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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains methods for *row modules*, that is,
## free left modules consisting of row vectors.
##
## Especially methods for *full row modules* $R^n$ are contained.
##
## (See the file `modulmat.gi' for the methods for matrix modules.)
##
#############################################################################
##
#F FullRowModule( <R>, <n> )
##
InstallGlobalFunction( FullRowModule, function( R, n )
local M,typ; # the free module record, result
if not ( IsRing( R ) and IsInt( n ) and 0 <= n ) then
Error( "usage: FullRowModule( <R>, <n> ) for ring <R>" );
fi;
typ:=IsFreeLeftModule and IsFullRowModule and IsAttributeStoringRep
and HasIsEmpty;
if IsDivisionRing( R ) then
typ:=typ and IsGaussianSpace;
fi;
if n=0 then
typ:=typ and IsTrivial;
else
typ:=typ and IsNonTrivial;
fi;
if n<>infinity and HasIsFinite(R) and IsFinite(R) then
typ:=typ and IsFinite;
elif n<>0 and HasIsFinite(R) and not IsFinite(R) then
typ:=typ and HasIsFinite;
fi;
M:= Objectify( NewType( CollectionsFamily( FamilyObj( R ) ),
typ ),
rec() );
SetLeftActingDomain( M, R );
SetDimensionOfVectors( M, n );
return M;
end );
#############################################################################
##
#M \^( <R>, <n> ) . . . . . . . . . . . . . . . full row module over a ring
##
InstallOtherMethod( \^,
"for ring and integer (delegate to `FullRowModule')",
[ IsRing, IsInt ],
FullRowModule );
#############################################################################
##
#M IsRowModule . return `false' for objects which are not free left modules
##
InstallOtherMethod( IsRowModule,
"return `false' for objects which are not free left modules",
true, [ IsObject ], 0,
function ( obj )
if not IsFreeLeftModule(obj) then
return false;
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M IsRowModule( <M> )
##
InstallMethod( IsRowModule,
"for a free left module",
[ IsFreeLeftModule ],
M -> IsRowVector( Zero( M ) ) );
#############################################################################
##
#M IsFullRowModule( M )
##
InstallMethod( IsFullRowModule,
"for free left (row) module",
[ IsFreeLeftModule ],
M -> IsRowModule( M )
and Dimension( M ) = DimensionOfVectors( M )
and ForAll( GeneratorsOfLeftModule( M ),
v -> IsSubset( LeftActingDomain( M ), v ) ) );
#############################################################################
##
#M Dimension( <M> )
##
InstallMethod( Dimension,
"for full row module",
[ IsFreeLeftModule and IsFullRowModule ],
DimensionOfVectors );
#############################################################################
##
#M Random( <M> )
##1
InstallMethodWithRandomSource( Random,
"for a random source and a full row module",
[ IsRandomSource, IsFreeLeftModule and IsFullRowModule ],
function( rs, M )
local R,v;
R:= LeftActingDomain( M );
v := List( [ 1 .. DimensionOfVectors( M ) ], x -> Random( rs, R ) );
if IsField(R) then
if IsHPCGAP then
if Size(R) <= 256 then
v := CopyToVectorRep(v,Size(R));
fi;
else
ConvertToVectorRep(v,R);
fi;
fi;
return v;
end );
#############################################################################
##
#M Representative( <M> )
##
InstallMethod( Representative,
"for full row module",
[ IsFreeLeftModule and IsFullRowModule ],
M -> ListWithIdenticalEntries( DimensionOfVectors( M ),
Zero( LeftActingDomain( M ) ) ) );
