#############################################################################
####
##
#W Strings.gi RADIROOT package Andreas Distler
##
## Installation file for the functions that generate Tex-strings
##
#Y 2006
##
#############################################################################
##
#F RR_Radikalbasis( <erw>, <elements>, <stream> )
##
## Produces a basis for the matrixfield in the record <erw> from the
## generating matrices <elements> and returns a Tex-readable strings
## for the basis as well
##
InstallGlobalFunction( RR_Radikalbasis, function( erw, elements, stream )
local k, basis, elm, mat, i, ll, basstr, elmstr, m, coeffs, scale;
k := DegreeOverPrimeField(erw.K) / Product(erw.degs);
basis := Basis(erw.K){[ 1..k ]};
if k =1 then
basstr := [""];
else
basstr := ["",Concatenation("\\zeta_{",String(Order(erw.unity)),"}")];
fi;
for i in [ 2..k-1 ] do
Add( basstr, Concatenation("\\zeta_{", String(Order(erw.unity)),
"}^{", String(i),"}"));
od;
AppendTo( stream,"\\\\\n");
for m in [ 1..Length(elements) ] do
mat := List( basis, Flat );;
elm := elements[m][1];
k := elements[m][2];
coeffs := SolutionMat(mat, Flat(elm^k));
scale := RR_Potfree(Concatenation(List(coeffs, ExtRepOfObj)), k);
elm := elm / scale; elements[m][1] := elm;
elmstr := RR_WurzelAlsString( k, coeffs / scale^k, basstr );
AppendTo( stream, "$\\omega_", String(m)," = ", elmstr, "$,\\\\\n");
basis := Concatenation( List( [1..k], i -> elm^(i-1) * basis));;
ll := [ basstr, List( basstr, str ->
Concatenation( str,"\\omega_",String(m) ) ) ];
for i in [ 3..k ] do
ll[i] := List( basstr, str -> Concatenation( str,
Concatenation( "\\omega_", String(m), "^", String(i-1) ) ) );
od;
basstr := Concatenation( ll );
od;
AppendTo( stream, "\\\\\n");
return [ basis, basstr ];
end );
#############################################################################
##
#F RR_BruchAlsString( <bruch> )
##
## Creates a Tex-readable String for the rational <bruch>
##
InstallGlobalFunction( RR_BruchAlsString, function( bruch )
local str, num, den, sgn;
if IsInt( bruch ) then
str := String( AbsInt( bruch ) );
else
num := String( AbsInt( NumeratorRat( bruch ) ) );
den := String( DenominatorRat( bruch ) );
str := Concatenation("\\frac{", num, "}{", den, "}" );
fi;
if IsNegRat( bruch ) then str := Concatenation( " - ", str ); fi;
return str;
end );
#############################################################################
##
#F RR_KoeffizientAlsString( <coeff>, <anf> )
##
## Creates a Tex-readable String for the cyclotomic <coeff>; if <anf>
## is true, positive signs of rationals will be omitted; if <coeff> is a
## sum, it will be included in brackets; finitely an empty string
## will be returned, if <coeff> is equal to 1
##
InstallGlobalFunction( RR_KoeffizientAlsString, function( coeff, anf )
local cstr;
if coeff = 1 then
cstr := "";
elif coeff = -1 then
cstr := "-";
else
cstr := RR_ZahlAlsString( coeff );
if not IsRat( coeff ) then
cstr := Concatenation( "\\left(",cstr,"\\right)" );
fi;
fi;
if not anf then
if IsPosRat( coeff ) then
cstr := Concatenation( " + ", cstr );
elif not IsRat( coeff ) then
cstr := Concatenation( " + ", cstr );
fi;
fi;
return cstr;
end );
#############################################################################
##
#F RR_WurzelAlsString( <k>, <coeffs>, <basstr> )
##
## Creates a Tex-readable String for the <k>-th root of the element
## described by <coeffs> and <basstr>
##
InstallGlobalFunction( RR_WurzelAlsString, function( k, coeffs, basstr )
local i, str, anf;
str := ""; anf := true;
for i in [ 1..Length(coeffs) ] do
if coeffs[i] in [ -1, 1 ] and basstr[i] = "" then
if not anf and coeffs[i] = 1 then
str := Concatenation( str, " + ", String( coeffs[i] ) );
else
str := Concatenation( str, String( coeffs[i] ) );
fi;
anf := false;
elif coeffs[i] <> 0 then
str := Concatenation( str,
RR_KoeffizientAlsString( coeffs[i], anf ),
basstr[i]);
anf := false;
fi;
od;
if k <> 1 then
str := Concatenation( "\\sqrt[", String(k), "]{", str, "}" );
fi;
return str;
end );
#############################################################################
##
#F RR_ZahlAlsString( <zahl> )
##
## Creates a Tex-readable String for the cyclotomic <zahl>
##
InstallGlobalFunction( RR_ZahlAlsString, function( zahl )
local bas, basstr, cond, i;
if IsRat( zahl ) then
return RR_BruchAlsString( zahl );
else
cond := Conductor( zahl );
bas := Basis( CF( cond ) );
basstr := [ Concatenation( "\\zeta_{", String(cond), "}" ) ];
for i in Filtered( [ 2..cond ], x -> Gcd( x, cond ) = 1 ) do
Add( basstr,
Concatenation("\\zeta_{",String(cond),"}^{", String(i), "}"));
od;
return RR_WurzelAlsString( 1, Coefficients( bas, zahl ), basstr );
fi;
end );
#############################################################################
##
#F RR_PolyAlsString( <poly> )
##
## Creates a Tex-readable String for the polynomial <poly>
##
InstallGlobalFunction( RR_PolyAlsString, function( poly )
local coeffs, polybasis, i;
coeffs := CoefficientsOfUnivariatePolynomial( poly );
polybasis := [ "", "x" ];
for i in [ 3..Length(coeffs) ] do
polybasis[i] := Concatenation( "x^{", String(i-1), "}" );
od;
return RR_WurzelAlsString( 1, Reversed(coeffs), Reversed(polybasis) );
end );
#############################################################################
##
#F RR_TexFile( <f>, <erw>, <elements>, <dir>, <file> )
