Quelle common.gi Sprache: unbekannt

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##
#A  common.gi                 Cryst library                      Bettina Eick
#A                                                              Franz G"ahler
#A                                                              Werner Nickel
##
#Y  Copyright 1997-1999  by  Bettina Eick,  Franz G"ahler  and  Werner Nickel
##
##  Common utility routines, most of which deal with integral matrices
##

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##
#F  MutableMatrix( M ) . . . . . . . . . . . . . . . . mutable copy of matrix
##
MutableMatrix := function( M )
    return List( M, ShallowCopy );
end;

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##
#F  AffMatMutableTrans( M ) . . . . . .affine matrix with mutable translation
##
AffMatMutableTrans := function( M )
    local l;
    if not IsMutable( M ) then
        M := ShallowCopy( M );
    fi;
    l := Length( M );
    if not IsMutable( M[l] ) then
        M[l] := ShallowCopy( M[l] );
    fi;
    return M;
end;

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##
#F  AugmentedMatrix( <matrix>, <trans> ). . . . .  construct augmented matrix
##
AugmentedMatrix := function( m, b )
   local g, t, x;
   g := MutableMatrix( m );
   for x in g do
       Add( x, 0 );
   od;
   t := ShallowCopy( b );
   Add( t, 1 );
   Add( g, t );
   return g;
end;

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##
#F  RowEchelonForm  . . . . . . . . . . row echelon form of an integer matrix
##
InstallMethod( RowEchelonForm,
        "for matrices",
        [ IsMatrix ],

  function( M )

    local a, i, j, k, m, n, r, Cleared;

    if M = [] then return []; fi;

    M := MutableMatrix( M );
    m := Length( M ); n := Length( M[1] );

    i := 1; j := 1;
    while i <= m and j <= n do
        k := i; while k <= m and M[k][j] = 0 do k := k+1; od;
        if k <= m then
            r := M[i]; M[i] := M[k]; M[k] := r;
            for k in [k+1..m] do
                a := AbsInt( M[k][j] );
                if a <> 0 and a < AbsInt( M[i][j] ) then
                    r := M[i]; M[i] := M[k]; M[k] := r;
                fi;
            od;
            if M[i][j] < 0 then M[i] := -1 * M[i]; fi;
            Cleared := true;
            for k in [i+1..m] do
                a := QuoInt(M[k][j],M[i][j]);
                if a <> 0 then  M[k] := M[k] - a * M[i]; fi;
                if M[k][j] <> 0 then Cleared := false; fi;
            od;
            if Cleared then i := i+1; j := j+1; fi;
        else
            j := j+1;
        fi;
    od;
    return M{[1..i-1]};
end );

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##
#F  RowEchelonFormVector  . . . . . . . . . . . .row echelon form with vector
##
RowEchelonFormVector := function( M, b )

    local a, i, j, k, m, n, r, Cleared;

    M := MutableMatrix( M );
    m := Length( M );
    if m = 0 then return M; fi;
    n := Length( M[1] );

    i := 1; j := 1;
    while i <= m and j <= n do
        k := i; while k <= m and M[k][j] = 0 do k := k+1; od;
        if k <= m then
            r := M[i]; M[i] := M[k]; M[k] := r;
            r := b[i]; b[i] := b[k]; b[k] := r;
            for k in [k+1..m] do
                a := AbsInt( M[k][j] );
                if a <> 0 and a < AbsInt( M[i][j] ) then
                    r := M[i]; M[i] := M[k]; M[k] := r;
                    r := b[i]; b[i] := b[k]; b[k] := r;
                fi;
            od;
            if M[i][j] < 0 then M[i] := -1 * M[i]; b[i] := -1 * b[i]; fi;
            Cleared := true;
            for k in [i+1..m] do
                a := QuoInt(M[k][j],M[i][j]);
                if a <> 0 then
                    M[k] := M[k] - a * M[i];
                    b[k] := b[k] - a * b[i];
                fi;
                if M[k][j] <> 0 then Cleared := false; fi;
            od;
            if Cleared then i := i+1; j := j+1; fi;
        else
            j := j+1;
        fi;
    od;
    return M{[1..i-1]};
end;

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##
#F  RowEchelonFormT . . . . . . . row echelon form with transformation matrix
##
RowEchelonFormT := function( M, T )

    local a, i, j, k, m, n, r, Cleared;

