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

16 Sep 2008

      Here is an example with sin(1),sin(2) and sin(3) (symbolic values
not NUMERIC).

1230145 sin(1) - 1121529 sin(2) - 108616 sin(3)

after 10 iterations : the error is 10^(-6),

BUT this is a pseudo-algorithm it will find vectors
but not necessarly the best ones for a precision of
0.000001

for exp(1), Pi and gamma I find :

Pi - 4 exp(1) + 3 gamma + 6 is almost 0.

for Pi, exp(1) and sqrt(2) I find :

-122*exp(1)+71*2^(1/2)+71+51*Pi = 0.518956568491*10^(-5).

which is not so bad.

I presume that this very naive algorithm is just the tip
of the iceberg, actualy to find approximations of the
order of 10^(-100) with small coefficients is another
game...!

Simon Plouffe