#############################################################################
##
#M GeneratorsOfLeftModule( <V> )
##
InstallMethod( GeneratorsOfLeftModule,
"for full row module",
[ IsFreeLeftModule and IsFullRowModule ],
M -> IdentityMat( DimensionOfVectors( M ), LeftActingDomain( M ) ) );
#############################################################################
##
#M ViewObj( <M> )
##
InstallMethod( ViewObj,
"for full row module",
[ IsFreeLeftModule and IsFullRowModule ],
function( M )
Print( "( " );
View( LeftActingDomain( M ) );
Print( "^", DimensionOfVectors( M ), " )" );
end );
#############################################################################
##
#M ViewString( <M> ) . . . . . . . . . . . . . . . . . for full row modules
##
InstallMethod( ViewString, "for full row modules", true,
[ IsFreeLeftModule and IsFullRowModule ], 0, String );
#############################################################################
##
#M PrintObj( <M> )
##
InstallMethod( PrintObj,
"for full row module",
[ IsFreeLeftModule and IsFullRowModule ],
function( M )
Print( "( ", LeftActingDomain( M ), "^", DimensionOfVectors( M ), " )" );
end );
#############################################################################
##
#M String( <M> ) . . . . . . . . . . . . . . . . . . . for full row modules
##
InstallMethod( String, "for full row modules", true,
[ IsFreeLeftModule and IsFullRowModule ], 0,
M -> Concatenation(List(["( ",LeftActingDomain(M),"^",
DimensionOfVectors(M)," )"], String)) );
#############################################################################
##
#M \in( <v>, <V> )
##
InstallMethod( \in,
"for full row module",
IsElmsColls,
[ IsRowVector, IsFreeLeftModule and IsFullRowModule ],
function( v, M )
return Length( v ) = DimensionOfVectors( M )
and IsSubset( LeftActingDomain( M ), v );
end );
#############################################################################
##
#M BasisVectors( <B> ) . . . . . for a canonical basis of a full row module
##
InstallMethod( BasisVectors,
"for canonical basis of a full row module",
[ IsBasis and IsCanonicalBasis and IsCanonicalBasisFullRowModule ],
function( B )
B:= UnderlyingLeftModule( B );
return IdentityMat( DimensionOfVectors( B ), LeftActingDomain( B ) );
end );
#############################################################################
##
#M CanonicalBasis( <V> )
##
InstallMethod( CanonicalBasis,
"for a full row module",
[ IsFreeLeftModule and IsFullRowModule ],
function( V )
local B;
B:= Objectify( NewType( FamilyObj( V ),
IsFiniteBasisDefault
and IsCanonicalBasis
and IsCanonicalBasisFullRowModule
and IsAttributeStoringRep ),
rec() );
SetUnderlyingLeftModule( B, V );
return B;
end );
#############################################################################
##
#M Basis( <M> ) . . . . . . . . . . . . . . . . . . . . for full row module
##
InstallMethod( Basis,
"for full row module (delegate to `CanonicalBasis')",
[ IsFreeLeftModule and IsFullRowModule ], CANONICAL_BASIS_FLAGS,
CanonicalBasis );
#############################################################################
##
#M Coefficients( <B>, <v> ) . . for a canonical basis of a full row module
##
InstallMethod( Coefficients,
"for canonical basis of a full row module",
IsCollsElms,
[ IsBasis and IsCanonicalBasisFullRowModule, IsRowVector ],
function( B, v )
local V, R;
V:= UnderlyingLeftModule( B );
R:= LeftActingDomain( V );
if Length( v ) = DimensionOfVectors( V ) and IsSubset( R, v ) then
