInfoForService( t, "DecideZeroColumns" );
IncreaseRingStatistics( R, "DecideZeroColumns" );
return C;
elif IsBound(RP!.DecideZeroRows) then
C := RP!.DecideZeroRows( Involution( A ), Involution( B ) );
SetNumberRows( C, l ); SetNumberColumns( C, m );
if not IsZero( C ) then
Assert( 6, not IsZero( A ) );
SetIsZero( A, false );
fi;
C := Involution( C );
C!.DecideZeroColumns := B;
if not ( IsMutable( A ) or IsMutable( B ) ) then
_AddTwoElmWPObj_ForHomalg( B!.DecideZeroColumns, A, C );
fi;
ColoredInfoForService( t, "DecideZeroColumns" );
DecreaseRingStatistics( R, "DecideZeroColumns" );
IncreaseRingStatistics( R, "DecideZeroRows" );
return C;
fi;
#=====# begin of the core procedure #=====#
if HasIsOne( A ) and IsOne( A ) then ## save as much new definitions as possible
id := A;
else
id := HomalgIdentityMatrix( l, R );
fi;
zz := HomalgZeroMatrix( l, n, R );
M := UnionOfColumns( UnionOfRows( id, A ), UnionOfRows( zz, B ) );
M := ReducedColumnEchelonForm( M );
C := CertainColumns( CertainRows( M, [ l + 1 .. l + m ] ), [ 1 .. l ] );
if not IsZero( C ) then
Assert( 6, not IsZero( A ) );
SetIsZero( A, false );
fi;
C!.DecideZeroColumns := B;
if not ( IsMutable( A ) or IsMutable( B ) ) then
_AddTwoElmWPObj_ForHomalg( B!.DecideZeroColumns, A, C );
fi;
ColoredInfoForService( t, "DecideZeroColumns" );
IncreaseRingStatistics( R, "DecideZeroColumns" );
return C;
end );
## <#GAPDoc Label="SyzygiesGeneratorsOfRows">
## <ManSection>
## <Oper Arg="M" Name="SyzygiesGeneratorsOfRows" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Let <M>R</M> be the ring over which <A>M</A> is defined (<M>R:=</M><C>HomalgRing</C>( <A>M</A> )).
## The matrix of row syzygies <C>SyzygiesGeneratorsOfRows</C>( <A>M</A> ) is a matrix whose rows span
## the left kernel of <A>M</A>, i.e. the <M>R</M>-submodule of the free left module <M>R^{(1 \times NumberRows( <A>M</A> ))}</M>
## consisting of all rows <M>X</M> satisfying <M>X<A>M</A>=0</M>.
## (&see; Appendix <Ref Chap="Basic_Operations"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( SyzygiesGeneratorsOfRows,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, RP, t, MI, C, B, nz;
## this is an attribute
R := HomalgRing( M );
RP := homalgTable( R );
t := homalgTotalRuntimes( );
ColoredInfoForService( "busy", "SyzygiesGeneratorsOfRows", NumberRows( M ), " x ", NumberColumns( M ), " : ", RingName( R ) );
if IsBound(R!.ComputeBasisBeforeSyzygies) and R!.ComputeBasisBeforeSyzygies = true then
BasisOfRowModule( M );
fi;
if IsBound(RP!.SyzygiesGeneratorsOfRows) then
C := RP!.SyzygiesGeneratorsOfRows( M );
if IsZero( C ) then
SetIsLeftRegular( M, true );
C := HomalgZeroMatrix( 0, NumberRows( M ), R ); ## most of the computer algebra systems cannot handle degenerated matrices
else
SetNumberColumns( C, NumberRows( M ) );
fi;
ColoredInfoForService( t, "SyzygiesGeneratorsOfRows", NumberRows( C ) );
IncreaseRingStatistics( R, "SyzygiesGeneratorsOfRows" );
return C;
elif IsBound(RP!.SyzygiesGeneratorsOfColumns) then
MI := Involution( M );
C := RP!.SyzygiesGeneratorsOfColumns( MI );
SetSyzygiesGeneratorsOfColumns( MI, C );
C := Involution( C );
if IsZero( C ) then
SetIsLeftRegular( M, true );
C := HomalgZeroMatrix( 0, NumberRows( M ), R ); ## most of the computer algebra systems cannot handle degenerated matrices
else
SetNumberColumns( C, NumberRows( M ) );
fi;
ColoredInfoForService( t, "SyzygiesGeneratorsOfRows", NumberRows( C ) );
