We document here various utility functions defined by VKCURVE package and which may be useful also in other contexts.
BigNorm(c)
Given a complex number c with real part r and imaginary part j,
returns a "cheap substitute" to the norm of c given by r+j.
gap> BigNorm(Complex(-1,-1));
2
DecimalLog(r)
Given a rational number r, returns an integer k such that 10k< r
≤ 10k+1.
gap> List([1,1/10,1/2,2,10],DecimalLog);
[ -1, -2, -1, 0, 1 ]
ComplexRational(c)
c is a cyclotomic or a Complex number with Decimal or real cyclotomic
real and imaginary parts. This function returns the corresponding rational
complex number.
gap> evalf(E(3)/3);
-0.1666666667+0.2886751346I
gap> ComplexRational(last);
-16666666667/100000000000+28867513459/100000000000I
gap> ComplexRational(E(3)/3);
-1/6+28867513457/100000000000I
Dispersal(v)
v is a list of complex numbers representing points in the real plane.
The result is a pair whose first element is the minimum distance between
two elements of v, and the second is a pair of indices [i,j] such that
v[i], v[j] achieves this minimum distance.
gap> Dispersal([Complex(1,1),Complex(0),Complex(1)]);
[ 1, [ 1, 3 ] ]
ConjugatePresentation(p ,conjugation)
This program modifies a presentation by conjugating a generator by another.
The conjugation to apply is described by a length-3 string of the same
style as the result of DisplayPresentation, that is "abA" means
replace the second generator by its conjugate by the first, and "Aba"
means replace it by its conjugate by the inverse of the first.
gap> F:=FreeGroup(4);;
gap> p:=PresentationFpGroup(F/[F.4*F.1*F.2*F.3*F.4*F.1^-1*F.4^-1*
> F.3^-1*F.2^-1*F.1^-1,F.4*F.1*F.2*F.3*F.4*F.2*F.1^-1*F.4^-1*F.3^-1*
> F.2^-1*F.1^-1*F.3^-1,F.2*F.3*F.4*F.1*F.2*F.3*F.4*F.3^-1*F.2^-1*
> F.4^-1*F.3^-1*F.2^-1*F.1^-1*F.4^-1]);
gap> DisplayPresentation(p);
1: dabcd=abcda
2: dabcdb=cabcda
3: bcdabcd=dabcdbc
gap> DisplayPresentation(ConjugatePresentation(p,"cdC"));
#I there are 4 generators and 3 relators of total length 36
1: cabdca=dcabdc
2: dcabdc=bdcabd
3: cabdca=abdcab
114.6 TryConjugatePresentation
TryConjugatePresentation(p [,goal [,printlevel]])
This program tries to simplify group presentations by applying conjugations to the generators. The algorithm depends on random numbers, and on tree-searching, so is not reproducible. By default the program stops as soon as a shorter presentation is found. Sometimes this does not give the desired presentation. One can give a second argument goal, then the program will only stop when a presentation of length less than goal is found. Finally, a third argument can be given and then all presentations the programs runs over which are of length less than or equal to this argument are displayed. Due to the non-deterministic nature of the program, it may be useful to run it several times on the same input. Upon failure (to improve the presentation), the program returns p.
gap> DisplayPresentation(p);
1: ba=ab
2: dbd=bdb
3: cac=aca
4: bcb=cbc
5: dAca=Acad
6: dcdc=cdcd
7: adad=dada
8: dcDbdc=bdcbdB
9: dcdadc=adcdad
10: adcDad=dcDadc
11: BcccbdcAb=dcbACdddc
gap> p:=TryConjugatePresentation(p);
#I there are 4 generators and 11 relators of total length 100
#I there are 4 generators and 11 relators of total length 120
#I there are 4 generators and 10 relators of total length 100
#I there are 4 generators and 11 relators of total length 132
#I there are 4 generators and 11 relators of total length 114
#I there are 4 generators and 11 relators of total length 110
#I there are 4 generators and 11 relators of total length 104
#I there are 4 generators and 11 relators of total length 114
#I there are 4 generators and 11 relators of total length 110
#I there are 4 generators and 11 relators of total length 104
#I there are 4 generators and 8 relators of total length 76
#I there are 4 generators and 8 relators of total length 74
#I there are 4 generators and 8 relators of total length 72
#I there are 4 generators and 8 relators of total length 70
#I there are 4 generators and 7 relators of total length 52
# d->adA gives length 52
<< presentation with 4 gens and 7 rels of total length 52 >>
gap> DisplayPresentation(p);
1: ba=ab
2: dc=cd
3: aca=cac
4: dbd=bdb
5: bcb=cbc
6: adad=dada
7: aBcADbdac=dBCacbdaB
gap> TryConjugatePresentation(p,48);
#I there are 4 generators and 7 relators of total length 54
#I there are 4 generators and 7 relators of total length 54
#I there are 4 generators and 7 relators of total length 60
#I there are 4 generators and 7 relators of total length 60
#I there are 4 generators and 7 relators of total length 48
# d->bdB gives length 48
<< presentation with 4 gens and 7 rels of total length 48 >>
gap> DisplayPresentation(last);
1: ba=ab
2: bcb=cbc
3: cac=aca
4: dbd=bdb
5: cdc=dcd
6: adad=dada
7: dAbcBa=bAcBad
FindRoots(p, approx)
p should be a univariate Mvp with cyclotomic or Complex rational or
decimal coefficients or a list of cyclotomics or Complex rationals or
decimals which represents the coefficients of a complex polynomial. The
function returns Complex rational approximations to the roots of p
which are better than approx (a positive rational). Contrary to the
functions SeparateRoots, etc... described in the previous chapter, this
function handles quite well polynomials with multiple roots. We rely on the
algorithms explained in detail in HSS01.
gap> FindRoots((x-1)^5,1/100000000000);
[ 6249999999993/6250000000000+29/12500000000000I,
12499999999993/12500000000000-39/12500000000000I,
12500000000023/12500000000000+11/6250000000000I,
12500000000023/12500000000000+11/6250000000000I,
312499999999/312500000000-3/6250000000000I ]
gap> evalf(last);
[ 1, 1, 1, 1, 1 ]
gap> FindRoots(x^3-1,1/10);
[ -1/2-108253175473/125000000000I, 1, -1/2+108253175473/125000000000I ]
gap> evalf(last);
[ -0.5-0.8660254038I, 1, -0.5+0.8660254038I ]
gap> List(last,x->x^3);
[ 1, 1, 1 ]
Cut(string s [, opt])
The first argument is a string, and the second one a record of options, if
not given taken equal to rec(). This function prints its string argument
s on several lines not exceeding opt.width; if not given opt.width is
taken to be equal SizeScreen[1]-2. This is similar to how GAP3 prints
strings, excepted no continuation line characters are printed. The user can
specify after which characters, or before which characters to cut the
string by giving fields opt.before and opt.after; the defaut is
opt.after:=",", but some other characters can be used --- for instance
a good choice for printing big polynomials could be opt.before:="+-".
If a field opt.file is given, the result is appended to that file instead
of written to standard output; this may be quite useful in conjunction with
FormatGAP for dumping some GAP3 values to a file for later re-reading.
gap> Cut("an, example, with, plenty, of, commas\n",rec(width:=10));
an,
example,
with,
plenty,
of,
commas
gap>
gap3-jm