return ShallowCopy( v );
else
return fail;
fi;
end );
#############################################################################
##
#M IsCanonicalBasisFullRowModule( <B> ) . . . . . . . . . . . . for a basis
##
InstallMethod( IsCanonicalBasisFullRowModule,
"for a basis",
[ IsBasis ],
B -> IsFullRowModule( UnderlyingLeftModule( B ) )
and IsCanonicalBasis( B ) );
#############################################################################
##
#M EnumeratorByBasis( <B> ) . . . . for canonical basis of full row module
##
BindGlobal( "NumberElement_FiniteFullRowModule", function( e, v )
local len, n, i, pos;
if not IsDenseList( v ) then
return fail;
fi;
len:= Length( v );
if len <> e!.dimension then
return fail;
fi;
n:= 0;
i:= 1;
while i <= len and v[i] = e!.coeffszero do
i:= i+1;
od;
while i <= len do
pos:= Position( e!.coeffsenum, v[i], 0 );
if pos = fail then
return fail;
fi;
n:= e!.q * n + pos - 1;
i:= i+1;
od;
return n + 1;
end );
BindGlobal( "PosVecEnumFF", function( enum, v )
local i,l;
if not IsCollsElms( FamilyObj( enum ), FamilyObj( v ) )
or not IsRowVector( v )
or Length( v ) <> enum!.dimension then
return fail;
fi;
# test whether the vector is indeed compact over a finite field
if not IsDataObjectRep(v) then
# the degree of the field extension q provides
l:= LogInt( enum!.q, Characteristic(v) );
for i in v do
if not (IsFFE(i) and IsInt(l/DegreeFFE(i))) then
# cannot convert, wrong type of object
return NumberElement_FiniteFullRowModule( enum, v );
fi;
od;
v := ImmutableVector( enum!.q, v );
if not IsDataObjectRep(v) then
# cannot convert, wrong type of object
return NumberElement_FiniteFullRowModule( enum, v );
fi;
fi;
# Problem with GF(4) vectors over GF(2)
if ( IsGF2VectorRep(v) and enum!.q <> 2 )
or ( Is8BitVectorRep(v) and enum!.q = 2 ) then
return NumberElement_FiniteFullRowModule( enum, v );
fi;
# compute index via number
v:= NumberFFVector( v, enum!.q );
if v <> fail then
v:= v+1;
fi;
return v;
end);
BindGlobal( "ElementNumber_FiniteFullRowModule", function( enum, n )
local v, i;
if Size( enum ) < n then
Error( "<enum>[", n, "] must have an assigned value" );
fi;
v:= ShallowCopy( enum!.zerovector );
i:= Length( v );
n:= n-1;
while 0 < n do
v[i]:= enum!.coeffsenum[ RemInt( n, enum!.q ) + 1 ];
n:= QuoInt( n, enum!.q );
i:= i-1;
od;
if IsFFE( enum!.coeffszero ) then
v := ImmutableVector( enum!.q, v );
fi;
MakeImmutable( v );
return v;
end );
BindGlobal( "NumberElement_InfiniteFullRowModule", function( enum, vector )
local n,
i,
max,
maxpos,
pos,
len;
if not IsCollsElms( FamilyObj( enum ), FamilyObj( vector ) )
or not IsList( vector ) then
return fail;
fi;
n:= Length( vector );
if n <> enum!.dimension then
return fail;
fi;
# Replace the entries of `vector' by their positions.
vector:= List( vector, x -> Position( enum!.coeffsenum, x, 0 ) );
if fail in vector then
return fail;
fi;
vector:= vector - 1;
# Find the maximal entry in the vector, and its number.
max:= vector[1];
for i in [ 2 .. n ] do
if max < vector[i] then
max:= vector[i];
fi;
od;
if max = 0 then
return 1;
fi;
# Compute the positions of `max' in `vector',
maxpos:= [];
for i in [ 1 .. n ] do
if vector[i] = max then
Add( maxpos, i-1 );
fi;
od;
# Compute the number of those elements with same distribution
# of `max' as in `vector' that come before `vector'.
pos:= 0;
for i in [ n, n-1 .. 1 ] do
if vector[i] <> max then
pos:= pos * max + vector[i];
fi;
od;
pos:= pos + 1;