DecreaseRingStatistics( R, "SyzygiesGeneratorsOfRows" );
IncreaseRingStatistics( R, "SyzygiesGeneratorsOfColumns" );
return C;
fi;
#=====# begin of the core procedure #=====#
C := HomalgVoidMatrix( R );
B := ReducedRowEchelonForm( M, C );
nz := Length( NonZeroRows( B ) );
C := CertainRows( C, [ nz + 1 .. NumberRows( C ) ] );
if IsZero( C ) then
SetIsLeftRegular( M, true );
C := HomalgZeroMatrix( 0, NumberRows( M ), R );
fi;
ColoredInfoForService( t, "SyzygiesGeneratorsOfRows", NumberRows( C ) );
IncreaseRingStatistics( R, "SyzygiesGeneratorsOfRows" );
return C;
end );
## <#GAPDoc Label="SyzygiesGeneratorsOfColumns">
## <ManSection>
## <Oper Arg="M" Name="SyzygiesGeneratorsOfColumns" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Let <M>R</M> be the ring over which <A>M</A> is defined (<M>R:=</M><C>HomalgRing</C>( <A>M</A> )).
## The matrix of column syzygies <C>SyzygiesGeneratorsOfColumns</C>( <A>M</A> ) is a matrix whose columns span
## the right kernel of <A>M</A>, i.e. the <M>R</M>-submodule of the free right module <M>R^{(NumberColumns( <A>M</A> ) \times 1)}</M>
## consisting of all columns <M>X</M> satisfying <M><A>M</A>X=0</M>.
## (&see; Appendix <Ref Chap="Basic_Operations"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( SyzygiesGeneratorsOfColumns,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, RP, t, MI, C, B, nz;
## this is an attribute
R := HomalgRing( M );
RP := homalgTable( R );
t := homalgTotalRuntimes( );
ColoredInfoForService( "busy", "SyzygiesGeneratorsOfColumns", NumberRows( M ), " x ", NumberColumns( M ), " : ", RingName( R ) );
if IsBound(R!.ComputeBasisBeforeSyzygies) and R!.ComputeBasisBeforeSyzygies = true then
BasisOfColumnModule( M );
fi;
if IsBound(RP!.SyzygiesGeneratorsOfColumns) then
C := RP!.SyzygiesGeneratorsOfColumns( M );
if IsZero( C ) then
SetIsRightRegular( M, true );
C := HomalgZeroMatrix( NumberColumns( M ), 0, R );
else
SetNumberRows( C, NumberColumns( M ) );
fi;
ColoredInfoForService( t, "SyzygiesGeneratorsOfColumns", NumberColumns( C ) );
IncreaseRingStatistics( R, "SyzygiesGeneratorsOfColumns" );
return C;
elif IsBound(RP!.SyzygiesGeneratorsOfRows) then
MI := Involution( M );
C := RP!.SyzygiesGeneratorsOfRows( MI );
SetSyzygiesGeneratorsOfRows( MI, C );
C := Involution( C );
if IsZero( C ) then
SetIsRightRegular( M, true );
C := HomalgZeroMatrix( NumberColumns( M ), 0, R );
else
SetNumberRows( C, NumberColumns( M ) );
fi;
ColoredInfoForService( t, "SyzygiesGeneratorsOfColumns", NumberColumns( C ) );
DecreaseRingStatistics( R, "SyzygiesGeneratorsOfColumns" );
IncreaseRingStatistics( R, "SyzygiesGeneratorsOfRows" );
return C;
fi;
#=====# begin of the core procedure #=====#
C := HomalgVoidMatrix( R );
B := ReducedColumnEchelonForm( M, C );
nz := Length( NonZeroColumns( B ) );
C := CertainColumns( C, [ nz + 1 .. NumberColumns( C ) ] );
if IsZero( C ) then
SetIsRightRegular( M, true );
C := HomalgZeroMatrix( NumberColumns( M ), 0, R );
fi;
ColoredInfoForService( t, "SyzygiesGeneratorsOfColumns", NumberColumns( C ) );
IncreaseRingStatistics( R, "SyzygiesGeneratorsOfColumns" );
return C;
end );
#### Relative:
## <#GAPDoc Label="RelativeSyzygiesGeneratorsOfRows">
## <ManSection>
## <Oper Arg="M, M2" Name="SyzygiesGeneratorsOfRows" Label="for pairs of matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Let <M>R</M> be the ring over which <A>M</A> is defined (<M>R:=</M><C>HomalgRing</C>( <A>M</A> )).