##
## Creates a Tex-file for a radical expression of the roots of the
## polynomial <f>.
##
InstallGlobalFunction( RR_TexFile, function( poly, erw, elements, dir, file )
local i,cstr,bas,root,coeffs,B,k,offset,str,min,stream;
# Create tex-Code and write to file, using stream because of linebreaks
Info( InfoRadiroot, 2, " creating tex-file." );
file := Filename( dir, file );
stream := OutputTextFile( file, false );
SetPrintFormattingStatus( stream, false );
AppendTo(stream, "\\documentclass[fleqn]{article} \n",
"\\setlength{\\paperwidth}{84cm} \n",
"\\setlength{\\textwidth}{80cm} \n",
"\\setlength{\\paperheight}{59.5cm} \n",
"\\setlength{\\textheight}{57cm} \n",
"\\begin{document} \n",
"\\noindent\n",
"An expression by radicals for the roots of the polynomial $",
RR_PolyAlsString( poly ),
"$ with the $n$-th root of unity $\\zeta_n$ and\n");
bas := RR_Radikalbasis( erw, elements, stream );;
AppendTo( stream, "is:\n\\\\\n\\noindent\n$" );
offset := CoefficientsOfUnivariatePolynomial(poly)[Degree(poly)] /
(Degree(poly) * LeadingCoefficient(poly));
k := Degree(poly) / Length(erw.roots);
if k <> 1 and offset <> 0 then
AppendTo( stream, RR_ZahlAlsString(-offset), "+");
fi;
B := Basis( erw.K, bas[1] );
coeffs := List([1..Length(erw.roots)], i->Coefficients(B, erw.roots[i]));
str := List([ 1..Length(erw.roots) ],
i -> RR_WurzelAlsString(k, coeffs[i], bas[2]));
min := First( [ 1..Length(erw.roots) ],
i -> Length(str[i]) = Minimum( List( str, Length )));
if Length( str[min] ) = 0 then
AppendTo( stream, "0" );
elif Length( str[min] ) < 1400 then
AppendTo( stream, str[min] );
else
AppendTo( stream, RR_NstInDatei( k, coeffs[min], bas[2] ));
fi;
AppendTo(stream, "$\n\\end{document}\n");
# "$\n\\\\$",String(Length(str[min])),
CloseStream( stream );
return file;
end );
#############################################################################
##
#F RR_Display( <file>, <dir> )
##
## Displays the latex-file <file> from the directory <dir>
##
InstallGlobalFunction( RR_Display, function( file, dir )
local dvi, latex;
# Execute latex and open the created document
latex := Filename( DirectoriesSystemPrograms( ), "latex" );
Process( dir, latex, InputTextNone( ), OutputTextNone( ),
[ Concatenation( file, ".tex" ) ] );
dvi := Filename( DirectoriesSystemPrograms( ), "xdvi" );
Process( dir, dvi, InputTextNone( ), OutputTextNone( ),
["-paper","a1r",Concatenation( file, ".dvi" ) ] );
end );
#############################################################################
##
#F RR_NstInDatei( <k>, <coeffs>, <basstr> )
##
## Creates a Tex-output containing a string for the <k>-th root of the
## element described by <coeffs> and <basstr>
##
InstallGlobalFunction( RR_NstInDatei, function( k, coeffs, basstr )
local str, i, anf;
str := "";
if k <> 1 then
str := Concatenation( str, "(" );
fi;
anf := true;
repeat
i := 0;
while Length( coeffs ) >= i+1 and
Length(RR_WurzelAlsString(1,coeffs{[1..i+1]},basstr{[1..i+1]}))
< 1400 do
i := i+1;
od;
if not anf then
str := Concatenation(str, "$\\\\\n$+" );
fi;
anf := false;
str := Concatenation(str, RR_WurzelAlsString(1, coeffs{[1..i]},
basstr{[1..i]}));
coeffs := coeffs{[i+1..Length(coeffs)]};
basstr := basstr{[i+1..Length(basstr)]};
until coeffs = [ ];
if k <> 1 then
str := Concatenation( str, ")^{\\frac{1}{",String(k),"}}");
fi;
return str;
end );
#############################################################################
##
#E
