[GAP Forum] Comparing real numbers

[Marek Mitros]

Hi again,

Since nobody answers I have created following function for comparing
real numbers from field CF(5). It works for numbers tau,
tau+1/10^4000. It does not for tau, tau + 1/10^5000 (tau is defined
below).

Regards,
Marek


# Function for comparing real numbers from CF(5) field
# The idea is to use fibonacci sequence approximating irrational
number tau= (Sqrt(5)-1)/2 = 0,61803398874989484820458683436564
# Define basis [1,tau] of the Field(tau). Any number x0 from this
field can be presented as a*1+b*tau
# for rational numbers a,b. Using approximation f(n)/f(n+1) for tau we
obtain approximation
# x(n)=a+b*f(n)/f(n+1) for x0.
# Case 1: compare irrational number x0 with rational number p0.
# In this case we look for n such that both x(n) and x(n+1) are
greater than p0 or both smaller than p0.
# Since all the time irrational x is between x(n) and x(n+1) we know
that x is smaller or greater than p0.
# Case 2: compare irrational numbers x0 with y0.
# Look for n such that we have both [x(n), x(n+1)] are smaller or
bigger than both [y(n), y(n+1)];

# function returns
tau:=(Sqrt(5)-1)/2;
F:=Field(tau);
bas:=Basis(F, [1,tau]);

# At first try do not look at efficiency; just obtain good result;
returns true when x<y
comp_real5:=function(x,y)
  local cc,a,b,c,d,n, maxn, nstep;
  maxn:=10000;   nstep:=100;
  if not ([x,y] in F^2) then Print ("Numbers outside the tau field,
cannot compare\n"); return fail; fi;
  if x=y then return false;
  elif IsRat(x) and IsRat(y) then return x<y; # both rationals
  elif IsRat(y) then
     cc:=Coefficients(bas, x); a:=cc[1]; b:=cc[2];
     n:=First(nstep*[1..maxn/nstep],
n->((a+b*Fibonacci(n)/Fibonacci(n+1)<y) and
(a+b*Fibonacci(n+1)/Fibonacci(n+2)<y))
         or ((a+b*Fibonacci(n)/Fibonacci(n+1)>y) and
(a+b*Fibonacci(n+1)/Fibonacci(n+2)>y)) );
     if n= fail then Print("Cannot compare using maxn
approximation\n"); return fail; fi;
     return (a+b*Fibonacci(n)/Fibonacci(n+1)<y) and
(a+b*Fibonacci(n+1)/Fibonacci(n+2)<y);
  elif IsRat(x) then
     cc:=Coefficients(bas, y); a:=cc[1]; b:=cc[2];
     n:=First(nstep*[1..maxn/nstep],
n->((a+b*Fibonacci(n)/Fibonacci(n+1)>x) and
(a+b*Fibonacci(n+1)/Fibonacci(n+2)>x))
         or ((a+b*Fibonacci(n)/Fibonacci(n+1)<x) and
(a+b*Fibonacci(n+1)/Fibonacci(n+2)<x)) );
     if n= fail then Print("Cannot compare using maxn
approximation\n"); return fail; fi;
     return (a+b*Fibonacci(n)/Fibonacci(n+1)>x) and
(a+b*Fibonacci(n+1)/Fibonacci(n+2)>x);
  else # both irrational
     cc:=Coefficients(bas, x); a:=cc[1]; b:=cc[2];
     cc:=Coefficients(bas, y); c:=cc[1]; d:=cc[2];
     n:=First(nstep*[1..maxn/nstep],
n->((a+b*Fibonacci(n)/Fibonacci(n+1)<c+d*Fibonacci(n)/Fibonacci(n+1))
                         and
(a+b*Fibonacci(n)/Fibonacci(n+1)<c+d*Fibonacci(n+1)/Fibonacci(n+2))
                         and
(a+b*Fibonacci(n+1)/Fibonacci(n+2)<c+d*Fibonacci(n)/Fibonacci(n+1))
                         and
(a+b*Fibonacci(n+1)/Fibonacci(n+2)<c+d*Fibonacci(n+1)/Fibonacci(n+2)))
         or    ((a+b*Fibonacci(n)/Fibonacci(n+1)>c+d*Fibonacci(n)/Fibonacci(n+1))
            and
(a+b*Fibonacci(n)/Fibonacci(n+1)>c+d*Fibonacci(n+1)/Fibonacci(n+2))
            and
(a+b*Fibonacci(n+1)/Fibonacci(n+2)>c+d*Fibonacci(n)/Fibonacci(n+1))
            and
(a+b*Fibonacci(n+1)/Fibonacci(n+2)>c+d*Fibonacci(n+1)/Fibonacci(n+2)))
);\
     if n= fail then Print("Cannot compare using maxn
approximation\n"); return fail; fi;
     return ((a+b*Fibonacci(n)/Fibonacci(n+1)<c+d*Fibonacci(n)/Fibonacci(n+1))
                         and
(a+b*Fibonacci(n)/Fibonacci(n+1)<c+d*Fibonacci(n+1)/Fibonacci(n+2))
                         and
(a+b*Fibonacci(n+1)/Fibonacci(n+2)<c+d*Fibonacci(n)/Fibonacci(n+1))
                         and
(a+b*Fibonacci(n+1)/Fibonacci(n+2)<c+d*Fibonacci(n+1)/Fibonacci(n+2)));
   fi;
end;




On 9/19/12, Marek Mitros <marek at mitros.org> wrote:
> Hi,
>
> I wonder how I could compare real numbers in GAP i.e. cyclotomics
> which are real. I try Float but it doesn't work for cyclotomics - any
> advice ? Say my numbers are limited to CF(5) field. Here is example:
> sigma:=(E(5)+E(5)^4)/2; # = RealPart(E(5)) = (Sqrt(5)-1)/4 = 0.309...
>  is real number. How I can compare whether it is bigger or smaller
> that 1/2 ? It is known that this number is equal to Another real
> number in CF(5) is
> tau:=-RealPart(E(5)^2); # =-(E(5)^2+E(5)^3)/2=(Sqrt(5)+1)/4 =
> 0.809...=sigma+1/2
>
> How could I develop function comparing elements of the field generated
> by sigma and tau ? I mean comparison as real numbers. I can obtain
> coefficients for given number x:
> Coefficients(Basis(Field(sigma)), x)
> Using this can I find the position of the number on real axis ? I.e.
> compare with another field element or rational number ?
>
> I would really be nice to have possibility of comparing real subset of
> cyclotomics are real numbers in GAP.
>
> Regards,
> Marek
>