## The matrix of <E>relative</E> row syzygies <C>SyzygiesGeneratorsOfRows</C>( <A>M</A>, <A>M2</A> ) is a matrix
## whose rows span the left kernel of <A>M</A> modulo <A>M2</A>, i.e. the <M>R</M>-submodule of the free left module
## <M>R^{(1 \times NumberRows( <A>M</A> ))}</M> consisting of all rows <M>X</M> satisfying <M>X<A>M</A>+Y<A>M2</A>=0</M>
## for some row <M>Y \in R^{(1 \times NumberRows( <A>M2</A> ))}</M>. (&see; Appendix <Ref Chap="Basic_Operations"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( SyzygiesGeneratorsOfRows, ### defines: SyzygiesGeneratorsOfRows (SyzygiesGenerators)
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ],
function( M1, M2 )
local R, RP, t, M, C, nz;
## a LIMAT method takes care of the case when M2 is _known_ to be zero
## checking IsZero( M2 ) causes too many obsolete calls
R := HomalgRing( M1 );
RP := homalgTable( R );
t := homalgTotalRuntimes( );
ColoredInfoForService( "busy", "RelativeSyzygiesGeneratorsOfRows", "( ", NumberRows( M1 ), " + ", NumberRows( M2 ), " ) x ", NumberColumns( M1 ), " : ", RingName( R ) );
if IsBound(R!.ComputeBasisBeforeSyzygies) and R!.ComputeBasisBeforeSyzygies = true then
BasisOfRowModule( M1 );
BasisOfRowModule( M2 );
fi;
if IsBound(RP!.RelativeSyzygiesGeneratorsOfRows) then
C := RP!.RelativeSyzygiesGeneratorsOfRows( M1, M2 );
if IsZero( C ) then
C := HomalgZeroMatrix( 0, NumberRows( M1 ), R );
else
SetNumberColumns( C, NumberRows( M1 ) );
fi;
ColoredInfoForService( t, "RelativeSyzygiesGeneratorsOfRows", NumberRows( C ) );
IncreaseRingStatistics( R, "RelativeSyzygiesGeneratorsOfRows" );
return C;
elif IsBound(RP!.RelativeSyzygiesGeneratorsOfColumns) then
C := Involution( RP!.RelativeSyzygiesGeneratorsOfColumns( Involution( M1 ), Involution( M2 ) ) );
if IsZero( C ) then
C := HomalgZeroMatrix( 0, NumberRows( M1 ), R );
else
SetNumberColumns( C, NumberRows( M1 ) );
fi;
ColoredInfoForService( t, "RelativeSyzygiesGeneratorsOfRows", NumberRows( C ) );
DecreaseRingStatistics( R, "RelativeSyzygiesGeneratorsOfRows" );
IncreaseRingStatistics( R, "RelativeSyzygiesGeneratorsOfColumns" );
return C;
fi;
#=====# begin of the core procedure #=====#
M := UnionOfRows( M1, M2 );
C := SyzygiesGeneratorsOfRows( M );
C := CertainColumns( C, [ 1 .. NumberRows( M1 ) ] );
## since we first compute the syzygies matrix of
## the stack of M1 and M2, and then keep only
## those columns C corresponding to M1,
## zero rows can potentially exist in C
## (and we like to remove them)
C := GetRidOfObsoleteRows( C );
if IsZero( C ) then
SetIsLeftRegular( M1, true );
fi;
## forgetting the original obsolete C may save memory
if HasEvalCertainColumns( C ) then
if not IsEmptyMatrix( C ) then
Eval( C );
fi;
ResetFilterObj( C, HasEvalCertainColumns );
Unbind( C!.EvalCertainColumns );
fi;
ColoredInfoForService( t, "RelativeSyzygiesGeneratorsOfRows", NumberRows( C ) );
DecreaseRingStatistics( R, "RelativeSyzygiesGeneratorsOfRows" );
return C;
end );
## <#GAPDoc Label="RelativeSyzygiesGeneratorsOfColumns">
## <ManSection>
## <Oper Arg="M, M2" Name="SyzygiesGeneratorsOfColumns" Label="for pairs of matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Let <M>R</M> be the ring over which <A>M</A> is defined (<M>R:=</M><C>HomalgRing</C>( <A>M</A> )).