# Compute the number of those elements with smaller distribution
# of `max'.
# Consider the following example.
# 1 3 4 7
# m ? m m ? ? m ? ... ? (`vector', the `?' mean entries < `m')
# * * * * * * ? ? ... ? gives (m+1)^6 m^{n-6}
# * * * ? ? ? m ? ... ? gives (m+1)^3 m^{n-3-1}
# * * ? m ? ? m ? ... ? gives (m+1)^2 m^{n-2-2}
# ? ? m m ? ? m ? ... ? gives (m+1)^0 m^{n-0-3}
len:= Length( maxpos );
for i in [ len, len-1 .. 1 ] do
pos:= pos + ( max + 1 )^maxpos[i] * max^( n - len - maxpos[i] + i );
od;
return pos;
end );
BindGlobal( "ElementNumber_InfiniteFullRowModule", function( enum, N )
local n,
val,
max,
maxpos,
pos,
vector,
i,
quorem;
# Catch the special case.
n:= enum!.dimension;
if N = 1 then
val:= enum!.coeffsenum[1];
return List( [ 1 .. n ], x -> val );
fi;
# Compute the maximal entry of the element.
max:= 1;
while max^n < N do
max:= max + 1;
od;
# Compute the positions of the maximal entry.
maxpos:= [];
repeat
pos:= 0;
val:= (max-1)^( n - Length( maxpos ) );
while val < N do
pos:= pos + 1;
val:= val * max / ( max - 1 );
od;
if 0 < pos then
N:= N - val * ( max - 1 ) / max;
Add( maxpos, pos );
fi;
until pos = 0;
maxpos:= Reversed( maxpos );
# Compute the values of the element that are strictly smaller than `max'.
vector:= [];
N:= N - 1;
for i in [ 1 .. n ] do
if i in maxpos then
vector[i]:= max;
else
quorem:= QuotientRemainder( Integers, N, max-1 );
vector[i]:= quorem[2] + 1;
N:= quorem[1];
fi;
od;
# Translate the positions to values.
for i in [ 1 .. n ] do
vector[i]:= enum!.coeffsenum[ vector[i] ];
od;
# Return the element.
return Immutable( vector );
end );
InstallMethod( EnumeratorByBasis,
"for enumerator via canonical basis of a full row module",
[ IsBasis and IsCanonicalBasis and IsCanonicalBasisFullRowModule ],
function( B )
local V, F, enum;
V:= UnderlyingLeftModule( B );
F:= LeftActingDomain( V );
if IsFinite( F ) then
enum:= EnumeratorByFunctions( V, rec(
ElementNumber := ElementNumber_FiniteFullRowModule,
NumberElement := NumberElement_FiniteFullRowModule,
coeffsenum := Enumerator( F ),
q := Size( F ),
coeffszero := Zero( F ),
zerovector := Zero( V ),
dimension := Dimension( V ) ) );
if IsField( F ) and Size( F ) < 256 and IsInternalRep( One( F ) ) then
# Use a more efficient method for `Position'.
enum!.NumberElement:= PosVecEnumFF;
SetFilterObj( enum, IsQuickPositionList );
fi;
SetFilterObj( enum, IsSSortedList );
return enum;
elif IsFiniteDimensional( V ) then
# The ring is infinite, use the canonical ordering of $\N_0^n$
# as defined for the iterator.
return EnumeratorByFunctions( V, rec(
ElementNumber := ElementNumber_InfiniteFullRowModule,
NumberElement := NumberElement_InfiniteFullRowModule,
dimension := Dimension( V ),
coeffsenum := Enumerator( F ) ) );
else
Error( "not implemented for infinite dimensional free modules" );
fi;
end );
#############################################################################
##
#M IteratorByBasis( <B> ) . . . . . . for canon. basis of a full row module
##
BindGlobal( "NextIterator_FiniteFullRowModule", function( iter )
local pos;
# Increase the counter.
pos:= iter!.dimension;
while iter!.counter[ pos ] = iter!.q do
iter!.counter[ pos ]:= 1;
pos:= pos - 1;
od;
iter!.counter[ pos ]:= iter!.counter[ pos ] + 1;
# Return the linear combination.
return iter!.ringelements{ iter!.counter };
end );
BindGlobal( "IsDoneIterator_FiniteFullRowModule",
iter -> iter!.counter = iter!.limit );
BindGlobal( "ShallowCopy_FiniteFullRowModule",
iter -> rec( dimension := iter!.dimension,
counter := ShallowCopy( iter!.counter ),
position := iter!.position,
q := iter!.q,
limit := ShallowCopy( iter!.limit ),
ringelements := iter!.ringelements ) );
BindGlobal( "NextIterator_InfiniteFullRowModule", function( iter )
local dim, # dimension of the free module
vector, # positions of the coefficients in `iter!.coeffsenum'
# of the previous element
result, # coefficients of the previous element
max1, # one less than the maximal entry in `vector'
max, # maximal entry in `vector'
firstval, # first entry in `iter!.coeffsenum'
