Re: [math-fun] Equation mixing integers and reals

16 Sep 2008

      Hello,

  Concerning the general problem of finding an integer relation.

Here is an algorithm for 3 values, that is :
if we are looking for aX + bY + Z ? 0

where a,b are 2 real numbers and X,Y,Z three integers.

do
{| a - b |} = c
a <- b
b <- c
od

{ } is the fractional part.

If we iterate this, after a certain amount of
iterations we find 3 integers where the equation is
near 0. I could generalize this example to more ;

Here is the program in Maple :

pseudoLLL:=proc(s)
local a, b, c, d, nn, k,i, aa, ss, g, h,m,z;
     aa:=s:
     nn:=nops(aa): 
k:=[seq(frac(abs(aa[i]-aa[i+1])),i=1..nn-1),frac(abs(aa[1]-aa[nn]))];
     ss:=evalf(k):
     g:=sort([seq([convert(1+ss[j],string),j],j=1..nn)],lexorder[2]):
     h:=[seq(g[i][3],i=1..nn)]:
     m:=[seq(k[h[z]],z=1..nn)]:
     return(m);
end:

This will work with a vector of real numbers up to a certain precision.
I could make it to work with Maple 11, but they changed the
syntax and the calling sequence again for the sort function (#@!$!@#($&!).

For n=3 there, it can be worked out this way :

L3:=proc(x, y, z)
local a, b, c, d, k1, k2, k3, ll;
     a := x;
     b := y;
     c := z;
     k1 := frac(abs(a - b));
     k2 := frac(abs(b - c));
     k3 := frac(abs(a - c));
     ll := sort([k1, k2, k3]);
     a := ll[1];
     b := ll[2];
     c := ll[3];
     RETURN(a, b, c)
end:

Simon Plouffe