## The matrix of <E>relative</E> column syzygies <C>SyzygiesGeneratorsOfColumns</C>( <A>M</A>, <A>M2</A> ) is a matrix
## whose columns span the right kernel of <A>M</A> modulo <A>M2</A>, i.e. the <M>R</M>-submodule of the free right module
## <M>R^{(NumberColumns( <A>M</A> ) \times 1)}</M> consisting of all columns <M>X</M> satisfying <M><A>M</A>X+<A>M2</A>Y=0</M>
## for some column <M>Y \in R^{(NumberColumns( <A>M2</A> ) \times 1)}</M>. (&see; Appendix <Ref Chap="Basic_Operations"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( SyzygiesGeneratorsOfColumns, ### defines: SyzygiesGeneratorsOfColumns (SyzygiesGenerators)
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgMatrix ],
function( M1, M2 )
local R, RP, t, M, C, nz;
## a LIMAT method takes care of the case when M2 is _known_ to be zero
## checking IsZero( M2 ) causes too many obsolete calls
R := HomalgRing( M1 );
RP := homalgTable( R );
t := homalgTotalRuntimes( );
ColoredInfoForService( "busy", "RelativeSyzygiesGeneratorsOfColumns", NumberRows( M1 ), " x ( ", NumberColumns( M1 ), " + ", NumberColumns( M2 ), " ) : ", RingName( R ) );
if IsBound(R!.ComputeBasisBeforeSyzygies) and R!.ComputeBasisBeforeSyzygies = true then
BasisOfColumnModule( M1 );
BasisOfColumnModule( M2 );
fi;
if IsBound(RP!.RelativeSyzygiesGeneratorsOfColumns) then
C := RP!.RelativeSyzygiesGeneratorsOfColumns( M1, M2 );
if IsZero( C ) then
C := HomalgZeroMatrix( NumberColumns( M1 ), 0, R );
else
SetNumberRows( C, NumberColumns( M1 ) );
fi;
ColoredInfoForService( t, "RelativeSyzygiesGeneratorsOfColumns", NumberColumns( C ) );
IncreaseRingStatistics( R, "RelativeSyzygiesGeneratorsOfColumns" );
return C;
elif IsBound(RP!.RelativeSyzygiesGeneratorsOfRows) then
C := Involution( RP!.RelativeSyzygiesGeneratorsOfRows( Involution( M1 ), Involution( M2 ) ) );
if IsZero( C ) then
C := HomalgZeroMatrix( NumberColumns( M1 ), 0, R );
else
SetNumberRows( C, NumberColumns( M1 ) );
fi;
ColoredInfoForService( t, "RelativeSyzygiesGeneratorsOfColumns", NumberColumns( C ) );
DecreaseRingStatistics( R, "RelativeSyzygiesGeneratorsOfColumns" );
IncreaseRingStatistics( R, "RelativeSyzygiesGeneratorsOfRows" );
return C;
fi;
#=====# begin of the core procedure #=====#
M := UnionOfColumns( M1, M2 );
C := SyzygiesGeneratorsOfColumns( M );
C := CertainRows( C, [ 1 .. NumberColumns( M1 ) ] );
## since we first computes the syzygies matrix of
## the augmentation of M1 and M2, and then keep
## only those rows C corresponding to M1,
## zero columns can potentially exist in C
## (and we like to remove them)
C := GetRidOfObsoleteColumns( C );
if IsZero( C ) then
SetIsRightRegular( M1, true );
fi;
## forgetting the original obsolete C may save memory
if HasEvalCertainRows( C ) then
if not IsEmptyMatrix( C ) then
Eval( C );
fi;
ResetFilterObj( C, HasEvalCertainRows );
Unbind( C!.EvalCertainRows );
fi;
ColoredInfoForService( t, "RelativeSyzygiesGeneratorsOfColumns", NumberColumns( C ) );
DecreaseRingStatistics( R, "RelativeSyzygiesGeneratorsOfColumns" );