    M := MutableMatrix( M );
    m := Length( M ); n := Length( M[1] );

    i := 1; j := 1;
    while i <= m and j <= n do
        k := i; while k <= m and M[k][j] = 0 do k := k+1; od;
        if k <= m then
            r := M[i]; M[i] := M[k]; M[k] := r;
            r := T[i]; T[i] := T[k]; T[k] := r;
            for k in [k+1..m] do
                a := AbsInt( M[k][j] );
                if a <> 0 and a < AbsInt( M[i][j] ) then
                    r := M[i]; M[i] := M[k]; M[k] := r;
                    r := T[i]; T[i] := T[k]; T[k] := r;
                fi;
            od;
            if M[i][j] < 0 then
                M[i] := -1 * M[i];
                T[i] := -1 * T[i];
            fi;
            Cleared := true;
            for k in [i+1..m] do
                a := QuoInt(M[k][j],M[i][j]);
                if a <> 0 then
                    M[k] := M[k] - a * M[i];
                    T[k] := T[k] - a * T[i];
                fi;
                if M[k][j] <> 0 then Cleared := false; fi;
            od;
            if Cleared then i := i+1; j := j+1; fi;
        else
            j := j+1;
        fi;
    od;
    return M{[1..i-1]};
end;

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##
#F  FractionModOne  . . . . . . . . . . . . . . . . . . a fraction modulo one
##
FractionModOne := function( q )
    q := q - Int(q);
    if q < 0 then q := q+1; fi;
    return q;
end;

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##
#F  VectorModL . . . . . . . . . . . . . . . . .vector modulo a free Z-module
##
VectorModL := function( v, L )

    local l, i, x, j;

    for l in L do
        i := PositionProperty( l, x -> x<>0 );
        x := v[i]/l[i];
        j := Int( x );
        if x < 0 and not IsInt( x ) then j := j-1; fi;
        v := v - j*l;
    od;
    return v;

end;

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##
#F  IntSolutionMat( M, b ) . . integer solution for inhom system of equations
##
IntSolutionMat := function( M, b )

    local Q, den, sol, i, x;

    if Concatenation(M) = [] then
        return fail;
    fi;

    # the trivial solution
    if RankMat( M ) = 0 then
        if b = 0 * M[1] then
            return List( M, x -> 0 );
        else
            return fail;
        fi;
    fi;

    den := Lcm( List( Flat( M ), x -> DenominatorRat( x ) ) );
    if den <> 1 then
        M := den*M;
        b := den*b;
    fi;

    b := ShallowCopy(b);
    Q := IdentityMat( Length(M) );

    M := TransposedMat(M);
    M := RowEchelonFormVector( M,b );
    while not IsDiagonalMat(M) do
        M := TransposedMat(M);
        M := RowEchelonFormT(M,Q);
        if not IsDiagonalMat(M) then
            M := TransposedMat(M);
            M := RowEchelonFormVector(M,b);
        fi;
    od;

    # are there integer solutions?
    sol:=[];
    for i in [1..Length(M)] do
        x := b[i]/M[i][i];
        if IsInt( x ) then
            Add( sol, x );
        else
            return fail;
        fi;
    od;

    # are there solutions at all?
    for i in [Length(M)+1..Length(b)] do
        if b[i]<>0 then
            return fail;
        fi;
    od;

    return sol*Q{[1..Length(sol)]};

end;

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##
#F  ReducedLatticeBasis . . . . . . . . . reduce lattice basis to normal form
##
ReducedLatticeBasis := function ( trans )

    local m, f, b, r, L;

    if trans = [] or ForAll( trans, x -> IsZero( x ) ) then
        return [];
    fi;

    m := ShallowCopy( trans );
    f := Flat( trans );

    if not ForAll( f, IsRat ) then
        b := Basis( Field( f ) );
        m := List( m, x -> BlownUpVector( b, x ) );
        f := Flat( m );
    fi;

    if not ForAll( f, IsInt ) then
        m := m * Lcm( List( f, DenominatorRat ) );
    fi;

    r := NormalFormIntMat( m, 6 );
    L := r.rowtrans{[1..r.rank]} * trans;
    return L;

end;

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##
#F  UnionModule( M1, M2 ) . . . . . . . . . . . . union of two free Z-modules
##
UnionModule := function( M1, M2 )
    return ReducedLatticeBasis( Concatenation( M1, M2 ) );
end;

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##
#F  IntersectionModule( M1, M2 ) . . . . . intersection of two free Z-modules
##
IntersectionModule := function( M1, M2 )

    local M, Q, r, T;

    if M1 = [] or M2 = [] then
        return [];
    fi;
    M := Concatenation( M1, M2 );
    M := M * Lcm( List( Flat( M ), DenominatorRat ) );
#    Q := IdentityMat( Length( M ) );
#    M := RowEchelonFormT( M, Q );
#    T := Q{[Length(M)+1..Length(Q)]}{[1..Length(M1)]};
#    if not IsEmpty(T) then T := T * M1; fi;
    r := NormalFormIntMat( M, 4 );
    T := r.rowtrans{[r.rank+1..Length(M)]}{[1..Length(M1)]};
    if not IsEmpty( T ) then
        T := T * M1;
    fi;
    return ReducedLatticeBasis( T );

end;

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