i; # loop variable
# (Increase the counter.)
dim := iter!.dimension;
vector := iter!.vector;
result := iter!.result;
max1 := iter!.maxentry - 1;
firstval := iter!.firstval;
# If not all entries in `vector' are `max1' or `max1 + 1' then
# increase the counter formed by the positions with entry
# different from `max1 + 1', and return the result.
for i in [ 1 .. dim ] do
if vector[i] < max1 then
vector[i]:= vector[i] + 1;
result[i]:= iter!.coeffsenum[ vector[i] ];
return ShallowCopy( result );
elif vector[i] = max1 then
vector[i]:= 1;
result[i]:= firstval;
fi;
od;
# Otherwise if all entries are `max1 + 1', increase the maximum.
max:= iter!.maxentry;
if dim < PositionNot( vector, max ) then
max:= max + 1;
iter!.maxentry:= max;
vector[1]:= max;
iter!.maxval:= iter!.coeffsenum[ max ];
result[1]:= iter!.maxval;
for i in [ 2 .. dim ] do
vector[i]:= 1;
result[i]:= firstval;
od;
return ShallowCopy( result );
fi;
# Otherwise get the next start configuration with maximum `max'.
# (The entries of `vector' are now either `1' or `max'.)
for i in [ 1 .. dim ] do
if vector[i] = max then
vector[i]:= 1;
result[i]:= firstval;
else
vector[i]:= max;
result[i]:= iter!.maxval;
return ShallowCopy( result );
fi;
od;
Assert( 2, true,
"there should be a position with value different from `max'" );
end );
BindGlobal( "ShallowCopy_InfiniteFullRowModule",
iter -> rec( dim := iter!.dimension,
vector := ShallowCopy( iter!.vector ),
result := ShallowCopy( iter!.result ),
coeffsenum := iter!.coeffsenum,
maxentry := iter!.maxentry,
firstval := iter!.firstval,
maxval := iter!.maxval ) );
InstallMethod( IteratorByBasis,
"for canonical basis of a full row module",
[ IsBasis and IsCanonicalBasisFullRowModule ],
function( B )
local V,
F,
dim,
counter,
q,
enum,
vector,
firstval,
result;
V:= UnderlyingLeftModule( B );
dim:= Dimension( V );
if dim = 0 then
return TrivialIterator( Zero( V ) );
elif IsFinite( LeftActingDomain( V ) ) then
F:= LeftActingDomain( V );
counter := List( [ 1 .. dim ], x -> 1 );
counter[ Length( counter ) ]:= 0;
q:= Size( F );
return IteratorByFunctions( rec(
IsDoneIterator := IsDoneIterator_FiniteFullRowModule,
NextIterator := NextIterator_FiniteFullRowModule,
ShallowCopy := ShallowCopy_FiniteFullRowModule,
dimension := dim,
counter := counter,
position := 1,
q := q,
limit := List( [ 1 .. dim ], x -> q ),
ringelements := EnumeratorSorted( F ) ) );
else
enum:= Enumerator( LeftActingDomain( V ) );
vector:= List( [ 1 .. dim ], x -> 0 );
# vector[1]:= -1;
firstval:= enum[1];
result:= List( [ 1 .. dim ], x -> firstval );
return IteratorByFunctions( rec(
IsDoneIterator := ReturnFalse,
NextIterator := NextIterator_InfiniteFullRowModule,
ShallowCopy := ShallowCopy_InfiniteFullRowModule,
dimension := dim,
vector := vector,
result := result,
coeffsenum := enum,
maxentry := 0,
firstval := firstval,
maxval := firstval ) );
fi;
end );
[ Dauer der Verarbeitung: 0.34 Sekunden
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