return C;
end );
#### Reduced:
## <#GAPDoc Label="ReducedBasisOfRowModule">
## <ManSection>
## <Oper Arg="M" Name="ReducedBasisOfRowModule" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Like <C>BasisOfRowModule</C>( <A>M</A> ) but where the matrix
## <C>SyzygiesGeneratorsOfRows</C>( <C>ReducedBasisOfRowModule</C>( <A>M</A> ) ) contains no units. This can easily
## be achieved starting from <M>B:=</M><C>BasisOfRowModule</C>( <A>M</A> )
## (and using <Ref Oper="GetColumnIndependentUnitPositions" Label="for matrices"/> applied to the matrix of row syzygies of <M>B</M>,
## etc). (&see; Appendix <Ref Chap="Basic_Operations"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( ReducedBasisOfRowModule,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, RP, t, nr, MI, B, S, unit_pos;
## this is an attribute
R := HomalgRing( M );
RP := homalgTable( R );
t := homalgTotalRuntimes( );
nr := NumberColumns( M );
ColoredInfoForService( "busy", "ReducedBasisOfRowModule", NumberRows( M ), " x ", nr, " : ", RingName( R ) );
if IsBound(RP!.ReducedBasisOfRowModule) then
B := RP!.ReducedBasisOfRowModule( M );
if HasRowRankOfMatrix( B ) then
SetRowRankOfMatrix( M, RowRankOfMatrix( B ) );
fi;
SetNumberColumns( B, nr );
## check assertion
Assert( 6, R!.asserts.BasisOfRowModule( B ) );
nr := NumberRows( B );
SetIsZero( B, nr = 0 );
if M = B then
B := M; ## we might know more about M
else
SetIsZero( M, nr = 0 );
## check assertion
Assert( 6, R!.asserts.ReducedBasisOfRowModule( M, B ) );
fi;
SetIsReducedBasisOfRowsMatrix( B, true );
if HasIsBasisOfRowsMatrix( M ) and IsBasisOfRowsMatrix( M ) then
SetBasisOfRowModule( B, M );
fi;
ColoredInfoForService( t, "ReducedBasisOfRowModule", NumberRows( B ) );
IncreaseRingStatistics( R, "ReducedBasisOfRowModule" );
return B;
elif IsBound(RP!.ReducedBasisOfColumnModule) then
MI := Involution( M );
B := RP!.ReducedBasisOfColumnModule( MI );
if HasColumnRankOfMatrix( B ) then
SetRowRankOfMatrix( M, ColumnRankOfMatrix( B ) );
fi;
SetNumberRows( B, nr );
SetIsReducedBasisOfColumnsMatrix( B, true );
SetReducedBasisOfColumnModule( MI, B );
B := Involution( B );
## check assertion
Assert( 6, R!.asserts.BasisOfRowModule( B ) );
nr := NumberRows( B );
SetIsZero( B, nr = 0 );
if M = B then
B := M; ## we might know more about M
else
SetIsZero( M, nr = 0 );
## check assertion
Assert( 6, R!.asserts.ReducedBasisOfRowModule( M, B ) );
fi;
SetIsReducedBasisOfRowsMatrix( B, true );
if HasIsBasisOfRowsMatrix( M ) and IsBasisOfRowsMatrix( M ) then
SetBasisOfRowModule( B, M );
fi;
ColoredInfoForService( t, "ReducedBasisOfRowModule", NumberRows( B ) );
DecreaseRingStatistics( R, "ReducedBasisOfRowModule" );
IncreaseRingStatistics( R, "ReducedBasisOfColumnModule" );
return B;
fi;
## try RP!.PartiallyReducedBasisOf*Module
if IsBound(RP!.PartiallyReducedBasisOfRowModule) then
B := RP!.PartiallyReducedBasisOfRowModule( M );
if HasRowRankOfMatrix( B ) then
SetRowRankOfMatrix( M, RowRankOfMatrix( B ) );
fi;
SetNumberColumns( B, nr );
## check assertion
Assert( 6, R!.asserts.BasisOfRowModule( B ) );
nr := NumberRows( B );
SetIsZero( B, nr = 0 );
if NumberRows( M ) <= nr then
B := M; ## we might know more about M
else
SetIsZero( M, nr = 0 );
## check assertion
Assert( 6, R!.asserts.ReducedBasisOfRowModule( M, B ) );
fi;
IncreaseRingStatistics( R, "PartiallyReducedBasisOfRowModule" );
elif IsBound(RP!.PartiallyReducedBasisOfColumnModule) then
MI := Involution( M );
B := RP!.PartiallyReducedBasisOfColumnModule( MI );
if HasColumnRankOfMatrix( B ) then
SetRowRankOfMatrix( M, ColumnRankOfMatrix( B ) );
fi;
SetNumberRows( B, nr );
B := Involution( B );
## check assertion
Assert( 6, R!.asserts.BasisOfRowModule( B ) );
nr := NumberRows( B );
SetIsZero( B, nr = 0 );
if NumberRows( M ) <= nr then
B := M; ## we might know more about M
else
SetIsZero( M, nr = 0 );
## check assertion
Assert( 6, R!.asserts.ReducedBasisOfRowModule( M, B ) );
fi;
DecreaseRingStatistics( R, "PartiallyReducedBasisOfRowModule" );
IncreaseRingStatistics( R, "PartiallyReducedBasisOfColumnModule" );
else
B := M;
fi;
#=====# begin of the core procedure #=====#
## iterate the syzygy trick
while true do
S := SyzygiesGeneratorsOfRows( B );
unit_pos := GetColumnIndependentUnitPositions( S );
unit_pos := List( unit_pos, a -> a[2] );
if NumberRows( S ) = 0 or unit_pos = [ ] then
break;
fi;
B := CertainRows( B, Filtered( [ 1 .. NumberRows( B ) ], j -> not j in unit_pos ) );
od;
## check assertion
Assert( 6, R!.asserts.BasisOfRowModule( B ) );
nr := NumberRows( B );
SetIsZero( B, nr = 0 );
SetIsZero( M, nr = 0 );
## check assertion
Assert( 6, R!.asserts.ReducedBasisOfRowModule( M, B ) );
SetIsReducedBasisOfRowsMatrix( B, true );
if HasIsBasisOfRowsMatrix( M ) and IsBasisOfRowsMatrix( M ) then
SetBasisOfRowModule( B, M );
fi;
ColoredInfoForService( t, "ReducedBasisOfRowModule", NumberRows( B ) );
DecreaseRingStatistics( R, "ReducedBasisOfRowModule" );
return B;
end );
## <#GAPDoc Label="ReducedBasisOfColumnModule">
## <ManSection>
## <Oper Arg="M" Name="ReducedBasisOfColumnModule" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Like <C>BasisOfColumnModule</C>( <A>M</A> ) but where the matrix
## <C>SyzygiesGeneratorsOfColumns</C>( <C>ReducedBasisOfColumnModule</C>( <A>M</A> ) ) contains no units. This can easily
## be achieved starting from <M>B:=</M><C>BasisOfColumnModule</C>( <A>M</A> )
## (and using <Ref Oper="GetRowIndependentUnitPositions" Label="for matrices"/> applied to the matrix of column syzygies of <M>B</M>,
## etc.). (&see; Appendix <Ref Chap="Basic_Operations"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( ReducedBasisOfColumnModule,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, RP, t, nr, MI, B, S, unit_pos;
## this is an attribute
R := HomalgRing( M );
RP := homalgTable( R );
t := homalgTotalRuntimes( );
nr := NumberRows( M );
ColoredInfoForService( "busy", "ReducedBasisOfColumnModule", nr, " x ", NumberColumns( M ), " : ", RingName( R ) );
if IsBound(RP!.ReducedBasisOfColumnModule) then
B := RP!.ReducedBasisOfColumnModule( M );
if HasColumnRankOfMatrix( B ) then
SetColumnRankOfMatrix( M, ColumnRankOfMatrix( B ) );
fi;
SetNumberRows( B, nr );
## check assertion
Assert( 6, R!.asserts.BasisOfColumnModule( B ) );
nr := NumberColumns( B );
SetIsZero( B, nr = 0 );
if M = B then
B := M; ## we might know more about M
else
SetIsZero( M, nr = 0 );
## check assertion
Assert( 6, R!.asserts.ReducedBasisOfColumnModule( M, B ) );
fi;
SetIsReducedBasisOfColumnsMatrix( B, true );
if HasIsBasisOfColumnsMatrix( M ) and IsBasisOfColumnsMatrix( M ) then
SetBasisOfColumnModule( B, M );
fi;
ColoredInfoForService( t, "ReducedBasisOfColumnModule", NumberColumns( B ) );
IncreaseRingStatistics( R, "ReducedBasisOfColumnModule" );
return B;
elif IsBound(RP!.ReducedBasisOfRowModule) then
MI := Involution( M );
B := RP!.ReducedBasisOfRowModule( MI );
if HasRowRankOfMatrix( B ) then
SetColumnRankOfMatrix( M, RowRankOfMatrix( B ) );
fi;
SetNumberColumns( B, nr );
SetIsReducedBasisOfRowsMatrix( B, true );
SetReducedBasisOfRowModule( MI, B );
B := Involution( B );
## check assertion
Assert( 6, R!.asserts.BasisOfColumnModule( B ) );
nr := NumberColumns( B );
SetIsZero( B, nr = 0 );
if M = B then
B := M; ## we might know more about M
else
SetIsZero( M, nr = 0 );
## check assertion
Assert( 6, R!.asserts.ReducedBasisOfColumnModule( M, B ) );
fi;
SetIsReducedBasisOfColumnsMatrix( B, true );
if HasIsBasisOfColumnsMatrix( M ) and IsBasisOfColumnsMatrix( M ) then
SetBasisOfColumnModule( B, M );
fi;
ColoredInfoForService( t, "ReducedBasisOfColumnModule", NumberColumns( B ) );
DecreaseRingStatistics( R, "ReducedBasisOfColumnModule" );
IncreaseRingStatistics( R, "ReducedBasisOfRowModule" );
return B;
fi;
## try RP!.PartiallyReducedBasisOf*Module
if IsBound(RP!.PartiallyReducedBasisOfColumnModule) then
B := RP!.PartiallyReducedBasisOfColumnModule( M );
if HasColumnRankOfMatrix( B ) then
SetColumnRankOfMatrix( M, ColumnRankOfMatrix( B ) );
fi;
SetNumberRows( B, nr );
## check assertion
Assert( 6, R!.asserts.BasisOfColumnModule( B ) );
nr := NumberColumns( B );
SetIsZero( B, nr = 0 );
if NumberColumns( M ) <= nr then
B := M; ## we might know more about M
else
SetIsZero( M, nr = 0 );
## check assertion
Assert( 6, R!.asserts.ReducedBasisOfColumnModule( M, B ) );
fi;
IncreaseRingStatistics( R, "PartiallyReducedBasisOfColumnModule" );
elif IsBound(RP!.PartiallyReducedBasisOfRowModule) then
MI := Involution( M );
B := RP!.PartiallyReducedBasisOfRowModule( MI );
if HasRowRankOfMatrix( B ) then
SetColumnRankOfMatrix( M, RowRankOfMatrix( B ) );
fi;
SetNumberColumns( B, nr );
B := Involution( B );
## check assertion
Assert( 6, R!.asserts.BasisOfColumnModule( B ) );
nr := NumberColumns( B );
SetIsZero( B, nr = 0 );
if NumberColumns( M ) <= nr then
B := M; ## we might know more about M
else
SetIsZero( M, nr = 0 );
## check assertion
Assert( 6, R!.asserts.ReducedBasisOfColumnModule( M, B ) );
fi;
DecreaseRingStatistics( R, "PartiallyReducedBasisOfColumnModule" );
IncreaseRingStatistics( R, "PartiallyReducedBasisOfRowModule" );
else
B := M;
fi;
#=====# begin of the core procedure #=====#
## iterate the syzygy trick
while true do
S := SyzygiesGeneratorsOfColumns( B );
unit_pos := GetRowIndependentUnitPositions( S );
unit_pos := List( unit_pos, a -> a[2] );
if NumberColumns( S ) = 0 or unit_pos = [ ] then
break;
fi;
B := CertainColumns( B, Filtered( [ 1 .. NumberColumns( B ) ], j -> not j in unit_pos ) );
od;
## check assertion
Assert( 6, R!.asserts.BasisOfColumnModule( B ) );
nr := NumberColumns( B );
SetIsZero( B, nr = 0 );
SetIsZero( M, nr = 0 );
## check assertion
Assert( 6, R!.asserts.ReducedBasisOfColumnModule( M, B ) );
SetIsReducedBasisOfColumnsMatrix( B, true );
if HasIsBasisOfColumnsMatrix( M ) and IsBasisOfColumnsMatrix( M ) then
SetBasisOfColumnModule( B, M );
fi;
ColoredInfoForService( t, "ReducedBasisOfColumnModule", NumberColumns( B ) );
DecreaseRingStatistics( R, "ReducedBasisOfColumnModule" );
return B;
end );
## <#GAPDoc Label="ReducedSyzygiesGeneratorsOfRows">
## <ManSection>
## <Oper Arg="M" Name="ReducedSyzygiesGeneratorsOfRows" Label="for matrices"/>
## <Returns>a &homalg; matrix</Returns>
## <Description>
## Like <C>SyzygiesGeneratorsOfRows</C>( <A>M</A> ) but where the matrix
## <C>SyzygiesGeneratorsOfRows</C>( <C>ReducedSyzygiesGeneratorsOfRows</C>( <A>M</A> ) ) contains no units.
## This can easily be achieved starting from <M>C:=</M><C>SyzygiesGeneratorsOfRows</C>( <A>M</A> )
## (and using <Ref Oper="GetColumnIndependentUnitPositions" Label="for matrices"/> applied to the matrix of row syzygies of <M>C</M>,
## etc.). (&see; Appendix <Ref Chap="Basic_Operations"/>)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( ReducedSyzygiesGeneratorsOfRows,
"for homalg matrices",
[ IsHomalgMatrix ],
function( M )
local R, RP, t, MI, C, B, nz;
if IsZero( M ) then
## redispatch to the specialized methods
return ReducedSyzygiesGeneratorsOfRows( M );
fi;
## this is an attribute
R := HomalgRing( M );
RP := homalgTable( R );
t := homalgTotalRuntimes( );
ColoredInfoForService( "busy", "ReducedSyzygiesGeneratorsOfRows", NumberRows( M ), " x ", NumberColumns( M ), " : ", RingName( R ) );
if IsBound(R!.ComputeBasisBeforeSyzygies) and R!.ComputeBasisBeforeSyzygies = true then
BasisOfRowModule( M );
fi;
if IsBound(RP!.ReducedSyzygiesGeneratorsOfRows) then
C := RP!.ReducedSyzygiesGeneratorsOfRows( M );
if IsZero( C ) then
SetIsLeftRegular( M, true );
C := HomalgZeroMatrix( 0, NumberRows( M ), R );
else
SetNumberColumns( C, NumberRows( M ) );
fi;
## check assertion
Assert( 6, R!.asserts.ReducedSyzygiesGeneratorsOfRows( M, C ) );
ColoredInfoForService( t, "ReducedSyzygiesGeneratorsOfRows", NumberRows( C ) );
IncreaseRingStatistics( R, "ReducedSyzygiesGeneratorsOfRows" );
return C;
elif IsBound(RP!.ReducedSyzygiesGeneratorsOfColumns) then
MI := Involution( M );
C := RP!.ReducedSyzygiesGeneratorsOfColumns( MI );
SetReducedSyzygiesGeneratorsOfColumns( MI, C );
--> --------------------
--> maximum size reached
--> --------------------
